Exponentiation

Exponentiation (from Latin potentia, power, power ', as a loan translation from gr δύναμις, Dynamis, which in the ancient geometry since Plato at the latest, the importance, square ' had ) is like multiplying its origin an abbreviation for a repeated mathematical arithmetic operation. As a summand is repeatedly added in multiplication, so potentising a factor is multiplied repeatedly.

5.1 Analysis
5.2 Set Theory

7.1 More general bases
7.2 Generalized exponents

8.1 multiplication
8.2 concatenation
8.3 derivation

Definition

Natural exponent

The potency is defined for real or complex numbers and natural numbers by

We speak these arithmetic operation as a power of n, a to the nth power or a short to the nth. In the case is also common a ( to) the square and in the case a ( to ) cubic.

Ie base (or radix ), is called the exponent (or exponent ) of the power. The result is the value of the power.

The above definition applies only to order that the identity also applies to is determined.

This definition can be applied not only to real or complex numbers, but also to arbitrary multiplicative monoids.

If superscript letter is not possible ( for example in an ASCII text), one often uses the notation a ^ b ( for example in Algol 60, in TeX source or computer algebra systems such as Maple ), and occasionally a ** b ( for example, in Fortran, Perl or Python).

Orders of magnitude are often presented in the electronic data processing with e or e. Example: 1.55 e 5: = 1.55 · 105 = 155,000th

The following modification facilitates the treatment of the special case:

The power of writing means " multiply the number 1 to the base number as frequently as the exponent indicates " so

The exponent of 0 indicates that the number 1 is not once multiplied by the base number and stand alone remains, so that one obtains the result is 1.

With a negative base and even-numbered exponents potency is positive,

With a negative base and odd exponents potency is negative,

Whole negative exponents

Negative exponents mean that you should perform the inverse operation for multiplication ( division). So " divide the number 1 by the base number as many times as the exponent indicates ."

Thus, for a real number and a natural number is defined

The analogous definition is also used in a more general context, whenever a multiplication and inverse elements are available, such as invertible matrices.

Rational exponents

Let be a rational number with the fraction representation with. For any positive real one defines

The value of the power does not depend on what fraction representation has been chosen.

The same definition applies to. It follows that for and that does not exist when is.

If you allow roots of negative numbers with odd exponents of the root, then we can extend this definition to negative bases and such rational exponents, whose abridged fraction models have odd denominators. These include powers with bases and all negative exponent because the denominator in this case are the same.

In the event you can use odd in calculations of all fraction models. But when using fraction models with straight errors can occur. For example, the following applies:

Real exponents

And any real number, and is a sequence of rational numbers that converges to, so we define:

This definition is correct, i.e., the limit value always exists, and does not depend on the selection of the sequence.

For example, is equal to the limit of the sequence

The definition can not be extended to the case, since in this case the limit does not need to exist and for different choices of the sequence results in various limits.

Another definition is possible of the natural exponential function and the natural logarithm:

For this, the exponential function can be defined by its series expansion:

Overall, the potencies with nonnegative bases for all real exponents are thus defined. Unlike them, the powers with negative bases are defined only for rational exponents, whose abridged fraction models have odd denominators. All powers with negative bases and all exponents belong.

Power laws

In order not to overload the table below, we consider only powers with real bases that are not equal. But if we consider one of the laws listed below with only positive exponents, then it is also valid for powers to base. When we speak of rational numbers with even or odd denominators, then the denominator of their reduced fraction representations are always meant.

For any rational with odd denominators, if is.

For any real case are; for any rational odd denominators, if at least one of the numbers is negative.

If at least one of the exponents are both irrational or rational, but has at least one of the numbers or a straight denominator, then one of the terms or undefined. Otherwise, both are defined and either agree or differ only by their sign. For any case, and for all if is, they always coincide. For not a whole, but rational, these two cases are possible. Which occurs, depends on the number of two in the prime factorization of the counter and of the denominator of from. In order to detect the correct sign on the right side of the formula, it is sufficient to use in this formula. The sign with which it is in valid then remains true for all and a given. Applies to, then for all (and for, if all exponents are positive).

For example applies and. That is why all real is valid for all and thus for.

The exponentiation is not commutative, for example, applies, by association, for example, applies.

The notation without brackets means exponentiation is therefore right-associative, cf operator precedence.

Powers of complex numbers

For integer exponents can be defined powers with complex bases, such as in the real case. For any real or complex exponents one must, however, take a different approach.

The first step for the definition of complex magnitude and the exponent is the base steady continuation of the function to the set of complex numbers. There are different possibilities. For example, you can set the

Use that converges for all and specifies the function for all. Using operations with rows proves one after that

For any and Euler's formula

Hold for arbitrary. It follows the formula

Which can be used for the definition of. This formula shows that the set of values of the same, and that the function is periodic with period.

Therefore its inverse function is ambiguous and defined for all. It can be specified by using the formula, where the amount that the set of values of the argument of and the usual real logarithm. The main value of this function is obtained if one uses the principal value instead. For real, according to the usual definition, so this function coincides on the set with the usual real logarithm.

For any with you then define:

This is also an ambiguous function whose main value is in the use of instead.

But for this ambiguity disappears and there are the usual powers with integral exponent, defined in the first section. Let and, then pulls the exponential representation

After that

Applies.

For a rational exponents with the reduced fraction representation, with the potency has exactly different values. This particularly applies to. Is odd and, then there is exactly one real number of them, and that's just the number from section 1.3. Is straight and then accepts any real values . If, however, are straight and then takes the power exactly two real values , which have different signs. The positive of this is in this case just equal to the number of the section 1.3.

As an example, we consider high potency.

Off and

Follows

It follows

The main value is equal to and is equal to

Special powers

Smooth powers of 10 are the basis of our number system, the decimal system. Written as a power, such as 10-9 for 0.000000001 or 1011 for 100 billion, they are used in the natural sciences for representing very large or very small numbers.

For the mathematics are particularly important powers to the base, the Euler number called.

Powers of two are obtained by repeated doubling, one considered individually plausible process. The still surprising rapid growth of the numbers makes them so popular for practical examples:

A sheet of paper can only be about seven fold to half the size. It then has 128 layers. If you (theoretically) could fold it 42 times, its thickness would correspond to the distance from the Earth to the Moon (about 384,000 km ).
Every person has two biological parents, and most have four grandparents and eight great-grandparents. Without loss of ancestors that would be before 70 generations at the time of Christ's birth, ancestors, although at that time less than 109 people lived.
The Legend of the inventor of the chess game, which doubled the number of grains of wheat on each square of the chessboard: WeizenkornLegende.

Digital processing of data at the computer, the dual system is used with the base 2. The size of memory units of digital systems are, therefore, the power of two, that is, the powers of the base 2 (that is, 1, 2, 4, 8, 16, ...). A kibibyte (abbreviated KiB) corresponds to bytes.

In pyramid schemes, for example, so-called Schenk circuits, systems are started for the part that provide not only a doubling, but for example a eight-fold increase of new members per step. Such consequences are growing at such a fast that systems inevitably collapse after a few steps.

High zero zero

Analysis

It has historically formed that one uses the symbol in mathematics in two completely different meanings: as the name for a type of indefinite expressions, and as the record of the potency, the base and exponent are the same.

In the first case it is nonsense, this symbol ascribe a numerical value. In the second case, the establishment of a value of powers is not a matter of true or false, but of appropriate or inappropriate. As a priori suitable values can, for example, either (because for any applies ) or ( because for any applies ) look. In today's Analysis textbooks and the Convention is common to have the potency undefined.

Can not be calculated directly on the basis of limit theorems and properties of continuous functions, a threshold, ie, the expression under the sign of the limit, an indefinite expression. These are, for example, etc. The indefinite expression results from calculations of the limits of the powers, whose bases and exponents go simultaneously against. The reason is that for any number (and also ) such sequences exist that, and apply. So the threshold arguments for fixing the value of power are unsuitable.

By the beginning of the 19th century, mathematicians have apparently set without question this definition of accurate. Augustin- Louis Cauchy, however, listed together with other terms such as in a table of indefinite expressions. 1833 Guillaume Libri published a paper in which he. Unconvincing arguments presented which were hotly debated in the sequence In defense of Libri published August Ferdinand Möbius proof of his teacher Johann Friedrich Pfaff, who showed, essentially, that is true, and an alleged proof, if applicable, delivered. This evidence was refuted by the counter-example and quickly.

Donald E. Knuth mentioned in 1992 in the American Mathematical Monthly, the history of the controversy and refused to conclude decisively that is left undefined. If one does not assume the value 1 for potency, require many mathematical theorems, such as the binomial theorem

Special treatment for the cases or or simultaneously.

Similarly, the potency immersed in power series, such as for the exponential function

Or in the sum of geometrical series formula for the

On. Again, the convention is useful.

Set theory

In set theory is a power of two sets is defined as the set of all functions from to, ie as a set of sets of ordered pairs, so that there is exactly one with each. If we denote by the cardinality of, then ( at least for finite sets, but also beyond), which justifies the power notation for quantities. Now there is exactly one defined on the empty set function, that is a set of pairs with the above property, namely. Therefore, on what remains to be right.

The natural numbers are defined recursively as follows in set theory (see von Neumann's model of the natural numbers ):

Accordingly applies in set theory

Inverse functions

Since the commutative law does not apply potentising, there are two reverse arithmetic:

The square root to resolve equations of the type according to, ie to determine the base when the exponent is known,
The logarithms of equations of the type, that is the determination of the exponent, if the base is fixed.

Generalizations

More general bases

Generally, there are powers with positive integer exponents in each half group. Did that a neutral element and becomes the monoid, as well exponent 0 makes sense, is then always the neutral element. Apply the power laws

If and exchange, that is, if applies.

(Everywhere. )

Is an invertible element, we can by means of

Define powers with any integer exponents. The calculation rules apply here. In the case of abelian groups, they suggest that the structure of a module is induced by the potentiation.

Generalized exponents

General exponent such matrices are mostly associated with the base, that is regarded as the values of the generalized exponential function.

In addition, the power notation is occasionally used for other natural continuations. So powers of elements of ( topological ) Galois groups with exponents in completions of example, consider the algebraic number theory occasionally; then it is each uniquely defined continuous extension of the mapping

For any cardinal numbers and can be defined by the power, the set of all functions denoted by ancient number set and image set, this generalization requires the power set axiom, whereby the handling of the cardinal numbers as a rule, the axiom of choice is assumed.

Ambiguity of the exponent notation for functions

A reminiscent of magnitude notation exists for functions. However, this notation can have different meanings. In general, it is clear from the context, which is precisely meant by the two meanings.

Multiplication

As a shorthand notation for the multiplication of several function values of trigonometric functions with the same arguments, as they often occur for example in the addition theorems for trigonometric functions, the following notation has naturalized:

General, however, does not apply:

Concatenation

On the other hand, the potency notation is often used as a shorthand notation for the concatenation of functions whose values are again in the domain.

Definition (id denotes the identity on the domain ):

For the function values , this means

As an extension of this definition is usually not defined as the inverse function of. In particular, this spelling is also found in many pocket calculators, eg is there and also otherwise called the Arkusfunktion with.

Derivation

If the exponent is written in brackets, as is usually the corresponding derivation meant, then denotes the -th derivative of f

In programming languages

The notation with superscript exponents xy is practically in handwritten formulas, but impractical for typewriters and terminals, in which the characters of a line are all on one level. Therefore, many programming languages use alternative ways to represent a power:

X ↑ y: ALGOL, Commodore BASIC
X ^ y: BASIC, J, MATLAB, R, Microsoft Excel, TeX (and its offshoots ), TI -Basic, bc ( for integer exponents ), Haskell ( for non-negative integer exponents ), Lua, ASP and most computer algebra systems
X ^ ^ y: Haskell ( for integer exponents and rational basis ), D
X ** y: Ada, Bash, COBOL, Fortran, FoxPro, Gnuplot, OCaml, Perl, PL / I, Python, REXX, Ruby, SAS, Seed7, Tcl, ABAP, Haskell ( for floating-point exponents ), Turing, VHDL
X ⋆ y: APL

In many programming languages , there is instead a power operator a corresponding library function.