Exponentiation

From Free net encyclopedia

In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition.

1. multiplication

   <math>{{a \times n = } \atop {\ }} {{\underbrace{a + \cdots + a}} \atop n}</math>

2. exponentiation

   <math>{{a^n = } \atop {\ }} {{\underbrace{a \times \cdots \times a}} \atop n}</math>

Exponentiation involves two numbers, the base and the exponent. The exponent is normally written as a superscript to the right of the base, and can be any real or complex number, as explained below. Exponentiation used to be called "involution". One inverse of exponentiation is the logarithm; exponentiation is sometimes called the antilogarithm, i.e. the inverse of z = x^y is that y = the logarithm of z base x. The other inverse of exponentiation is extracting a root, i.e. the inverse of z = x^{y</sub> is that x = the y-th root of z. Extracting a root used to be called "evolution".}

The next generalized operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function. Image:Exponentials.png

Exponentiation is a basic mathematical operation that is used pervasively in other fields as well, including physics, chemistry, biology, computer science and economics, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

Contents 1 Exponentiation with integer exponents 1.1 Positive integer exponents 1.2 Exponents one and zero 1.3 Negative integer exponents 1.4 Identities and properties 1.5 Powers of ten 1.6 Powers of two 1.7 Powers of zero 1.8 Powers of e 2 Exponentiation with real exponents 2.1 Fractional exponents method 2.2 Logarithm method 3 Exponentiation with complex exponents 3.1 Complex powers of positive real numbers 3.2 Complex powers of complex numbers 3.3 Real powers of unity 4 Solving polynomial equations 5 Advanced topics 5.1 Exponents on function names 5.2 Exponentiation in abstract algebra 5.3 Exponentiation over sets 6 Syntax in computer languages and applications 7 Table of powers 8 See also 9 External links

Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires basic algebra.

Positive integer exponents

The simplest case involves a positive integer exponent. The exponent then says how many times the base is to be multiplied. For example, 3⁵ = 3 × 3 × 3 × 3 × 3 = 243. Here, 3 is the base, 5 is the exponent, and 243 is 3 raised to the 5th power or 3 raised to the power 5. (The word "raised" is usually omitted, and most often "power" as well, so 3⁵ is typically pronounced "three to the fifth" or "three to the five".) Notice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5.

Traditionally a² = a×a is called the square and a³ = a×a×a is called the cube. 3² is pronounced "three squared," and 3³ is "three cubed."

The work required for computing, say, 2¹³, can be reduced like this.

2¹³ = 2×2¹² = 2×(2²)⁶ = 2×4⁶ = 2×(4²)³ = 2×16³ = 2×16×(16²)¹ = 2×16×256 = 8192.

The number of multiplications performed was 5, rather than 12. See exponentiation by squaring.

Formally, powers with positive integer exponents are defined by the initial condition a¹=a and the recurrence relation aⁿ⁺¹=a·aⁿ .

Exponents one and zero

The meaning of 3⁵ may also be viewed as 1 × 3 × 3 × 3 × 3 × 3: the starting value 1 (the identity element of multiplication) is multiplied by the base as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to exponents one and zero:

• any number to the power 1 is itself.

a¹ = a

• any number to the 0 power is 1.

a⁰ = 1

Negative integer exponents

Raising a nonzero number to the -1 power produces its reciprocal.

a⁻¹ = 1/a

Thus:

a⁻ⁿ = (aⁿ)⁻¹ = 1/aⁿ

Raising 0 to a negative power would imply division by 0, and so is undefined.

A negative integer exponent can also be seen as repeated division by the base. Thus 3⁻⁵ = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243 = 1/3⁵.

Identities and properties

The most important identity satisfied by exponentiation is:

<math> a^{m + n} = a^m \cdot a^n </math>

It has the following consequences:

<math> a^{m - n} = \begin{matrix}\frac{a^m}{a^n}\end{matrix} </math>

<math> (a^m)^n = a^{mn} \!\, </math>

Whereas addition or multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2×3 = 6 = 3×2), exponentiation is not commutative: 2³ = 8 while 3² = 9. Similarly, whereas addition or multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2×3)×4 = 24 = 2×(3×4)), exponentiation is not associative either: 2³ to the 4th power is 8⁴ or 4096, while 2 to the 3⁴ power is 2⁸¹ or 2417851639229258349412352.

Powers of ten

Powers of 10 are easy to compute because we use a base ten number system: for example 10⁶ = 1 million, which is 1 followed by 6 zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458 × 10⁸ and then approximated as 2.998 × 10⁸ if this is useful. SI 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 two

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. They occur so commonly that SI prefixes are commonly reinterpreted to refer to them: 1 kilobyte = 2¹⁰ = 1024 bytes. As the standard meanings of the prefixes also occur, confusion may result, and in 1998 the International Electrotechnical Commission approved a set of binary prefixes. For instance, the prefix for multiples of 1024 is kibi-, so 1024 bytes is 1 kibibyte. Other prefixes are mebi-, gibi-, and tebi-.

Powers of zero

If the exponent is positive, the power of zero is zero: 0^x = 0, where x > 0.

If the exponent is negative, the power of zero (0^−x, where x > 0) is undefined, because division by zero is implied .

If the exponent is zero, the power is defined as one. 0⁰ = 1. See Empty Product.

Powers of e

The number e is defined as the limit of a sequence of integer powers

<math>\ e=\lim_{n \rightarrow \infty} \left(1+\frac{1}{n} \right) ^n</math>.

An integer power of e is

<math>e^x

= \left( \lim_{m \rightarrow \infty} \left(1+\frac{1}{m} \right) ^m\right) ^x = \lim_{m \rightarrow \infty} \left(1+\frac{x}{mx} \right) ^{mx} = \lim_{n \rightarrow \infty} \left(1+\frac{x}{n} \right) ^n </math> .

The right hand side generalizes the meaning of e^x so that x does not have to be an integer but can be a fraction, a real number, a complex number, or a square matrix. See exponential function.

Exponentiation with real exponents

The idea of raising a number to a power that is not an integer can also be explained in other ways:

• Defining fractional exponents in terms of (n^th roots). See radicals. This method is perhaps the way most widely taught in schools.
• Defining the natural logarithm as the area under the curve 1/x.

The identities and properties shown above are true for non-integer exponents as well.

Fractional exponents method

Image:Root graphs.png Exponentiation with a simple fractional exponent 1/n, with integer n > 2, can be defined as taking n^th roots.

<math> a^{\frac{1}{n}} =\sqrt[n]{a}</math>

For example: 8^1/3 = 2.

Taking n^th roots means solving the equation

<math>\ x^n = a.</math>

For <math>a \ne 0</math> this equation has <math>n</math> solutions. If <math>\ a</math> is a positive real number, then one of the solutions is also a positive real number, called the radical. This is the value given to <math> \sqrt[n]{a}</math>. (If <math>a</math> is a complex number which is not a positive real number, then <math>\sqrt[n]{a}</math> is a multivalued function of <math>a</math>.)

Exponentiation with a rational exponent <math>m/n</math> can now be defined as

<math>a^{\frac{m}{n}} = \left( \sqrt[n]{a}\right)^m</math>

For example: 8^2/3 = 4.

Since any real number can be approximated by rational numbers, exponentiation to an arbitrary real exponent can be defined by continuity.

For example, since <math>\sqrt[2]{3} \approx 1.732 </math>, we can assume <math>5^{\sqrt[2]{3}} \approx 5^{1.732} </math>

Logarithm method

Define the natural logarithm of x as the area under the curve 1/t between from t = 1 to t = x. (The area is counted as negative if x < 1.) In terms of integral calculus:

<math>\ln x = \log_e x = \int_1^x \frac {1}{t} dt </math>

The exponential function e^x is then defined as the inverse function to the natural logarithm. One must prove the exponential identities mentioned above. Exponentiation in any base can be expressed as:

<math>\!\, x^y=e^{y \ln x}</math>

Exponentiation with complex exponents

Complex powers of positive real numbers

The definition of e^x also applies when x is a complex number, (that is x = a+ib, where a and b are real numbers, positive, negative, or zero, and i satisfies the equation i² = −1. i is neither positive, nor negative, nor zero).

A complex power of a positive real number, a, can be expressed as a complex power of e :

a^x = e^bx

where b is the real solution to the equation

e^b = a .

For example,

2^1/2 = e^b/2

where

e^b = 2.

The real solution b=0.693147 is the natural logarithm of 2.

2^1/2 = e^{0.693147 / 2} = e^0.34657359 = 1.414213562

This is the square root of two.

Complex powers of complex numbers

A complex power of a complex number can still be expressed as a complex power of e :

a^x = e^bx

where b is any solution to the equation

e^b = a ,

but when it is not possible to single out one of the solutions, then the complex power of a complex number is not uniquely defined, but is multivalued.

Examples

(−1)^1/2 = (e^{i·π + i·2π·n})^1/2 = e^i·π/2·(e^i·π·)ⁿ = i·(−1)ⁿ = i·{+1, −1} = {+i, −i} .

1ⁱ = (e^i·2π·n)ⁱ = (e^−2π)ⁿ = 0.001867443ⁿ for n in Z .

Real powers of unity

If the exponent is an imaginary number, i·x, then e^i·x is a complex number on the unit circle, a direction. (See Euler's formula). The real number x is an angle measured in radians.

The angle 2π radian is a turn: e^i·2π = 1 . So b = i·2π is a solution to the equation e^b = 1 . It is not the only one. The solutions are i·2π·n , where n is an integer.

Choosing n=0 gives the trivial solution : 1^x = 1 .

Choosing n=1 gives the important expression for any real power of one:

1^x = e^i·2π·x

If x is an integer the result is 1. If x is a rational number the result is a root of unity. If x is a real number the result is a direction.

Example: 1^1/4 = e^i·π/2 = i

Choosing n=0,1,2,3 gives the four fourth roots of unity: 1 = e^i·(π/2)·0, i = e^i·(π/2)·1, −1 = e^i·(π/2)·2, −i = e^i·(π/2)·3

Solving polynomial equations

It was once conjectured that the roots of any polynomial could be expressed in terms of exponentiation with fractional exponents. (See Quadratic equation).

That this is not true in general is the assertion of the Abel-Ruffini theorem.

For example, the solutions of the equation x⁵ = x+1 cannot be expressed in terms of fractional exponents.

For solving any equation of the n^th degree, see the Durand-Kerner method.

Advanced topics

Exponents on function names

When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication. Thus f³(x) may mean f(f(f(x))); in 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. That is, sin²x is just a shorthand way to write (sin x)² without using parentheses, whereas sin^-1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)^-1 is normally just written as csc x.

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. In 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. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0.

Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). Then we can define x^-n to be the inverse of xⁿ when n is a natural number. Now xⁿ is defined for any integer n.

Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):

• <math>\ x^{m+n}=x^mx^n </math>
• <math>\ x^{m-n}=x^m/x^n </math>
• <math>\ x^{-n}=1/x^n </math>
• <math>\ x^0=1 </math>
• <math>\ x^1=x </math>
• <math>\ x^{-1}=1/x </math>
• <math>\ (x^m)^n=x^{mn} </math>

Here, 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. However, 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 (so that the magma is an abelian group), then we have some additional laws:

• (x'y)ⁿ = xⁿyⁿ
• (x/y)ⁿ = xⁿ/yⁿ

Notice that in this algebraic context, 0⁰ is always equal to 1. When 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. For 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". Thus, 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. Thus, 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. That is, g^h = h^-1g'h, 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.

Exponentiation over sets

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In 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

<math>\prod_{i \in I} k_{i}</math>

for any index set I and cardinal numbers k_i. By taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^I. In 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. 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

<math>\bigoplus_{i \in I} V_{i},</math>

where each V_i is a vector space. Then 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. We 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. Taking 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. In that case, S^I becomes simply the set of all functions from I to S. This fits in with the exponentiation of cardinal numbers once gain, in the sense that |S^I| = |S|^|I|, where |X| is the cardinality of X. When I=2={0,1}, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X. (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. In 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.

Syntax in computer languages and applications

Most programming languages have a facility for exponentiation, however they have different ways of expressing this:

• x ^ y: Basic, Matlab and many others
• x ** y: Fortran, Perl, Python, Ruby
• Power(x, y): Excel, Pascal
• pow(x, y): C, C++, PHP
• Math.pow(x, y): Java, JavaScript

Note that in C, C++, Java and JavaScript, the symbol ^ represents bitwise XOR.

Table of powers

Table of kⁿ, with k on the left and n at the top.

	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	2	4	8	16	32	64	128	256	512	1,024
3	3	9	27	81	243	729	2,187	6,561	19,683	59,049
4	4	16	64	256	1,024	4,096	16,384	65,536	262,144	1,048,576
5	5	25	125	625	3,125	15,625	78,125	390,625	1,953,125	9,765,625
6	6	36	216	1,296	7,776	46,656	279,936	1,679,616	10,077,696	60,466,176
7	7	49	343	2,401	16,807	117,649	823,543	5,764,801	40,353,607	282,475,249
8	8	64	512	4,096	32,768	262,144	2,097,152	16,777,216	134,217,728	1,073,741,824
9	9	81	729	6,561	59,049	531,441	4,782,969	43,046,721	387,420,489	3,486,784,401
10	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

See also

Template:Commons

• List of exponential topics
• Exponential growth
• Exponential decay
• Exponentiating by squaring
• Logarithm
• Modular exponentiation
• Addition chain exponentiation using an addition chain

External links

• sci.math FAQ: What is 0⁰?
• Template:Planetmath reference
• Larger Table of Powers
• Laws of Exponents with derivation and examples
• 1058 Powers of Twocs:Umocňování

da:Potens (matematik) de:Potenz (Mathematik) fr:Exposant he:חזקה is:Veldi (stærðfræði) it:Potenza (matematica) nl:Machtsverheffen no:Potens (matematikk) ja:冪乗 pt:Exponenciação pl:Potęga zh:冪 es:Potenciación