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Exponentiation
From OeisWiki
Exponentiation with an integer exponent is repetitive multiplication (a 3^{rd} 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 2^{nd} 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
\ | 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
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.
-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
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 n = 5
- 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]}
0^{th} iteration
- Successor S(n)
- Predecessor P(n)
1^{st} iteration
- Addition, S(S(... s times ...(S(n)))), the sum n+s
- Subtraction, P(P(... s times ...(P(n)))), the difference n-s
2^{nd} iteration
- Multiplication, n+(n+(... m times ...(n+(n)))), the product n∗m
- Division, the ratio n/d
3^{rd} iteration
- Exponentiation (d as "dimension", b as "base", n as "variable")
- Powers, n∗(n∗(... d times ...(n∗(n)))), written n^d
- Exponentials, b∗(b∗(... n times ...(b∗(b)))), written b^n
- Exponential function e^n, where e is Euler's number
- Exponentiation inverses
- Roots,
- Logarithms, log_{b}n
- Natural logarithm function log n, or log_{e}, where e is Euler's number
4^{th} iteration
- Tetration (d as "dimension", b as "base", n as "variable")
- Tetra-powers (super-powers), n^(n^(... d times ...(n^(n)))), written n^^d or n↑↑d
- Tetra-exponentials (super-exponentials), b^(b^(... n times ...(b^(b)))), written b^^n or b↑↑n
- Tetration inverses
5^{th} iteration
- Pentation (d as "dimension", b as "base", n as "variable")
- Penta-powers, n^^(n^^(... d times ...(n^^(n^^(n))))), written n^^^d or n↑↑↑d
- Penta-exponentials, b^^(b^^(... n times ...(b^^(b^^(b))))), written b^^^n or b↑↑↑n
- Pentation inverses
6^{th} iteration
- Hexation (d as "dimension", b as "base", n as "variable")
- Hexa-powers, n^^^(n^^^(... d times ...(n^^^(n)))), written n^^^^d or n↑↑↑↑d
- Hexa-exponentials, b^^^(b^^^(... n times ...(b^^^(b)))), written b^^^^n or b↑↑↑↑n
- Hexation inverses
7^{th} iteration
- Heptation (d as "dimension", b as "base", n as "variable")
- Hepta-powers, n^^^^(n^^^^(... d times ...(n^^^^(n)))), written n^^^^^d or n↑↑↑↑↑d
- Hepta-exponentials, b^^^^(b^^^^(... n times ...(b^^^^(b)))), written b^^^^^n or b↑↑↑↑↑n
- Heptation inverses
8^{th} iteration
- Octation (d as "dimension", 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
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Cf. 0^0 or The special case of zero to the zeroeth power.
- ↑ Weisstein, Eric W., Multinomial Coefficient, From MathWorld--A Wolfram Web Resource.
- ↑ 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.)
- ↑ Herbert S. Wilf, generatingfunctionology, 2^{nd} ed., 1994.
- ↑ Hyperoperation—Wikipedia.org.
- ↑ Grzegorczyk hierarchy—Wikipedia.org.
Operator precedence |
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Parenthesization — Factorial — Exponentiation — Multiplication and division — Addition and subtraction |