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 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

0^th iteration

• Successor: 

  S(n)

  .
• Predecessor: 

  P(n)

  .

1^st 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.

2^nd iteration

• Multiplication: 

  n + (n + (⋯ "k times" ⋯ (n + (n))))

  , the product 

  m  ⋅   k

  , where 

  m

  is the multiplicand and 

  k

  is the multiplier. (When multiplication is commutative both are simply called factors.)
• Division: the ratio 

  n  /  d

  , where 

  n

  is the dividend and 

  d

  is the divisor.
  • Quotient: (integer division).
  • Remainder: (modulo and congruences).

3^rd iteration

• Exponentiation ( 

  d

  as "degree", 

  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 ( 

  d

  as "degree", 

  b

  as "base", 

  n

  as "variable").
  • Roots: 

    d √  n

    .
  • Logarithms: 

    logb n

    .
    • Natural logarithm function: 

      log n

      , or 

      loge n

      , where 

      e

      is Euler's number.

4^th iteration

• Tetration ( 

  d

  as "degree", 

  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 ( 

  d

  as "degree", 

  b

  as "base", 

  n

  as "variable").
  • Tetra-roots (super-roots)
  • Tetra-logarithms (super-logarithms): 

    slogb n

    .
    • Iterated logarithm: 

      log ⁎b n = ⌈slogb n⌉

      .

5^th iteration

• Pentation ( 

  d

  as "degree", 

  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
  • Penta-roots
  • Penta-logarithms

6^th iteration

• Hexation ( 

  d

  as "degree", 

  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
  • Hexa-roots
  • Hexa-logarithms

7^th iteration

• Heptation ( 

  d

  as "degree", 

  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
  • Hepta-roots
  • Hepta-logarithms

8^th 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
  • Octa-roots
  • Octa-logarithms

Notes

Operator precedence
Parenthesization — Factorial — Exponentiation — Multiplication and division — Addition and subtraction