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Addition is the act of adding up two quantities. Addition is the core binary operation of arithmetic. All other arithmetic operations can be defined, directly or indirectly, in terms of addition: subtraction is addition with additive inverse of second term (the subtrahend,) multiplication with an integer multiplier is repetitive addition (a 2^{nd} iteration "hyper-addition",) division is multiplication with multiplicative inverse of second term (the divisor,) exponentiation with an integer exponent is repetitive multiplication (a 3^{rd} iteration "hyper-addition",) root extraction is exponentiation with multiplicative inverse of second term, etc.
Addition with an integer addend $\scriptstyle n\,$ is repetitive successor operation (a 2 nd iteration "hyper-succession"): a given number $\scriptstyle m\,$ is repeatedly incremented a number of times $\scriptstyle n\,$; this can be notated $\scriptstyle m+n\,$ and read "$\scriptstyle m\,$ plus $\scriptstyle n\,$." For example, $\scriptstyle 7+4\,=\,(((7^{+})^{+})^{+})^{+}\,=\,11\,$. In most computer programming languages, and in TeX source, the plus character + is used as the addition operator: m+n. Addition is commutative. Thus, $\scriptstyle 4+7\,=\,((((((4^{+})^{+})^{+})^{+})^{+})^{+})^{+}\,=\,11\,$.
Addition table
A003056 n appears n+1 times. Also table T(n,k)=n+k read by antidiagonals.
- {0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, ...}
Iterated addition
Iterated addition, can be abbreviated by the use of the summation operator (denoted with the capital letter sigma of the Greek alphabet), i.e.
- $\sum _{i=1}^{n}a_{i}\equiv a_{1}+a_{2}+a_{3}+\ldots +a_{n}.\,$
Additive identity
The additive identity is 0.
Additive inverse
The additive inverse (denoted $\scriptstyle -n\,$) of $\scriptstyle n\,$ is defined by
- $(-n)+n=(0-n)+n=0\,$
Subtraction is addition with additive inverse of second term (the subtrahend,) which by definition makes it non-commutative.
See also
Hierarchical list of operations pertaining to numbers ^{[1]} ^{[2]}
0^{th} iteration
1^{st} iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2^{nd} iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.^{[3]} (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3^{rd} iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4^{th} iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5^{th} iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6^{th} iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7^{th} iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8^{th} iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes
Notes