document_title: Half dollar (United States coin)

question_text: when did they stop making all silver half dollars

text: Half dollar (United States coin)

text: Circulation[edit]

text: Half dollar coins saw heavy use, particularly in the first half of the twentieth century. For many years, they were commonly (and in many areas still are) used by the gambling community. Rolls of half dollars may still be kept on hand in cardrooms for games requiring 50-cent antes or bring-in bets, for dealers to pay winning naturals in blackjack, or where the house collects a rake in increments. Additionally, some concession vendors at sporting events distribute half dollar coins as change for convenience.

text: By the early 1960s, the rising price of silver was nearing the point where the bullion value of U.S. silver coins would exceed face value. In 1965, the U.S. introduced layered composition coins made of a copper core laminated between two cupronickel outer faces. The silver content of dimes and quarters was eliminated, but the Kennedy half dollar composition still contained silver (reduced from 90 to 40 percent) from 1965 to 1970.

text: Half dollars issued through the end of the 1960s were hoarded as the only precious metal U.S. coins remaining in production, and as the price of silver continued to rise, pre-1964 halves disappeared from circulation as well. By the time that the coin's composition was changed to match that of the clad dimes and quarters in 1971, both businesses and the public had began to adapt to a country in which the half dollar did not generally circulate. The half saw a moderate increase in usage during the mid 1970s, only to fall again by the end of the decade. The quarter then took over the half's role as the highest-value component of change in general use.

text: Most coins enter circulation through the change drawers of businesses. Few businesses stock their change drawers with half dollars, and some banks do not stock them at all. Most banks do not hand them out as normal business practice, so the coins see little circulation. Most U.S. vending machines do not accept half dollars, nor do payphones, which further curtails their circulation; Many transit systems however, accept halves and the coin is the most common denomination used for U.S. commemorative coins.[3]

text: Since 2002, half dollars have been minted only for collectors, due to large Federal Reserve and government inventories on hand of pre-2001 pieces; this is mostly due to lack of demand and large quantity returns from casino slot machines that now operate "coinless". Eventually, when the reserve supply runs low, the mint will again fill orders for circulation half dollars.[citation needed] It took about 18 years (1981–1999) for the large inventory stockpile of a similar low-demand circulation coin, the $1 coin, to reach reserve levels low enough to again produce circulation pieces.[citation needed] Modern-date half dollars can be purchased in proof sets, mint sets, rolls, and bags from the U.S. Mint, and existing inventory circulation pieces can be ordered through most U.S. banks. All collector issues since 2001 have had much lower mintages than in previous years. Although intended only for collectors, these post-2001 half dollars sometimes find their way into circulation.[4]

text: Aspects of early history[edit]

text: On December 1, 1794, the first half dollars, approximately 5,300 pieces, were delivered. Another 18,000 were produced in January 1795 using dies of 1794, to save the expense of making new ones.[5] Another 30,000 pieces were struck by the end of 1801. The coin had the Heraldic Eagle, based on the Great Seal of the United States on the reverse.[5] 150,000 were minted in 1804 but struck with dies from 1803, so no 1804 specimens exist, though there were some pieces dated 1805 that carried a "5 over 4" overdate.[5]

text: In 1838, half dollar dies were produced in the Philadelphia Mint for the newly established New Orleans Mint, and ten test samples of the 1838 half dollars were made at the main Philadelphia mint. These samples were put into the mint safe along with other rarities like the 1804 silver dollar. The dies were then shipped to New Orleans for the regular production of 1838 half dollars. However, New Orleans production of the half dollars was delayed due to the priority of producing half dimes and dimes. The large press for half dollar production was not used in New Orleans until January 1839 to produce 1838 half dollars, but the reverse die could not be properly secured, and only ten samples were produced before the dies failed. Rufus Tyler, chief coiner of the New Orleans mint, wrote to Mint Director Patterson of the problem on February 25, 1839.[6] The Orleans mint samples all had a double stamped reverse as a result of this production problem and they also showed dramatic signs of die rust, neither of which are present on the Philadelphia produced test samples. While eight Philadelphia minted samples survive to this day, there is only one known New Orleans minted specimen with the tell-tale double stamped reverse and die rust. This is the famous coin that Rufus Tyler presented to Alexander Dallas Bache (great grandson of Benjamin Franklin) in the summer of 1839 and was later purchased in June 1894 by A. G. Heaton, the father of mint mark coin collecting.[7] The 1838 Philadelphia-produced half dollars are extremely rare, with two separate specimens having sold for $632,500 in Heritage auctions in 2005 and 2008[8] respectively. The sole surviving Orleans minted 1838 is one of the rarest of all American coins.[9][10] In 1840 this mint produced nearly 180,000 half dollars.[5]

text: In 1861, the New Orleans mint produced coins for three different governments. A total of 330,000 were struck under the United States government, 1,240,000 for the State of Louisiana after it seceded from the Union, and 962,633 after it joined the Confederacy. Since the same die was used for all strikings, the output looks identical. However the Confederate States of America actually minted four half dollars with a CSA (rather than USA) reverse and the obverse die they used had a small die crack. Thus "regular" 1861 half dollars with this crack probably were used by the Confederates for some of the mass striking.[11]

text: There are two varieties of Kennedy half dollars in the proof set issues of 1964. Initially, the die was used with accented hair, showing deeper lines than the president's widow, Jacqueline Kennedy Onassis, liked. New dies were prepared to smooth out some of the details. It is estimated that about 1 to 3% (40,000 to 100,000) of the proof halves are of the earlier type, making them somewhat more expensive for collectors.[12]

text: List of designs[edit]

text:
Silver half dollars

Flowing Hair 1794–1795[13]
Draped Bust 1796–1807

Draped Bust, Small Eagle 1796–1797[14]
Draped Bust, Heraldic Eagle 1801–1807[15]

Capped Bust 1807–1839

Capped Bust (Large Size), With Motto 1807–1836[16]
Capped Bust (Small Size), No Motto 1836–1839[17]

Seated Liberty 1839–1891

Seated Liberty, No Motto 1839–1866[18]
Seated Liberty, With Motto 1866–1891[19]

Barber 1892–1915[20]
Walking Liberty 1916–1947[21]
Franklin 1948–1963[22]
Kennedy 1964 (General circulation issue)[23] (the last 90% silver half dollar for circulation, contains 0.36169 oz. net silver per coin, or 7.234 oz. silver per roll)
Kennedy 1992–present (silver proof sets available)

40% silver half dollars

Kennedy 1965–1969
Kennedy 1970 (collectors sets only)
Kennedy 1976 (only collectors sets produced with 40% silver)

Copper-nickel clad half dollars

Kennedy 1971–1974, 1977–1986, 1988–2001 (general circulation issues)
Kennedy 1987, 2002–present (collectors only)
Kennedy Bicentennial 1975–1976 (all dated 1776–1976.)

Various half dollar designs

text: List of early commemorative issues[edit]

text:
Columbian half dollar (1892–1893)
Panama–Pacific half dollar (1915)
Illinois Centennial half dollar (1918)
Maine Centennial half dollar (1920)
Pilgrim Tercentenary half dollar (1920–1921)
Missouri Centennial half dollar (1921)
Alabama Centennial half dollar (1921)
Grant Memorial half dollar (1922)
Monroe Doctrine Centennial half dollar (1923)
Huguenot-Walloon half dollar (1924)
Lexington-Concord Sesquicentennial half dollar (1925)
Stone Mountain Memorial half dollar (1925)
California Diamond Jubilee half dollar (1925)
Fort Vancouver Centennial half dollar (1925)
United States Sesquicentennial half dollar (1926)
Oregon Trail Memorial half dollar (1926–1939)
Vermont Sesquicentennial half dollar (1927)
Hawaii Sesquicentennial half dollar (1928)
Maryland Tercentenary half dollar (1934)
Texas Centennial half dollar (1934–1938)
Daniel Boone Bicentennial half dollar (1934–1938)
Connecticut Tercentenary half dollar (1935)
Arkansas Centennial half dollar (1935–1939)
Arkansas-Robinson half dollar (1936)

Hudson Sesquicentennial half dollar (1935)
California Pacific International Exposition half dollar (1935–1936)
Old Spanish Trail half dollar (1935)
Rhode Island Tercentenary half dollar (1936)
Cleveland Centennial half dollar (1936)
Wisconsin Territorial Centennial half dollar (1936)
Cincinnati Musical Center half dollar (1936)
Long Island Tercentenary half dollar (1936)
York County, Maine Tercentenary half dollar (1936)
Bridgeport, Connecticut Centennial half dollar (1936)
Lynchburg Sesquicentennial half dollar (1936)
Elgin, Illinois, Centennial half dollar (1936)
Albany Charter half dollar (1936)
San Francisco–Oakland Bay Bridge half dollar (1936)
Columbia, South Carolina Sesquicentennial half dollar (1936)
Delaware Tercentenary half dollar (1936)
Battle of Gettysburg half dollar (1936)
Norfolk, Virginia Bicentennial half dollar (1936)
Roanoke Island half dollar (1937)
Battle of Antietam half dollar (1937)
New Rochelle, New York half dollar (1938)
Iowa Centennial half dollar (1946)
Booker T. Washington Memorial half dollar (1946–1954)


document_title: Exponentiation

question_text: what is the value of any number raised to the zero power

question_tokens: what is the value of any number raised to the zero power

root

text: Exponentiation

text: History of the notation[edit]

text: The term power was used by the Greek mathematician Euclid for the square of a line.[1] Archimedes discovered and proved the law of exponents, 10a ⋅ 10b = 10a+b, necessary to manipulate powers of 10.[2][better source needed] In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kahb for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.[3]

tag: p

text: In the late 16th century, Jost Bürgi used Roman numerals for exponents.[4]

text: Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.[5]

text: Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word "exponent" was coined in 1544 by Michael Stifel.[6] Samuel Jeake introduced the term indices in 1696.[1] In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).[7] Biquadrate has been used to refer to the fourth power as well.

text: Some mathematicians (e.g., Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d.

text: Another historical synonym, involution,[8] is now rare and should not be confused with its more common meaning.

text: In 1748 Leonhard Euler wrote "consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant."[9] With this introduction of transcendental functions, Euler laid the foundation for the modern introduction of natural logarithm as the inverse function for the natural exponential function, f(x) = ex.

text: Terminology[edit]

text: The expression b2 = b ⋅ b is called "the square of b" or "b squared" because the area of a square with side-length b is b2.

text: The expression b3 = b ⋅ b ⋅ b  is called "the cube of b" or "b cubed" because the volume of a cube with side-length b is b3.

text: When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is the power or, more specifically, 3 raised to the 5th power.

text: The word "raised" is usually omitted, and sometimes "power" as well, so 35 can also be read "3 to the 5th" or "3 to the 5". Therefore, the exponentiation bn can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most briefly as "b to the n".

text: Integer exponents[edit]

text: The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.

text: Positive exponents[edit]

text: Formally, powers with positive integer exponents may be defined by the initial condition[10]

text: and the recurrence relation

text: From the associativity of multiplication, it follows that for any positive integers m and n,

text: Zero exponent[edit]

text: Any nonzero number raised to the 0 power is 1:[11]

text: One interpretation of such a power is as an empty product.

text: The case of 00 is more complicated, and the choice of whether to assign it a value and what value to assign may depend on context. For more details, see Zero to the power of zero.

text: Negative exponents[edit]

text: The following identity holds for an arbitrary integer n and nonzero b:

text: Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity (∞).

text: The identity above may be derived through a definition aimed at extending the range of exponents to negative integers.

text: For non-zero b and positive n, the recurrence relation above can be rewritten as

text: By defining this relation as valid for all integer n and nonzero b, it follows that

text: and more generally for any nonzero b and any nonnegative integer n,

text: This is then readily shown to be true for every integer n.

text: Identities and properties[edit]

text: The following identities hold for all integer exponents, provided that the base is non-zero:

text: Unlike addition and multiplication:

text:
Exponentiation is not commutative. For example, 2 + 3 = 3 + 2 = 5 and 2 ⋅ 3 = 3 ⋅ 2 = 6, but 23 = 8, whereas 32 = 9.Exponentiation is not associative. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 ⋅ 3) ⋅ 4 = 2 ⋅ (3 ⋅ 4) = 24, but 23 to the 4 is 84 or 4096, whereas 2 to the 34 is 281 or 7024241785163922925♠2417851639229258349412352. Without parentheses to modify the order of operations, the conventional order is top-down (or right-associative), not bottom-up[12] (or left-associative):

b

p

q

=

b

(

p

q

)

≠
(

b

p

)

q

=

b

(
p
⋅
q
)

=

b

p
⋅
q

.

{\displaystyle b^{p^{q}}=b^{(p^{q})}\neq (b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}.}

tag: ul

text: Combinatorial interpretation[edit]

text: For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet).  Some examples for particular values of m and n are given in the following table:

text:
nm

The nm possible m-tuples of elements from the set {1,...,n}

0

5

=
0

{\displaystyle 0^{5}=0}

none

1

4

=
1

{\displaystyle 1^{4}=1}

(
1
,
1
,
1
,
1
)

{\displaystyle (1,1,1,1)}

2

3

=
8

{\displaystyle 2^{3}=8}

(
1
,
1
,
1
)
,
(
1
,
1
,
2
)
,
(
1
,
2
,
1
)
,
(
1
,
2
,
2
)
,
(
2
,
1
,
1
)
,
(
2
,
1
,
2
)
,
(
2
,
2
,
1
)
,
(
2
,
2
,
2
)

{\displaystyle (1,1,1),(1,1,2),(1,2,1),(1,2,2),(2,1,1),(2,1,2),(2,2,1),(2,2,2)}

3

2

=
9

{\displaystyle 3^{2}=9}

(
1
,
1
)
,
(
1
,
2
)
,
(
1
,
3
)
,
(
2
,
1
)
,
(
2
,
2
)
,
(
2
,
3
)
,
(
3
,
1
)
,
(
3
,
2
)
,
(
3
,
3
)

{\displaystyle (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

4

1

=
4

{\displaystyle 4^{1}=4}

(
1
)
,
(
2
)
,
(
3
)
,
(
4
)

{\displaystyle (1),(2),(3),(4)}

5

0

=
1

{\displaystyle 5^{0}=1}

(
)

{\displaystyle ()}

tag: table

text: Particular bases[edit]

text: Powers of ten[edit]

text: In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent.  For example, 7003100000000000000♠103 = 7003100000000000000♠1000 and 6996100000000000000♠10−4 = 6996100000000000000♠0.0001.

text: Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 7008299792458000000♠299792458 m/s (the speed of light in vacuum, in metres per second) can be written as 7008299792458000000♠2.99792458×108 m/s and then approximated as 7008299800000000000♠2.998×108 m/s.

text: SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 7003100000000000000♠103 = 7003100000000000000♠1000, so a kilometre is 7003100000000000000♠1000 m.

text: Powers of two[edit]

text: The first negative powers of 2 are commonly used, and have special names, e.g.: half and quarter.

text: Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2n members.

text: Integer powers of 2 are important in computer science. The positive integer powers 2n give the number of possible values for an n-bit integer binary number; for example, a byte may take 28 = 256 different values. The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.

text: Powers of one[edit]

text: The powers of one are all one: 1n = 1.

text: Powers of zero[edit]

text: If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.

text: If the exponent is negative, the power of zero (0n, where n < 0) is undefined, because division by zero is implied.

text: If the exponent is zero, 00 may be defined as 0, 1, or left undefined (see Zero to the power of zero).

text: Powers of minus one[edit]

text: If n is an even integer, then (−1)n = 1.

text: If n is an odd integer, then (−1)n = −1.

text: Because of this, powers of −1 are useful for expressing alternating sequences.  For a similar discussion of powers of the complex number i, see § Powers of complex numbers.

text: Large exponents[edit]

text: The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:

text: This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".

text: Powers of a number with absolute value less than one tend to zero:

text: Any power of one is always one:

text: Powers of –1 alternate between 1 and –1 as n alternates between even and odd, and thus do not tend to any limit as n grows.

text: If b < –1, bn, alternates between larger and larger positive and negative numbers as n alternates between even and odd, and thus does not tend to any limit as n grows.

text: If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

text: See § The exponential function below.

text: Other limits, in particular those of expressions that take on an indeterminate form, are described in § Limits of powers below.

text: Power functions[edit]

text:
Real functions of the form

f
(
x
)
=
c

x

n

{\displaystyle f(x)=cx^{n}}

 with

c
≠
0

{\displaystyle c\neq 0}

 are sometimes called power functions.[citation needed] When

n

{\displaystyle n}

 is an integer and

n
≥
1

{\displaystyle n\geq 1}

, two primary families exist: for

n

{\displaystyle n}

 even, and for

n

{\displaystyle n}

 odd.  In general for

c
>
0

{\displaystyle c>0}

, when

n

{\displaystyle n}

 is even

f
(
x
)
=
c

x

n

{\displaystyle f(x)=cx^{n}}

 will tend towards positive infinity with increasing

x

{\displaystyle x}

, and also towards positive infinity with decreasing

x

{\displaystyle x}

.  All graphs from the family of even power functions have the general shape of

y
=
c

x

2

{\displaystyle y=cx^{2}}

, flattening more in the middle as

n

{\displaystyle n}

 increases.[13]  Functions with this kind of symmetry (

f
(
−
x
)
=
f
(
x
)

{\displaystyle f(-x)=f(x)}

) are called even functions.

text:
When

n

{\displaystyle n}

 is odd,

f
(
x
)

{\displaystyle f(x)}

's asymptotic behavior reverses from positive

x

{\displaystyle x}

 to negative

x

{\displaystyle x}

.  For

c
>
0

{\displaystyle c>0}

,

f
(
x
)
=
c

x

n

{\displaystyle f(x)=cx^{n}}

 will also tend towards positive infinity with increasing

x

{\displaystyle x}

, but towards negative infinity with decreasing

x

{\displaystyle x}

. All graphs from the family of odd power functions have the general shape of

y
=
c

x

3

{\displaystyle y=cx^{3}}

, flattening more in the middle as

n

{\displaystyle n}

 increases and losing all flatness there in the straight line for

n
=
1

{\displaystyle n=1}

. Functions with this kind of symmetry (

f
(
−
x
)
=
−
f
(
x
)

{\displaystyle f(-x)=-f(x)}

) are called odd functions.

text:
For

c
<
0

{\displaystyle c<0}

, the opposite asymptotic behavior is true in each case.[14]

text: List of whole-number powers[edit]

text:
n
n2
n3
n4
n5
n6
n7
n8
n9
n10

2
4
8
16
32
64
128
256
512
1,024

3
9
27
81
243
729
2,187
6,561
19,683
59,049

4
16
64
256
1,024
4,096
16,384
65,536
262,144
1,048,576

5
25
125
625
3,125
15,625
78,125
390,625
1,953,125
9,765,625

6
36
216
1,296
7,776
46,656
279,936
1,679,616
10,077,696
60,466,176

7
49
343
2,401
16,807
117,649
823,543
5,764,801
40,353,607
282,475,249

8
64
512
4,096
32,768
262,144
2,097,152
16,777,216
134,217,728
1,073,741,824

9
81
729
6,561
59,049
531,441
4,782,969
43,046,721
387,420,489
3,486,784,401

10
100
1,000
10,000
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000
10,000,000,000

text: Rational exponents[edit]

text: An nth root of a number b is a number x such that xn = b.

text: If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal nth root of b. It is denoted n√b, where √   is the radical symbol; alternatively, the principal root may be written b1/n. For example: 41/2 = 2, 81/3 = 2.

text:
The fact that

x
=

b

1

/

n

{\displaystyle x=b^{1/n}}

 solves

x

n

=
b

{\displaystyle x^{n}=b}

  follows from noting that

text: If n is even, then xn = b has two real solutions if b is positive, which are the positive and negative nth roots, i.e., b1/n > 0 and -(b1/n) < 0. If b is negative, the equation has no solution in real numbers for even n.

text: If n is odd, then xn = b has exactly one real solution. The solution b1/n is positive if b is positive and negative if b is negative.

text: Taking a positive real number b to a rational exponent u/v, where u is an integer and v is a positive integer, and considering principal roots only, yields

text: Taking a negative real number b to a rational power u/v, where u/v is in lowest terms, yields a positive real result if u is even, and hence v is odd, because then bu is positive; and yields a negative real result, if u and v are both odd, because then bu is negative. The case of odd u and even v cannot be treated this way within the reals, since there is no real number x such that x2k = −1, the value of bu/v in this case must use the imaginary unit i, as described more fully in the section § Powers of complex numbers.

text: Thus we have (−27)1/3 = −3 and (−27)2/3 = 9. The number 4 has two 3/2th powers, namely 8 and −8; however, by convention the notation 43/2 employs the principal root, and results in 8. For employing the v-th root the u/v-th power is also called the u/v-th root, and for even v the term principal root denotes also the positive result.

text: This sign ambiguity needs to be taken care of when applying the power identities. For instance:

text: is clearly wrong. The problem starts already in the first equality by introducing a standard notation for an inherently ambiguous situation –asking for an even root– and simply relying wrongly on only one, the conventional or principal interpretation. The same problem occurs also with an inappropriately introduced surd-notation, inherently enforcing a positive result:

text: instead of

text: In general the same sort of problems occur for complex numbers as described in the section § Failure of power and logarithm identities.

text: Real exponents[edit]

text: The identities and properties shown above for integer exponents are true for positive real numbers with non-integer exponents as well. However the identity

text: cannot be extended consistently to cases where b is a negative real number (see § Real exponents with negative bases). The failure of this identity is the basis for the problems with complex number powers detailed under § Failure of power and logarithm identities.

text: Exponentiation to real powers of positive real numbers can be defined either by extending the rational powers to reals by continuity, or more usually as given in § Powers via logarithms below.

text: Limits of rational exponents[edit]

text: Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule[15]

text: where the limit as r gets close to x is taken only over rational values of r. This limit only exists for positive b. The (ε, δ)-definition of limit is used; this involves showing that for any desired accuracy of the result bx one can choose a sufficiently small interval around x so all the rational powers in the interval are within the desired accuracy.

text: For example, if x = π, the nonterminating decimal representation π = 3.14159... can be used (based on strict monotonicity of the rational power) to obtain the intervals bounded by rational powers

text:
The bounded intervals converge to a unique real number, denoted by

b

π

{\displaystyle b^{\pi }}

. This technique can be used to obtain the power of a positive real number b for any irrational exponent. The function fb(x) = bx is thus defined for any real number x.

text: The exponential function[edit]

text: The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. Although exponentiation of e could, in principle, be treated the same as exponentiation of any other real number, such exponentials turn out to have particularly elegant and useful properties. Among other things, these properties allow exponentials of e to be generalized in a natural way to other types of exponents, such as complex numbers or even matrices, while coinciding with the familiar meaning of exponentiation with rational exponents.

text: As a consequence, the notation ex usually denotes a generalized exponentiation definition called the exponential function, exp(x), which can be defined in many equivalent ways, for example by:

text: Among other properties, exp satisfies the exponential identity

text: The exponential function is defined for all integer, fractional, real, and complex values of x. In fact, the matrix exponential is well-defined for square matrices (in which case this exponential identity only holds when x and y commute), and is useful for solving systems of linear differential equations.

text: Since exp(1) is equal to e and exp(x) satisfies this exponential identity, it immediately follows that exp(x) coincides with the repeated-multiplication definition of ex for integer x, and it also follows that rational powers denote (positive) roots as usual, so exp(x) coincides with the ex definitions in the previous section for all real x by continuity.

text: Powers via logarithms[edit]

text: The natural logarithm ln(x) is the inverse of the exponential function ex. It is defined for b > 0, and satisfies

text: If bx is to preserve the logarithm and exponent rules, then one must have

text: for each real number x.

text: This can be used as an alternative definition of the real number power bx and agrees with the definition given above using rational exponents and continuity.  The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.

text: Real exponents with negative bases[edit]

text: Powers of a positive real number are always positive real numbers. The solution of x2 = 4, however, can be either 2 or −2. The principal value of 41/2 is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well-behaved.

text: Neither the logarithm method nor the rational exponent method can be used to define br as a real number for a negative real number b and an arbitrary real number r. Indeed, er is positive for every real number r, so ln(b) is not defined as a real number for b ≤ 0.

text: The rational exponent method cannot be used for negative values of b because it relies on continuity. The function f(r) = br has a unique continuous extension[15] from the rational numbers to the real numbers for each b > 0.  But when b < 0, the function f is not even continuous on the set of rational numbers r for which it is defined.

text: For example, consider b = −1. The nth root of −1 is −1 for every odd natural number n. So if n is an odd positive integer, (−1)(m/n) = −1 if m is odd, and (−1)(m/n) = 1 if m is even.  Thus the set of rational numbers q for which (−1)q = 1 is dense in the rational numbers, as is the set of q for which (−1)q = −1. This means that the function (−1)q is not continuous at any rational number q where it is defined.

text: On the other hand, arbitrary complex powers of negative numbers b can be defined by choosing a complex logarithm of b.

text: Irrational exponents[edit]

text: If a is a positive algebraic number, and b is a rational number, it has been shown above that ab is algebraic. This remains true even if one accepts any algebraic number for a, with the only difference that ab may take several values (see below), all algebraic. The Gelfond–Schneider theorem provides some information on the nature of ab when b is irrational (that is, not rational). It states:

text: If a is an algebraic number different from 0 and 1, and b an irrational algebraic number, then all the values of ab are transcendental numbers (that is, not algebraic).

text: Complex exponents with positive real bases[edit]

text: Imaginary exponents with base e[edit]

text:
A complex number is an expression of the form

z
=
x
+
i
y

{\displaystyle z=x+iy}

, where x and y are real numbers, and i is the so-called imaginary unit, a number that satisfies the rule

i

2

=
−
1

{\displaystyle i^{2}=-1}

.  A complex number can be visualized as a point in the (x,y) plane.  The polar coordinates of a point in the (x,y) plane consist of a non-negative real number r and angle θ such that x = r cos θ and  y = r sin θ.  So

text:
The product of two complex numbers z1 = x1 + iy1, z2 = x2 + iy2 is obtained by expanding out the product of the binomials and simplifying using the rule

i

2

=
−
1

{\displaystyle i^{2}=-1}

:

text: As a consequence of the angle sum formulas of trigonometry, if z1 and z2 have polar coordinates (r1, θ1), (r2, θ2), then their product z1z2 has polar coordinates equal to (r1r2, θ1 + θ2).

text: Consider the right triangle in the complex plane which has 0, 1, 1 + ix/n as vertices. For large values of n, the triangle is almost a circular sector with a radius of 1 and a small central angle equal to x/n radians. 1 + ix/n may then be approximated by the number with polar coordinates (1, x/n). So, in the limit as n approaches infinity, (1 + ix/n)n approaches (1, x/n)n = (1n, nx/n) = (1, x), the point on the unit circle whose angle from the positive real axis is x radians. The Cartesian coordinates of this point are (cos x, sin x). So e ix = cos x + isin x; this is Euler's formula, connecting algebra to trigonometry by means of complex numbers.

text: The solutions to the equation ez = 1 are the integer multiples of 2πi:

text: More generally, if ev = w, then every solution to ez = w can be obtained by adding an integer multiple of 2πi to v:

text: Thus the complex exponential function is a periodic function with period 2πi.

text: More simply: eiπ = −1; ex + iy = ex(cos y + i sin y).

text: Trigonometric functions[edit]

text: It follows from Euler's formula stated above that the trigonometric functions cosine and sine are

text: Before the invention of complex numbers, cosine and sine were defined geometrically. The above formula reduces the complicated formulas for trigonometric functions of a sum into the simple exponentiation formula

text: Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.

text: Complex exponents with base e[edit]

text: The power z = ex + iy can be computed as ex ⋅ eiy. The real factor ex is the absolute value of z and the complex factor eiy identifies the direction of z.

text: If b is a positive real number, and z is any complex number, the power bz is defined as ez ⋅ ln(b), where x = ln(b) is the unique real solution to the equation ex = b. So the same method working for real exponents also works for complex exponents.

text: For example:

text: The identity (bz)u=bzu is not generally valid for complex powers. The power bz is a complex number and any power of it has to follow the rules for powers of complex numbers below. A simple counterexample is given by:

text:
The identity is, however, valid for arbitrary complex

z

{\displaystyle z}

 when

u

{\displaystyle u}

 is an integer.

text: Powers of complex numbers[edit]

text: Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then in equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4.

text: Complex powers of positive reals are defined via ex as in section Complex exponents with positive real bases above. These are continuous functions.

text: Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory.

text: The rational power of a complex number must be the solution to an algebraic equation. Therefore, it always has a finite number of possible values. For example, w = z1/2 must be a solution to the equation w2 = z. But if w is a solution, then so is −w, because (−1)2 = 1. A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers.

text: Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.

text: Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the complex logarithm. The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as does the rule defined above for the corresponding real base.

text: Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However, in the common case of a positive real number the principal value is the same.

text: The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.

text: Complex exponents with complex bases[edit]

text: For complex numbers w and z with w ≠ 0, the notation wz is ambiguous in the same sense that log w is.

text: To obtain a value of wz, first choose a logarithm of w; call it log w.  Such a choice may be the principal value Log w (the default, if no other specification is given), or perhaps a value given by some other branch of log w fixed in advance.  Then, using the complex exponential function one defines

text: because this agrees with the earlier definition in the case where w is a positive real number and the (real) principal value of log w is used.

text: If z is an integer, then the value of wz is independent of the choice of log w, and it agrees with the earlier definition of exponentiation with an integer exponent.

text: If z is a rational number m/n in lowest terms with z > 0, then the countably infinitely many choices of log w yield only n different values for wz; these values are the n complex solutions s to the equation sn = wm.

text: If z is an irrational number, then the countably infinitely many choices of log w lead to infinitely many distinct values for wz.

text: The computation of complex powers is facilitated by converting the base w to polar form, as described in detail below.

text: A similar construction is employed in quaternions.

text: Complex roots of unity[edit]

text: A complex number w such that wn = 1 for a positive integer n is an nth root of unity.  Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1.

text: If wn = 1 but wk ≠ 1 for all natural numbers k such that 0 < k < n, then w is called a primitive nth root of unity.  The negative unit −1 is the only primitive square root of unity. The imaginary unit i is one of the two primitive 4th roots of unity; the other one is −i.

text: The number e2πi/n is the primitive nth root of unity with the smallest positive argument.  (It is sometimes called the principal nth root of unity, although this terminology is not universal and should not be confused with the principal value of n√1, which is 1.[16])

text: The other nth roots of unity are given by

text: for 2 ≤ k ≤ n.

text: Roots of arbitrary complex numbers[edit]

text: Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power wq in the important special case where q = 1/n and n is a positive integer. These are the nth roots of w; they are solutions of the equation zn = w.  As with real roots, a second root is also called a square root and a third root is also called a cube root.

text: It is usual in mathematics to define w1/n as the principal value of the root, which is, conventionally, the nth root whose argument has the smallest absolute value. When w is a positive real number, this is coherent with the usual convention of defining w1/n as the unique positive real nth root. On the other hand, when w is a negative real number, and n is an odd integer, the unique real nth root is not one of the two nth roots whose argument has the smallest absolute value. In this case, the meaning of w1/n may depend on the context, and some care may be needed for avoiding errors.

text: The set of nth roots of a complex number w is obtained by multiplying the principal value w1/n by each of the nth roots of unity. For example, the fourth roots of 16 are 2, −2, 2i, and −2i, because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, i, and −i.

text: Computing complex powers[edit]

text: It is often easier to compute complex powers by writing the number to be exponentiated in polar form.  Every complex number z can be written in the polar form

text: where r is a nonnegative real number and θ is the (real) argument of z.  The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates.  That is, r is the "radius" r2 = u2 + v2 and θ is the "angle" θ = atan2(v, u).  The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point.  Each choice of θ gives in general a different possible value of the power.  A branch cut can be used to choose a specific value.  The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.

text: In order to compute the complex power wz, write w in polar form:

text: Then

text: and thus

text: If z is decomposed as c + di, then the formula for wz can be written more explicitly as

text: This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form.  It is shown here both in polar form and in Cartesian form (via Euler's identity).

text: The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute ii, write i in polar and Cartesian forms:

text: Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields:

text: Similarly, to find (−2)3 + 4i, compute the polar form of −2,

text: and use the formula above to compute

text: The value of a complex power depends on the branch used.  For example, if the polar form i = 1e5πi/2 is used to compute ii, the power is found to be e−5π/2; the principal value of ii, computed above, is e−π/2. The set of all possible values for ii is given by:[17]

text: So there is an infinity of values which are possible candidates for the value of ii, one for each integer k. All of them have a zero imaginary part so one can say ii has an infinity of valid real values.

text: Failure of power and logarithm identities[edit]

text: Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

text:
The identity log(bx) = x ⋅ log b holds whenever b is a positive real number and x is a real number.  But for the principal branch of the complex logarithm one has

i
π
=
log

(
−
1
)
=
log

[

(
−
i

)

2

]

≠
2
log

(
−
i
)
=
2

(

−

i
π

2

)

=
−
i
π

{\displaystyle i\pi =\log(-1)=\log \left[(-i)^{2}\right]\neq 2\log(-i)=2\left(-{\frac {i\pi }{2}}\right)=-i\pi }

Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:

log

(

w

z

)
≡
z
⋅
log

(
w
)

(
mod

2
π
i
)

{\displaystyle \log(w^{z})\equiv z\cdot \log(w){\pmod {2\pi i}}}

This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) contain those of z ⋅ log w as a subset. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:

{

log

(

w

z

)

}

=

{

z
⋅
Log

(
w
)
+
z
⋅
2
π
i
n
+
2
π
i
m

}

{

z
⋅
log

(
w
)

}

=

{

z
⋅
Log

(
w
)
+
z
⋅
2
π
i
n

}

{\displaystyle {\begin{aligned}\left\{\log(w^{z})\right\}&=\left\{z\cdot \operatorname {Log} (w)+z\cdot 2\pi in+2\pi im\right\}\\\left\{z\cdot \log(w)\right\}&=\left\{z\cdot \operatorname {Log} (w)+z\cdot 2\pi in\right\}\end{aligned}}}

The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x is a real number.  But a calculation using principal branches shows that

1
=
(
−
1
⋅
−
1

)

1
2

≠
(
−
1

)

1
2

(
−
1

)

1
2

=
−
1

{\displaystyle 1=(-1\cdot -1)^{\frac {1}{2}}\not =(-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=-1}

and

i
=
(
−
1

)

1
2

=

(

1

−
1

)

1
2

≠

1

1
2

(
−
1

)

1
2

=

1
i

=
−
i

{\displaystyle i=(-1)^{\frac {1}{2}}=\left({\frac {1}{-1}}\right)^{\frac {1}{2}}\not ={\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}

On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.
If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1)1/2 are {1, −1}. The identity holds, but saying {1} = {(−1 ⋅ −1)1/2} is wrong.
The identity (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:[18]
For any integer n, we have:

e

1
+
2
π
i
n

=

e

1

e

2
π
i
n

=
e
⋅
1
=
e

{\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}

(

e

1
+
2
π
i
n

)

1
+
2
π
i
n

=
e

{\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad }

  (taking the

(
1
+
2
π
i
n
)

{\displaystyle (1+2\pi in)}

-th power of both sides)

e

1
+
4
π
i
n
−
4

π

2

n

2

=
e

{\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad }

 (using

(

e

x

)

y

=

e

x
y

{\displaystyle (e^{x})^{y}=e^{xy}}

 and expanding the exponent)

e

1

e

4
π
i
n

e

−
4

π

2

n

2

=
e

{\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad }

 (using

e

x
+
y

=

e

x

e

y

{\displaystyle e^{x+y}=e^{x}e^{y}}

)

e

−
4

π

2

n

2

=
1

{\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad }

 (dividing by e)
but this is false when the integer n is nonzero.
The error is the following: by definition,

e

y

{\displaystyle e^{y}}

 is a notation for

exp

(
y
)
,

{\displaystyle \exp(y),}

 a true function, and

x

y

{\displaystyle x^{y}}

 is a notation for

exp

(
y
log

x
)
,

{\displaystyle \exp(y\log x),}

 which is a multi-valued function. Thus the notation is ambiguous when x = e. Here, before expanding the exponent, the second line should be

text: Generalizations[edit]

text: Monoids[edit]

text: Exponentiation with integer exponents can be defined in any multiplicative monoid.[19]  A monoid is an algebraic structure consisting of a set X together with a rule for composition ("multiplication") satisfying an associative law and a multiplicative identity, denoted by 1.  Exponentiation is defined inductively by:

text:
x

0

=
1

{\displaystyle x^{0}=1}

 for all

x
∈
X

{\displaystyle x\in X}

x

n
+
1

=

x

n

x

{\displaystyle x^{n+1}=x^{n}x}

 for all

x
∈
X

{\displaystyle x\in X}

 and non-negative integers n
If n is a negative integer then

x

n

{\displaystyle x^{n}}

 is only defined[20] if

x

{\displaystyle x}

 has an inverse in X.

text: Monoids include many structures of importance in mathematics, including groups and rings (under multiplication), with more specific examples of the latter being matrix rings and fields.

text: Matrices and linear operators[edit]

text:
If A is a square matrix, then the product of A with itself n times is called the matrix power.  Also

A

0

{\displaystyle A^{0}}

 is defined to be the identity matrix,[21] and if A is invertible, then

A

−
n

=
(

A

−
1

)

n

{\displaystyle A^{-n}=(A^{-1})^{n}}

.

text:
Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system.[22]  This is the standard interpretation of a Markov chain, for example.  Then

A

2

x

{\displaystyle A^{2}x}

 is the state of the system after two time steps, and so forth:

A

n

x

{\displaystyle A^{n}x}

 is the state of the system after n time steps.  The matrix power

A

n

{\displaystyle A^{n}}

 is the transition matrix between the state now and the state at a time n steps in the future.  So computing matrix powers is equivalent to solving the evolution of the dynamical system.  In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

text:
Apart from matrices, more general linear operators can also be exponentiated.  An example is the derivative operator of calculus,

d

/

d
x

{\displaystyle d/dx}

, which is a linear operator acting on functions

f
(
x
)

{\displaystyle f(x)}

 to give a new function

(
d

/

d
x
)
f
(
x
)
=

f
′

(
x
)

{\displaystyle (d/dx)f(x)=f'(x)}

.  The n-th power of the differentiation operator is the n-th derivative:

text: These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents.  This is the starting point of the mathematical theory of semigroups.[23]  Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics.  Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution.  The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

text: Finite fields[edit]

text:
A field is an algebraic structure in which multiplication, addition, subtraction, and division are all well-defined and satisfy their familiar properties.  The real numbers, for example, form a field, as do the complex numbers and rational numbers.  Unlike these familiar examples of fields, which are all infinite sets, some fields have only finitely many elements.  The simplest example is the field with two elements

F

2

=
{
0
,
1
}

{\displaystyle F_{2}=\{0,1\}}

 with addition defined by

0
+
1
=
1
+
0
=
1

{\displaystyle 0+1=1+0=1}

 and

0
+
0
=
1
+
1
=
0

{\displaystyle 0+0=1+1=0}

, and multiplication

0
⋅
0
=
1
⋅
0
=
0
⋅
1
=
0

{\displaystyle 0\cdot 0=1\cdot 0=0\cdot 1=0}

 and

1
⋅
1
=
1

{\displaystyle 1\cdot 1=1}

.

text: Exponentiation in finite fields has applications in public key cryptography.  For example, the Diffie–Hellman key exchange uses the fact that exponentiation is computationally inexpensive in finite fields, whereas the discrete logarithm (the inverse of exponentiation) is computationally expensive.

text:
Any finite field F has the property that there is a unique prime number p such that

p
x
=
0

{\displaystyle px=0}

 for all x in F; that is, x added to itself p times is zero.  For example, in

F

2

{\displaystyle F_{2}}

, the prime number p = 2 has this property.  This prime number is called the characteristic of the field.  Suppose that F is a field of characteristic p, and consider the function

f
(
x
)
=

x

p

{\displaystyle f(x)=x^{p}}

 that raises each element of F to the power p.  This is called the Frobenius automorphism of F.  It is an automorphism of the field because of the Freshman's dream identity

(
x
+
y

)

p

=

x

p

+

y

p

{\displaystyle (x+y)^{p}=x^{p}+y^{p}}

.  The Frobenius automorphism is important in number theory because it generates the Galois group of F over its prime subfield.

text: In abstract algebra[edit]

text: Exponentiation for integer exponents can be defined for quite general structures in  abstract algebra.

text: Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by

text: One has the following properties

text: If the operation has a two-sided identity element 1, then x0 is defined to be equal to 1 for any x.

text: If the operation also has two-sided inverses and is associative, then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents.

text: If the multiplication operation is commutative (as for instance in abelian groups), then  the following holds:

text: If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

text: When there are several power-associative binary operations defined on a set, any of which might be iterated, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x∗n is x ∗ ... ∗ x, while x#n is x # ... # x, whatever the operations ∗ and # might be.

text: Superscript notation is also used, especially in group theory, to indicate conjugation. That is, gh = h−1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

text: Over sets[edit]

text: If n is a natural number and A is an arbitrary set, the expression An is often used to denote the set of ordered n-tuples of elements of A. This is equivalent to letting An denote the set of functions from the set {0, 1, 2, ..., n−1} to the set A; the n-tuple (a0, a1, a2, ..., an−1) represents the function that sends i to ai.

text: For an infinite cardinal number κ and a set A, the notation Aκ is also used to denote the set of all functions from a set of size κ to A.   This is sometimes written κA to distinguish it from cardinal exponentiation, defined below.

text: This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of

text: where each Vi is a vector space.

text: Then if Vi = V for each i, the resulting direct sum can be written in exponential notation as V⊕N, or simply VN with the understanding that the direct sum is the default. We can again replace the set N with a cardinal number n to get Vn, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and n to be some natural number, we get the vector space that is most commonly studied in linear algebra, the real vector space Rn.

text: If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an n-tuple, which can be represented by a function on a set of appropriate cardinality, SN becomes simply the set of all functions from N to S in this case:

text: This fits in with the exponentiation of cardinal numbers, in the sense that |SN| = |S||N|, where |X| is the cardinality of X. When "2" is defined as {0, 1}, we have |2X| = 2|X|, where 2X, usually denoted by P(X), is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.

text: In category theory[edit]

text: In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object.  This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 00 is isomorphic to any terminal object 1.

text: Of cardinal and ordinal numbers[edit]

text: In set theory, there are exponential operations for cardinal and ordinal numbers.

text: If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.[24]  If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation.  For example, the set of 3-tuples of elements from a 2-element set has cardinality 8 = 23. In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero).

text: Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.

text: Repeated exponentiation[edit]

text: Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration.  Iterating tetration leads to another operation, and so on, a concept named hyperoperation.  This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than  multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7012762559748498700♠7625597484987 (= 327 = 333 = 33) respectively.

text: Limits of powers[edit]

text: Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.

text: More precisely, consider the function f(x, y) = xy defined on D = {(x, y) ∈ R2 : x > 0}. Then D can be viewed as a subset of R2 (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.

text: In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).[25] Accordingly, this allows one to define the powers xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

text: Under this definition by continuity, we obtain:

text:
x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞.
x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1.
0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞.
0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 0.

text: These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.

text: On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.

text: Efficient computation with integer exponents[edit]

text: Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example.  To compute 2100, note that 100 = 64 + 32 + 4.  Compute the following in order:

text:
22 = 4
(22)2 = 24 = 16
(24)2 = 28 = 256
(28)2 = 216 = 65,536
(216)2 = 232 = 4,294,967,296
(232)2 = 264 = 18,446,744,073,709,551,616
264 232 24 = 2100 = 1,267,650,600,228,229,401,496,703,205,376

text: This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).

text: In general, the number of multiplication operations required to compute bn can be reduced to Θ(log n) by using exponentiation by squaring or (more generally) addition-chain exponentiation.  Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bn is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.[26]

text: Exponential notation for function names[edit]

text: Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus, f 3(x) may mean f(f(f(x))); in particular, f −1(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root f 1/2(x).

text: For historical reasons, this notation applied to the trigonometric and hyperbolic functions has a specific and diverse interpretation: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of −1 denotes the inverse function. That is, sin2 x is just a shorthand way to write (sin x)2 without using parentheses, whereas sin−1 x refers to the inverse function of the sine, also called arcsin x. Each trigonometric and hyperbolic has its own name and abbreviation both for the reciprocal; for example, 1/(sin x) = (sin x)−1 = csc x, as well as for its inverse, for example cosh−1 x = arcosh x.  A similar convention applies to logarithms, where log2 x usually means (log x)2, not log log x.

text: In programming languages[edit]

text: Programming languages generally express exponentiation either as an infix operator or as a function, as they are linear notations which do not support superscripts:

text:
x ↑ y: Algol, Commodore BASIC
x ^ y: AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua and most computer algebra systems. Conflicting uses of the symbol ^ include: XOR (in POSIX Shell arithmetic expansion, AWK, C, C++, C#, D, Go, Java, JavaScript, Perl, PHP, Python, Ruby and Tcl), indirection (Pascal), and string concatenation (OCaml and Standard ML).
x ^^ y: Haskell (for fractional base, integer exponents), D
x ** y: Ada, Z shell, Korn shell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, VHDL
pown x y: F# (for integer base, integer exponent)
x⋆y: APL

text: Many other programming languages lack syntactic support for exponentiation, but provide library functions:

text:
pow(x, y): C, C++
Math.Pow(x, y): C#
math:pow(X, Y): Erlang

text: For certain exponents there are special ways to compute xy much faster than through generic exponentiation. These cases include small positive and negative integers (prefer x*x over x2; prefer 1/x over x−1) and roots (prefer sqrt(x) over x0.5, prefer cbrt(x) over x1/3).

text: Not all programming languages adhere to the convention of right-associativity for exponentiation: while the Wolfram language and many others follow this convention, some computer programs such as Microsoft Office Excel and Matlab associate to the left (bottom-up) instead (i.e. a^b^c is evaluated as (a^b)^c).


document_title: Toto (band)

question_text: where did the band name toto come from

question_tokens: where did the band name toto come from

text: Toto (band)

text: History[edit]

text: Formation (1976)[edit]

text: The members of Toto were regulars on albums by Steely Dan,[4] Seals and Crofts,[5] Boz Scaggs,[6] Sonny and Cher,[7] and many others, contributing to many of the most popular records of the 1970s. Keyboardist David Paich, son of musician and session player/arranger Marty Paich, rose to fame after having co-written much of Scaggs's Silk Degrees album. Having played on many sessions with drummer Jeff Porcaro (the son of noted session percussionist Joe Porcaro), whom he met while attending Grant High School in Van Nuys, California, where they formed the band Rural Still Life, Paich began to discuss seriously with Porcaro the possibility of them forming their own band. They brought in bassist and fellow session vet David Hungate, having played with him in the backing band for Scaggs. In addition, the duo asked fellow Grant High School students, guitarist Steve Lukather (who also played in Scaggs's band as a replacement for Les Dudek) and Jeff Porcaro's brother Steve Porcaro (keyboards) to join the team. Lukather and Steve Porcaro were in the same year at Grant and continued the band Rural Still Life (the name shortened to Still Life) after Paich and Jeff graduated. With the addition of former S.S. Fools singer Bobby Kimball, the group began to work on their first album in 1976 after signing with Columbia Records.

text: Band name and debut album (1976–1979)[edit]

text: Once the band came together, David Paich began composing what would become the eponymous debut album, Toto. According to popular myth, at the first recording sessions, in order to distinguish their own demo tapes from other bands' in the studio, Jeff Porcaro wrote the word "Toto" on them. In the early 1980s, band members told the press that the band was named after the dog in The Wizard of Oz.[8] One popular rumor is that the name came from Bobby Kimball's "true" last name, Toteaux. This rumor was in fact only a joke popularized by former bassist David Hungate.[9] After the completion of the first album, the band and record were still unnamed. David Hungate, after viewing the name on the demo tapes, explained to the group that the words "In Toto" in Latin translated to "all-encompassing." Because the band members played on so many records and so many musical genres, they adopted the name "Toto" as their own.[10]

text: After its release, Toto climbed the charts quickly, earning popularity with the hit single "Hold the Line", as well as the charting "I'll Supply the Love" and "Georgy Porgy", featuring Cheryl Lynn. The band garnered international acclaim and was nominated for a Grammy Award for Best New Artist. Shortly thereafter, in early 1979, Toto embarked on their first American tour in support of the debut album.[11] For the tour, Toto brought along two additional musicians, Tom Kelly (guitar, backup vocals) and Lenny Castro (percussion), to increase the depth of the sound, and continued to bring additional touring musicians for all subsequent tours. (See the "Tour Musicians" section below).

text: Hydra and Turn Back (1980–1981)[edit]

text: At the close of the first tour, the band began work on their next album, entitled Hydra, which was released later that year and featured the single "99", inspired by George Lucas' cult film THX 1138.[12] Nearly 30 years later, Steve Lukather confessed that, despite the song's popularity, he hated "99" and that it's one of his least favorite Toto songs, which is why it was rarely performed after Hydra's tour.[13] The band also released four promotional music videos for the album, including the title track. The other two were "St George and The Dragon" and "All Us Boys". They were directed by Bruce Gowers and produced by Paul Flattery for Jon Roseman Productions International. Although the album Hydra failed to achieve the commercial success of Toto's first release, it still went gold. Following the album's release, the band set out on the "Hydra Tour", which featured both American and international dates. The tour lasted from February until June 1980.[11]

text: In early 1981, Toto released their third album, Turn Back. The album was a venture into arena rock[14] and featured heavier guitar and fewer keyboards than on the previous two records.

text: On December 15, 1981, Bobby Kimball was arrested for allegedly selling cocaine to an undercover police officer.[15]

text: Toto IV (1982)[edit]

text:
Africa

Problems playing this file? See media help.

text: 1982 marked the beginning of Toto's most successful era. After the disappointing sales of Turn Back, the band was under a great deal of pressure from the record company to produce a new smash record. With the Triple Platinum-certified Toto IV, the band delivered one of the most commercially successful records of the 1980s. The album featured three singles that reached the Top 10 on the Billboard Hot 100 chart: "Rosanna", "Africa" and "I Won't Hold You Back". The album also appeared on several worldwide charts, introducing the band to new audiences around the globe. "Africa" topped the charts in February 1983 and was a constant presence on radios around the world, but it was "Rosanna" that earned the band multiple Grammy nominations. Toto IV earned seven Grammy Awards, including "Record of the Year" for "Rosanna", "Album of the Year" for Toto IV and "Producer of the Year". At the time Steve Porcaro was dating actress Rosanna Arquette, but the song is not about her, according to writer David Paich.[16] In the music video for the song, Cynthia Rhodes plays the title character. In addition to "Africa" and "Rosanna", Toto IV continued its successful run with the release of another single, "Make Believe". Toto toured throughout 1982 in support of Toto IV.[11] Also, during this time, Steve Porcaro co-wrote and co-composed "Human Nature," which Michael Jackson recorded for his best-selling album Thriller (1982), turning the song into a smash hit.

text: Isolation (1982–1985)[edit]

text: Subsequent to the Toto IV release, bassist David Hungate left the band. Hungate, who had relocated to Nashville in 1980 to pursue a session/production career, felt that the fame surrounding Toto IV would prevent him from spending time with his family. A third Porcaro brother, Mike Porcaro, who had performed cello on a track from Toto IV, replaced Hungate on bass. Lead singer Bobby Kimball spent the early part of 1983 facing prosecution for drug-related charges. Kimball was ordered to stand trial, but pleaded not guilty. The charges were dismissed on May 28 of that year.[17] However, Kimball was fired from the band in 1984.[18] Later that year, Toto composed most of the music for the soundtrack to the film Dune.

text: At one point, Richard Page of the band Mr. Mister was offered the lead singer spot, but turned it down to continue with his band. Fergie Frederiksen (formerly of Angel, Trillion and Le Roux) was brought in as the new vocalist and the band recorded Isolation, released in November 1984. While Isolation did not achieve the acclaim or sales of Toto IV, it did achieve Gold status, largely on the strength of the single "Stranger in Town". "Isolation"'s tour began in February 1985 and concluded three months later.[11]

text: Fahrenheit and The Seventh One with Joseph Williams (1985–1988)[edit]

text: At the close of the Isolation tour in 1985, Fergie Frederiksen was let go. Lukather claimed that the band was not meshing well with Frederiksen because he had a difficult time recording with them in the studio.[19] The band held an audition and Joseph Williams, son of film composer John Williams and 1950s singer/actress Barbara Ruick, was chosen to take over lead vocals in early 1986.

text: With Joseph Williams now onboard officially, Toto wrote and recorded Fahrenheit, released in October 1986. While Williams is credited as lead vocalist, Frederiksen had begun recording a few tracks and is featured as a background vocalist on the track "Could This Be Love".

text: Fahrenheit brought the band back from the heavier sound of Isolation to their pop/rock roots. "I'll Be Over You" and "Without Your Love", which were both ballads sung by Lukather, were the two hit singles. The band recruited several guest musicians for the album. They recorded an instrumental piece entitled "Don't Stop Me Now" with legendary jazz trumpeter Miles Davis. In addition, a then-unknown Paula Abdul appeared as a dancer in their "Till the End" music video. Michael McDonald provided backup vocals on the song "I'll Be Over You" (and appeared in the accompanying music video), while Eagles founder and songwriting giant Don Henley appeared on the Steve Porcaro penned track "Lea".

text: After its release, the band embarked on another world tour. Upon its conclusion in 1987, Steve Porcaro left the band to pursue a career in film and television scoring. Fahrenheit eventually went Gold on October 3, 1994.[20] Steve Porcaro was never replaced and Toto decided to continue with only five members. Although Porcaro occasionally assisted the band on synthesizers for their subsequent studio albums (and appeared on their 1988 tour), David Paich handled most of the live keyboard work (with keyboard technician John Jessel assisting on certain dates) post 1988.

text: In 1988 Toto released their next album The Seventh One, featuring Jon Anderson of Yes on backup vocals on the single "Stop Loving You". The album's other single, "Pamela", became very popular and would be the band's last to hit the US Top 40. The Seventh One became the band's most successful release since Toto IV.[21] The band toured from February through July 1988.[11]

text: Past To Present and Jean-Michel Byron (1988–1990)[edit]

text: Although "The Seventh One Tour" was very successful, after it was finished the band decided to replace lead singer Joseph Williams. Originally, the band wanted to reunite with original vocalist Bobby Kimball to record new songs for a greatest hits record, but the record company instead insisted they hire South African singer Jean-Michel Byron. Before Byron was brought in, the band recorded "Goin' Home" with Kimball. This song was later featured on the Toto XX album as an 'unreleased song.' When Byron was brought in (in 1989) he and Toto recorded four new songs which were included on their greatest hits album Past to Present 1977–1990, released in 1990. Toto then embarked upon the "Planet Earth" tour that lasted from September until December 1990. The band didn't get along with Byron, whose diva-like behavior and flamboyant stage presence caused friction during the tour. He was demoted to background vocals before ultimately being fired at the conclusion of the tour. During this time-frame, the band also found that former singer, Bobby Kimball, was booking shows and billing himself and his backing band as "Toto".[22] In April, mirroring the situation that had happened with Kimball seven years previously, former singer Joseph Williams was arrested on drug-related charges.[23]

text: Kingdom of Desire and Jeff Porcaro's death (1991–1992)[edit]

text: Once again without a lead vocalist, guitarist Steve Lukather sang lead vocals and became the new front man. Toto played at the Montreux Jazz Festival in 1991 and the band recorded Kingdom of Desire, which was released on Columbia Records in most parts of the world and on Clive Davis' label Relativity Records in the United States.

text: Jeff Porcaro died in an accident on August 5, 1992, at the age of 38 while working in his garden. According to the LA Times Report, the Los Angeles County Coroner's office lists the cause of death to be a heart attack from the hardening of the arteries caused by cocaine use.[24][25] Facing the prospect of a tour without Jeff, Toto almost broke up. However, Jeff Porcaro's family insisted the band continue. Englishman Simon Phillips was the only drummer ever contacted to replace Jeff Porcaro since the band knew that Porcaro liked Phillips and because Lukather worked with Phillips on a previous tour with Santana and Jeff Beck in Japan in 1986.

text: Phillips joined the band and they went on the tour, which they dedicated to Jeff's memory. In 1993 they released a live album called Absolutely Live. From 1991 on, Steve Lukather would handle a majority of the vocals (until Bobby Kimball's return in 1998), but some older songs originally sung by Kimball, Fergie Frederiksen, and Joseph Williams were put in the set list and sung by new backup singers Fred White (who was replaced by John James in 1992), Jackie McGee (who'd joined for the 1990 tour and was replaced by Donna McDaniel in 1992) and Jenny Douglas-McRae (who'd also come aboard in 1990.) John sang "Stop Loving You" and Bobby's part on "Rosanna", Donna sang "Home of the Brave" and "Angel Don't Cry", and Jenny sang "Hold the Line".

text: On December 14, 1992, the Tribute to Jeff Porcaro Concert was held at Universal City's Universal Amphitheatre. Performers, including Don Henley, Eddie Van Halen, Donald Fagen, Walter Becker, Boz Scaggs, James Newton Howard, Michael McDonald, Richard Marx and special guest George Harrison, performed various Toto songs along with the four remaining Toto members.[26] At the close of the tour, the band then took a break to pursue individual projects as well as to keep up their session schedules.

text: Addition of Simon Phillips and Tambu (1995–1997)[edit]

text: In 1995 Toto recorded Tambu, their first album with Simon Phillips, which saw the band back with CBS (now Sony). A departure from Toto's sound of the late 1970s and 1980s, Tambu was a very organic release and featured the single "I Will Remember", which received moderate radio play. Other singles released were "Drag Him To The Roof" and "The Turning Point". Tambu also featured John James and Jenny Douglas-McRae as backup singers on some of the tracks. Douglas-McRae even sang lead on the album's bonus track, "Blackeye", and also in a duet with Steve Lukather on "Baby He's Your Man". Tambu sold 600,000 copies worldwide.[citation needed]

text: The "Tambu Tour" proved to be another success, although there were no North American dates. Simon Phillips suffered from a back problem, so Gregg Bissonette had to fill in for him during the first leg of the tour in late 1995. The tour concluded in 1996. The rest of the tour personnel remained the same, with the exception of Donna McDaniel who'd left in 1994 shortly after the "Night of the Proms" performances (which Douglas-McRae had missed since she was out touring with Joe Cocker). The song "Hold the Line" was now sung as a duet between James and Douglas-McRae.[27] Both James and Douglas-McRae were dropped from the band at the conclusion of the 1997 tour.

text: Toto XX, Bobby Kimball's return and Mindfields (1997–2001)[edit]

text: 1997 marked the band's 20th anniversary, and in order to commemorate it, David Paich and Steve Lukather started to go through several old tapes and demos for a special record of unreleased songs. In 1998 they released Toto XX with the single "Goin Home". Toto went on a small promotional tour with former members Bobby Kimball, Steve Porcaro and Joseph Williams.

text: After the "Toto XX" tour, Bobby Kimball rejoined the band as lead singer after 14 years. The band released Mindfields in early 1999 and embarked on the "Reunion" tour, touring worldwide and returning to the United States for the first time in six years. The new album featured three singles, "Melanie", "Cruel" and "Mad About You", a song co-written by David Paich and former Toto vocalist Joseph Williams. Later that year, a live album titled Livefields was released. The tour officially concluded in 2000, but the band played a few shows throughout 2001. David Paich briefly took a break from touring in 2000, so Jeff Babko filled in on keyboards.[28] Paich then resumed touring with Toto in 2001.

text: Through the Looking Glass and Toto's 25th anniversary (2002–2003)[edit]

text: In 2002, in celebration of Toto's 25th anniversary, the band released Through the Looking Glass, a covers album that paid tribute to the band's musical influences, such as Bob Marley, Steely Dan, George Harrison and Elton John. Two singles were released, "Could You Be Loved", a Bob Marley cover, and "While My Guitar Gently Weeps", a Beatles cover. The album was not a commercial success and many fans were upset about the release, thinking that the band should have written new material instead.[29] However, the record gave the band material to promote their "25th Anniversary Tour", which started in 2002 and concluded in 2003. After the tour, Toto released a live album and DVD of the show titled Live in Amsterdam. Both the live album and the DVD were released in late 2003.

text: Greg Phillinganes and David Paich's semi-retirement (2003–2005)[edit]

text: Beginning in June 2003, near the end of their "25th Anniversary Tour", keyboardist David Paich took a leave of absence from touring to spend time with a sick family member. Veteran keyboardist Greg Phillinganes filled in for Paich for the rest of the tour.

text: In late 2003, Toto headlined Night of the Proms for two months straight. Paich returned but was only able to play for a few weeks before he had to leave again, so Phillinganes once again filled in for the remainder of the shows. In early 2004, the band embarked on a world tour that went throughout 2004 and 2005. Paich only occasionally appeared, with Phillinganes playing at most of the shows. In 2005, Phillinganes was asked to become a regular member of the band and Paich retired from touring. Paich was still a member of Toto, however, as he continued to record and produce on all of Toto's releases.[30][citation needed]

text: Falling In Between and Falling In Between Live (2006–2008)[edit]

text: In early 2006, Toto released Falling in Between on the Italian label Frontiers, their first album of new material since 1999. The release featured extensive keyboard work from Steve Porcaro and a duet with Joseph Williams on the first single, "Bottom of Your Soul". Following the record's release, Toto embarked on an extensive worldwide tour in 2006, which continued into 2007 for a second leg. The 2007 leg featured Leland Sklar filling in on bass for Mike Porcaro due to an (at the time) undisclosed illness. 2007 featured extensive dates in both Europe and the United States, including an appearance at Moondance Jam in Walker, Minnesota. Former lead singer Fergie Frederiksen made a guest appearance at the Minneapolis date on May 5, 2007[31] and Joseph Williams also made a few guest appearances with the band in June 2007.[32]

text: Toto released a two-CD set Falling in Between Live on Eagle Records to commemorate the tour. This live set marks the fourth for the band, following 1993's Absolutely Live, 1999's Livefields and 2003's Live In Amsterdam.

text: In 2008, a companion DVD of the show which was recorded in Paris in March 2007 was released.

text: Hiatus (2008)[edit]

text: After a period of rumors and some allusions, on June 5, 2008, Lukather posted a message on his official website, stating, "The fact is yes I have left Toto. There is no more Toto. I just can't do it anymore and at 50 years old I wanted to start over and give it one last try on my own."[33]

text: As the main motivation for his decision, Lukather said:

text: When Dave [Paich] retired that was REAL hard for me 'cause we started the band together. Hell, it's 35 years if you count High School where the core all met. When Mike [Porcaro] fell ill and had to leave that was it for me. If there isn't Paich or at least one Porcaro how can we even call it Toto? … Honestly, I have just had enough. This is NOT a break. It is over. I really can't go out and play Hold The Line with a straight face anymore.

text: Reformation (2010)[edit]

text: On February 26, 2010, the band reformed and reunited for a brief tour of Europe in the summer to benefit Mike Porcaro, who had been diagnosed with ALS. The lineup featured David Paich, Steve Lukather, Steve Porcaro, Simon Phillips, Joseph Williams and special guest Nathan East.[34]

text: Steve Lukather stated that the band still exists:

text:
“
Edit to 2010.. well with Mike not being well and some of us missing each other the REAL high school friends (yes that includes Joe!!) got back together for the RIGHT reason.
People want to hear the hits played the way they were recorded, Joe has his voice back 100% and I loved having that power in front again and all REAL, BIG vocals again strong and seeing Paich AND Steve Porcaro with THOSE real keyboard sounds, Simon back and as Lee was busy having old pal Nate East step in.. it was so much fun and so successful we thought "Hmm, maybe a few gigs once in awhile would be fun" and we can make some money etc.. please some of the hard core fans. We ALL have outside careers that keep us all busy, not to mention families etc.. no one really wants to make new music, not a full record anyway. MAYBE a track someday but not in the near future. I am on tour for a year anyway and enjoying a very successful sold out solo tour and my record sales are better than they have ever been so I dont wanna mess THAT up and everyone has their own very busy successful lives and once in awhile.. why not right? There is no covert BS goin on here.[35]

”

text: Another tour in the summer of 2011 took place with former backup singer Jenny Douglas once again joining. Their show on July 17, 2011 in Verona, Italy was recorded for a live DVD but has yet to be released because of a contractual issue with their former label. They once again toured Europe in 2012.[36]

text: Toto's 35th anniversary, death of Fergie Frederiksen, Simon Phillips' departure, and Keith Carlock (2013–2014)[edit]

text: In 2013, celebrating their 35th anniversary, the band embarked on tour across Europe and North America, along with Japanese dates to follow in 2014. Their show on June 25, 2013 in Łódź, Poland was recorded for a live release and was released on April 29, 2014. On November 5, it was confirmed both on Toto's and David Paich's official Facebook pages that a new studio album was in the works and that the band planned to go into the studio early 2014.

text: On January 18, 2014, former vocalist Fergie Frederiksen died after a long battle with liver cancer.[37][38] After the 2013 leg of the 35th anniversary tour, it was revealed via the band's official website on January 23, 2014 that Simon Phillips had departed the band to pursue a solo career. Phillips was then replaced by Steely Dan drummer Keith Carlock.[39]

text: David Hungate's return and North American tour with Michael McDonald (2014)[edit]

text: While Keith Carlock was now part of the band, he did not join them for at least the North American tour and was filled in by Shannon Forrest. Bass player Nathan East, who has been touring with them since 2010, left after the Japanese tour to pursue his own projects. Instead, original bass player David Hungate rejoined the band. They once again toured the United States in spring 2014 as co-headliners with singer Michael McDonald. It was their most comprehensive tour on the continent in years.[40][41][42]

text: Toto XIV, Mike Porcaro's death & 40 Trips Around The Sun (2015–present)[edit]

text: Toto released their fourteenth studio album and their first in nine years titled Toto XIV on March 20 (Europe), March 23 (UK and Oceania), and March 24, 2015 (North America). To promote the newly finished project, the band started a world tour running with an extensive European headline arena tour including appearances at key festivals, along with a North American tour to follow in the summer of 2015 and Asia later that year.[43][44]

text: On March 15, 2015, former bassist Mike Porcaro died, due to complications from his battle with ALS, in his sleep at his home in Los Angeles.[45]

text: On 6 April 2015, Toto announced that they would embark on 7 August 2015 in Mashantucket, Connecticut with veteran progressive band Yes on a joint summer tour of North America due to end on 12 September 2015 in Coquitlam, British Columbia.[46][47][48] Shannon Forrest has continued to perform as the band's drummer.

text: On 29 September 2015, Toto announced the first leg of their 2016 Tour in support of Toto XIV consisting of European and Japanese dates. Leland Sklar, who joined them on their 2007 and 2008 tours, replaced founding member Hungate.[49] Sklar left Toto in early 2017 and was replaced by Shem von Schroeck.

text: Toto released their anniversary album "40 Trips Around The Sun" on February 9, 2018. They'll embark on their world tour promoting their album and celebrating 40 years of music.

text: Band members[edit]

text: Current members[edit]

text:
Steve Lukather – guitars, backing and lead vocals, occasional bass, occasional keyboards (1976–2008, 2010–present)
David Paich – keyboards, backing and lead vocals (1976–2008, 2010–present)
Steve Porcaro – keyboards, backing and lead vocals (1976–1987, 1998, 2010–present)
Joseph Williams – lead and backing vocals, occasional keyboards (1986–1989, 1998, 2010–present)

text: Timeline[edit]

text: Discography[edit]

text: Studio albums[edit]

text:
Toto (1978)
Hydra (1979)
Turn Back (1981)
Toto IV (1982)
Isolation (1984)
Dune (1984)
Fahrenheit (1986)
The Seventh One (1988)
Kingdom of Desire (1992)
Tambu (1995)
Mindfields (1999)
Through the Looking Glass (2002)
Falling in Between (2006)
Toto XIV (2015)

text: Awards and nominations[edit]

text:
Year
Nominee/work
Award
Result

1979
Toto
Best New Artist
Nominated

1983
Producer of the Year
Won

"Rosanna"
Record of the Year
Won

Song of the Year
Nominated

Best Vocal Arrangement for Two or More Voices
Won

Best Instrumental Arrangement Accompanying Vocal(s)
Won

Toto IV
Album of the Year
Won

Best Engineered Recording, Non-Classical
Won

1997
Tambu
Nominated

2000
Mindfields
Nominated

text: Tours[edit]

text:
1979 – Toto Tour
1980 – Hydra Tour
1982 – Toto IV Tour
1985–1986 – Isolation Tour
1986–1987 – Fahrenheit Tour
1988 – The Seventh One Tour
1990 – Planet Earth Tour (Past to Present Tour)
1991 – Summer Tour
1992–1993 – Kingdom of Desire Tour
1993 – Summer Tour
1995–1996 – Tambu Tour
1997 – South African Tour
1998 – Toto XX tour
1999–2000 – Mindfields Tour
2001–2002 – Summer Tour
2002–2004 – 25th Anniversary Tour (Through the Looking Glass Tour)
2004–2005 – Summer Tour
2006–2008 – Falling in Between Tour
2010 – Mike Porcaro Honor Tour
2011 – In the Blink of an Eye Tour
2012 – Summer Tour
2013–2014 – 35th Anniversary Tour
2015–2016 – Toto XIV Tour (Yes & Toto Co-Headlining North American Summer Tour 2015) [50][51]
2017 - An Evening With Toto Tour
2018 – 40 Trips Around the Sun Tour (40th Anniversary tour) [52]