This site is supported by donations to The OEIS Foundation.

Exponentiation

From OeisWiki
(Redirected from B↑n)
Jump to: navigation, search

Exponentiation with an integer exponent is repetitive multiplication (a 3rd iteration "hyper-addition"): a given number (called the base) is repeatedly multiplied by itself a number of times (called the exponent); this is usually notated and read " exponent ." For example, .

Exponentiation

Exponentiation operator

In most computer programming languages, and in TeX source, the caret character ^ is used as the exponentiation operator (e.g. b^d,) although sometimes two asterisk characters ** are used as the exponentiation operator (e.g. b**d,) implying a 2nd iteration "hyper-multiplication."

You may also use Knuth's up-arrow notation to represent exponentiation.

Exponentiation table

The columns of the table, with fixed exponent , are powers . The rows of the table, with fixed base , are exponentials . The diagonal of the table (entries in bold) are

Exponentiation table
\ 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 [1] 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
3 1 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441
4 1 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216
5 1 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 244140625
6 1 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336
7 1 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 1977326743 13841287201
8 1 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736
9 1 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481
10 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 100000000000 1000000000000
11 1 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937424601 285311670611 3138428376721
12 1 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256

Base and exponent

Base

Cf. exponentials, exponentiation and fixed integer base positional numeral systems and logarithms.

Exponent

When is positive and is negative, the exponentiations are the reciprocals of the exponentiations of with exponent .

For example, the exponentials (base 2) give .

We have the following rules

  • with is .
  • for any real, imaginary or complex (including if is interpreted as the empty product, e.g. 1.)

0^0

If is interpreted as the empty product, which equals the multiplicative identity, i.e. 1 for numbers, this should be the result for any , including 0.

In algebra, for the binomial expansion

we need the conventions

for the constant term to be 1 for any value of , including .

In regards to , see 0^0 or the special case of zero to the zeroeth power.

Powers

When the exponent is fixed, the exponentiation operations are considered powers (n^d or n**d)

Table of powers

A sequence of integers is called "the powers to the degree ." Some sequences of powers in the OEIS are given in the following table

Table of powers
sequences A-number
0[1] {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012
1 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, ...} A000027
2 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, ...} A000290
3 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, ...} A000578
4 {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, ...} A000583
5 {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, ...} A000584
6 {0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, ...} A001014
7 {0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, ...} A001015
8 {0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, ...} A001016
9 {0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, ...} A001017
10 {0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, ...} A008454
11 {0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, ...} A008455
12 {0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, ...} A008456

Powers as figurate numbers

Powers may be considered as -dimensional regular orthotopic numbers.

Regular orthotopic numbers
-dimensional orthotopic numbers
0 Point numbers
1 Segment numbers (Cf. triangular gnomonic numbers)
2 Square numbers
3 Cube numbers
4 Tesseract numbers
5 Penteract numbers
6 Hexeract numbers
7 Hepteract numbers
8 Octeract numbers
9 Enneract numbers
10 Dekeract numbers
11 Hendekeract numbers
12 Dodekeract numbers

Formulae

Cf. Formulae for regular orthotopic numbers.

Recurrence relation for powers

Cf. Recurrence relation for regular orthotopic numbers.

Generating function for powers

Cf. Generating function for regular orthotopic numbers.

Order of basis of powers

Cf. Order of basis of regular orthotopic numbers.

Differences of powers

Cf. Differences of regular orthotopic numbers.

Partial sums of powers

Cf. Partial sums of regular orthotopic numbers.

Partial sums of reciprocals of powers

Cf. Partial sums of reciprocals of regular orthotopic numbers.

Sum of reciprocals of powers

Cf. Sum of reciprocals of regular orthotopic numbers.

Exponentials

When the base is fixed, the exponentiation operations are considered exponentials (b^n or b**n)

Table of exponentials

A sequence of integers is called "the exponentials base ." Some sequences of exponentials in the OEIS are given in the following table

Table of exponentials
sequences A-number
0[1] {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...} A000007
1 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012
2 {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, ...} A000079
3 {1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, ...} A000244
4 {1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, ...} A000302
5 {1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, ...} A000351
6 {1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...} A000400
7 {1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, ...} A000420
8 {1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...} A001018
9 {1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...} A001019
10 {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...} A011557
11 {1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...} A001020
12 {1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...} A001021

Exponentials as figurate numbers

The exponentials may be interpreted as the regular orthotopic numbers read cross-dimensionally, although there is a disagreement about 0^0,[1] between the figurate number interpretation (which has to be 0 for ) and the exponentiation interpretation (which is 1.)

Exponentials as a sum of multinomial coefficients

For any positive integer and any nonnegative integer , the multinomial formula tells us how a polynomial expands when raised to an arbitrary power

where

are the multinomial coefficients.[2]

Letting all the equal 1, we get

Thus:

Recurrence relation for exponentials

Generating function for exponentials

Since , the generating function of 1 is then[3] [4]

Substituting for , we get

which is thus the generating function for exponentials.

Setting gives

which generates the desired sequence for

{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}

Order of basis of exponentials

Any can be uniquely represented, i.e. a representation exists and it is unique, as a sum of powers of a base , i.e.

where the are the digits (i.e. multipliers, or repeated additions, of powers of ) of the base representation.

This is a consequence of the fact that

or equivalently

which says that is the successor of , where all the digits reached their maximal allowed values.

The number of powers to add reaches local maxima when is of the form , i.e. we need to add powers of .

The order of basis of powers is thus infinite, since to represent any we need to add powers of .

Differences of exponentials

Partial sums of exponentials

Partial sums of reciprocals of exponentials

Sum of reciprocals of exponentials

N^n

When the base is equal to the exponent we get n^n (or n**n), i.e.

using Knuth's up-arrow notation.

A000312 n^n: number of labeled mappings from points to themselves (endofunctions), . (For we get 1 mapping, the empty mapping.)

{1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, ...}

For example, with

0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4

one labeled mapping is (0, 0, 2, 3, 0), among of them.

Exponentiation inverses

There are two distinct exponentiation inverses, root extraction and logarithm.

Root extractions

The th root of is . Root extraction is exponentiation with multiplicative inverse of second term (the exponent, which is the multiplicative inverse of the root index)

Logarithms

The logarithm base of is

N//n

When the base is equal to the root index we get n//n (inverse operation of n**n,) i.e. n^(1/n)

Iterated exponentiation

Iterated exponentiation could be abbreviated by the use of a power tower operator (tentatively denoted with the capital letter epsilon of the Greek alphabet,) i.e.

where it should be noted that the power tower is to be evaluated top down.

Tetration

The particular case (using Knuth's up-arrow notation)

where for we get the empty tower (actually the empty product, giving the multiplicative identity, i.e. 1,) is called tetration.

It has been attempted to generalize tetration to heights other than nonnegative integers (up to complex numbers.) Some aspects of its formal approach as well as some constants have relations into the integer-sequence-space.

As is the case with exponentiation, we may distinguish between tetra-powers (where the tower height is fixed)

and tetra-exponentials (where the tower base is fixed)

Exponentiation identities

Power identity

The power identity is 1, since for all .

Exponential identity

There is no such thing as an exponential identity, since there is no base such that , for all .

Exponentiation and fixed integer base positional numeral systems

The concept of exponentiation is crucial to our modern place-value systems of numeration; indeed it is the combination of exponentiation (with fixed integer base ) and addition that represents the advantage of the binary numeral system and the decimal numeral system over non place-value systems of numeration such as Greek numerals, Roman numerals, etc. When in decimal we say "1729," we're in fact saying . Since the exponents for the base can get arbitrarily large simply by adding more places, there is no need to invent more than symbols as needs to be done with ancient additive systems.

See also

Hierarchical list of operations pertaining to numbers [5] [6]

0th iteration
1st iteration
  • Addition:
    S(S( "a times" (S(n))))
    , the sum
    n  +  a
    , where
    n
    is the augend and
    a
    is the addend. (When addition is commutative both are simply called terms.)
  • Subtraction:
    P(P( "s times" (P(n))))
    , the difference
    n  −  s
    , where
    n
    is the minuend and
    s
    is the subtrahend.
2nd iteration
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
8th iteration
  • Octation (
    d
    as "degree",
    b
    as "base",
    n
    as "variable").
    • Octa-powers:
      n ^^^^^ (n ^^^^^ ( "d times" (n ^^^^^ (n))))
      , written
      n ^^^^^^ d or n ↑↑↑↑↑↑ d
      .
    • Octa-exponentials:
      b ^^^^^ (b ^^^^^ ( "n times" (b ^^^^^ (b))))
      , written
      b ^^^^^^ n or b ↑↑↑↑↑↑ n
      .
  • Octation inverses

Notes

  1. 1.0 1.1 1.2 1.3 Cf. 0^0 or The special case of zero to the zeroeth power.
  2. Weisstein, Eric W., Multinomial Coefficient, From MathWorld--A Wolfram Web Resource.
  3. Since the power series associated with generating functions are only formal, i.e. used as placeholders for the as coefficients of , we need not worry about convergence (as long as it converges for some range of , whatever that range.)
  4. Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.
  5. HyperoperationWikipedia.org.
  6. Grzegorczyk hierarchyWikipedia.org.
  7. There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:
    ω  ×  2 := ω  +  ω
    .
Operator precedence

Formula Operator Precedence Demo.png

Parenthesization — FactorialExponentiationMultiplication and divisionAddition and subtraction


Notes