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Addition

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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 2nd iteration "hyper-addition",) division is multiplication with multiplicative inverse of second term (the divisor,) exponentiation with an integer exponent is repetitive multiplication (a 3rd iteration "hyper-addition",) root extraction is exponentiation with multiplicative inverse of second term, etc.

Addition with an integer addend is repetitive successor operation (a 2 nd iteration "hyper-succession"): a given number is repeatedly incremented a number of times ; this can be notated and read " plus ." For example, . In most computer programming languages, and in TeX source, the plus character + is used as the addition operator: m+n. Addition is commutative. Thus, .

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.

Additive identity

The additive identity is 0.

Additive inverse

The additive inverse (denoted ) of is defined by

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]

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. HyperoperationWikipedia.org.
  2. Grzegorczyk hierarchyWikipedia.org.
  3. 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