Exponentiation

In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. (The next operation after exponentiation is tetration. ) The simplest case involves a positive integer exponent:\nFor example, 3⁵ = 3 × 3 × 3 × 3 × 3 = 243.\nHere, 3 is the base, 5 is the exponent (written as a superscript), and 243 is 3 raised to the 5th power or 3 raised to the power 5.\n(The word "raised" is usually omitted, and most often "power" as well, so 3⁵\nis typically pronounced "three to the fifth" or "three to the five".)\nNotice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5.\nIn contexts where superscripts are not available, such as computer languages and e-mail, 3⁵ is commonly written "3^5", and sometimes as "3**5". The exponent 1 is not normally written, since any number to the power 1 is itself.\nThe exponents 2 and 3 occur so commonly that there are short words for them: the powers are called\nthe square and cube of the base, respectively.\n3² is pronounced "three squared," and 3³ is "three cubed." The meaning of 3⁵ may also be viewed as 1 × 3 × 3 × 3 × 3 × 3:\nthe starting value 1 (the identity element of multiplication)\nis multiplied by the base, as many times as indicated by the exponent.\nWith this definition in mind, it is easy to see how to generalize exponentiation\nto zero and negative exponents: any number to the 0 power is 1, and a negative\nexponent indicates repeated division by the base.\nThus 3^-5 = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243, and raising any nonzero number to the -1\npower produces its reciprocal. Raising 0 to a negative power would imply division by 0, and so is undefined.\n0⁰ is sometimes taken as undefined, but is sensibly defined as 1;\nsee the reference below. Important identities satisfied by exponentiation include:

• x^m+n = x^mxⁿ\n* x^m-n = x^m/xⁿ\n* (x^m)ⁿ = x^mn

Whereas addition or multiplication are commutative (for example,\n2+3 = 5 = 3+2 and\n2×3 = 6 = 3×2), this is not true of exponentiation:\n2³ = 8 while 3² = 9.\nSimilarly, whereas addition or multiplication are associative (for example,\n(2+3)+4 = 9 = 2+(3+4) and\n(2×3)×4 = 24 = 2×(3×4)), this is not true of exponentiation either:\n2³ to the 4th power is 8⁴ or 4,096 ,while 2 to the 3⁴ power is\n2⁸¹ or 2,417,851,639,229,258,349,412,352. Powers of 10 are easy to compute: for example 10⁶ = 1 million, which is 1 followed by 6 zeros.\nExponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 can be written as 2.99792458 × 10⁸ and then approximated as 2.998 × 10⁸ if this is useful.\nSI prefixes are also used to describe small or large quantities, and these are also based on powers of 10; for example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres. Powers of 2 are important in computer science; for example, there are 2ⁿ possible values for a variable that takes n bits to store in memory.\nThey occur so commonly that SI prefixes are commonly reinterpreted to refer to them: 1 kilobyte = 2¹⁰ = 1024 bytes.\nAs the standard meanings of the prefixes also occur, confusion may result, and the International Electrotechnical Commission has declared that the term for 1024 bytes should be kibibyte; but this has seen little acceptance. Exponentiation with a fractional exponent is defined as\n: \nSo for example 8^2/3 = 4, and the 1/2 power of a non-negative number is its square root (positive or negative). Exponentiation to an arbitrary real exponent can then be defined by continuity. The exponential function exp is the same as raising the\ntranscendental number e to the indicated\npower: exp x = e^x. Exponentiation of real numbers, and even complex numbers, can be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define\n: x^y = exp(y ln x). For more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function.\nThat article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations. When the name or symbol of a function is given an integer superscript, as if being raised to a power,\nthis commonly refers to repeated function composition rather than repeated multiplication.\nThus f³(x) may mean f(f(f(x)));\nin particular, f^-1(x) usually denotes f's inverse function. A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function.\nThat is, sin²x is just a shorthand way to write (sin x)² without using parentheses,\nwhereas sin^-1x refers to the inverse function of the sine, also called arcsin x.\nThere is no need for a shorthand of this kind for reciprocal trigonometric\nfunctions since they each have their own name and abbreviation already:\n(sin x)^-1 is normally just written as csc x.\n \n

Table of contents

1 Exponentiation in abstract algebra 2 Exponentiation over sets 3 See also 4 External link

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers. Specifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively.\nIn this very general situation, we can define xⁿ for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications. Now additionally suppose that the operation has an identity element 1.\nThen we can define x⁰ to be equal to 1 for any x.\nNow xⁿ is defined for any natural number n, including 0. Finally, suppose that the operation has inverses.\nThen we can define x^-n to be the inverse of xⁿ when n is a natural number.\nNow xⁿ is defined for any integer n. In particular, xⁿ is defined for any integer n and any element x of a group.\nHowever, because we need only power associativity and not general associativity, the concept of exponentiation also makes sense in some other useful situations, such as the nonzero octonions. Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):\n* x^m+n = x^mxⁿ\n* x^m-n = x^m/xⁿ\n* x^-n = 1/xⁿ = (1/x)ⁿ\n* x⁰ = 1\n* x¹ = x\n* x^-1 = 1/x\n* (x^m)ⁿ = x^mn\nHere, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x^-1 for raising x to the power -1, rather than the inverse of x.\nHowever, as one of the laws above states, x^-1 is always equal to the inverse of x, so the notation doesn't matter in the end. If in addition the multiplication operation is commutative and alternative, then we have some additional laws:\n* (xy)ⁿ = xⁿyⁿ\n* (x/y)ⁿ = xⁿ/yⁿ\nHere, alternativity is a condition stronger than power associativity but weaker than general associativity.\nSo in particular, this law is satisfied in an Abelian group, such as the multiplicative group of elements from a given field that are distinct from zero. Notice that in this algebraic context, 0⁰ is always equal to 1.\nWhen 0⁰ is attained as a limit, however, it may be more useful to leave 0⁰ undefined. However, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it is generally most useful to let 0⁰ be 1, just like every other case of x⁰.\nFor example, if you expand (0 + x)ⁿ using the binomial theorem, you'll want to use 0⁰ = 1. If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition".\nThus, each of the laws of exponentiation above has an analogue among laws of multiplication. When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript.\nThus, x^*n is x * ··· * x, while x^#n is x # ··· # x, whatever the operations * and # might be. Exponential notation is also used, especially in group theory, to indicate conjugation.\nThat is, g^h = h^-1gh, where g and h are elements of some group.\nAlthough conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense.\nA quandle is an algebraic structure in which these laws of conjugation play a central role.

Exponentiation over sets

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation.\nIn more general contexts, one may be able to define an infinitary operation directly on an indexed set. For example, in the arithmetic of cardinal numbers, it makes sense to say\n: \nfor any index set I and cardinal numbers k_i.\nBy taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^I.\nIn fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that k^l is k^I for any set I whose cardinality is l. This can be done even for operations on sets or sets with extra structure.\nFor example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets.\nThat is, we can speak of\n: \nwhere each V_i is a vector space.\nThen if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^(+)I, or simply V^I with the understanding that the direct sum is the default.\nWe can again replace the set I with a cardinal number k to get V^k, although without choosing a specific standard set with cardinality k, this is defined only up to isomorphism.\nTaking V to be the field R of real numbers (thought of as a vector space over itself) and k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rⁿ. If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product.\nIn that case, S^I becomes simply the set of all functionss from I to S.\nThis fits in with the exponentation of cardinal numbers once gain, in the sense that |S^I| = |S|^|I|, where |X| is the cardinality of X.\nWhen I=2={0,1}, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X.\n(This is where the term "power set" comes from.) Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process.\nIn the ordinal numbers, a^b is the smallest ordinal number greater than a^c for c < b when b is a limit ordinal, and of course a^b+1 := a^ba. In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.

See also

Exponentiating by squaring

External link

• sci.math FAQ: What is 0⁰?

\n\n\n\n