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Addition and Sum

Definition

Addition combines two numbers aa and bb to produce the number a+b,a+b\text{,} called their sum. Math I: Calculus takes the usual addition operations on N\mathbb{N}, Z\mathbb{Z}, Q\mathbb{Q}, and R\mathbb{R} as given and uses their standard arithmetic rules.

Interpretation

For natural numbers, addition combines counts: 2+32+3 is the number of objects in a group of 22 and a group of 33 together. On the number line, adding a positive number moves to the right and adding a negative number moves to the left.

The Key Results are the rules behind everyday algebra. Zero leaves a number unchanged, while adding the negation of a number returns to zero. Cancellation justifies removing the same term from both sides of an equation, and compatibility with order justifies adding the same number to both sides of an inequality. By commutativity and associativity, a sum of finitely many numbers has the same value in any order and with any brackets.

Sums of many terms return later in the summation notation of finite series.

Key Results

  • Closure: if aa and bb belong to the same one of N\mathbb{N}, Z\mathbb{Z}, Q\mathbb{Q}, or R,\mathbb{R}\text{,} then a+ba+b belongs to that set.
  • Identity: a+0=0+a=a.a+0=0+a=a\text{.}
  • Commutativity: a+b=b+a.a+b=b+a\text{.}
  • Associativity: (a+b)+c=a+(b+c).(a+b)+c=a+(b+c)\text{.}
  • Cancellation: if a+b=a+c,a+b=a+c\text{,} then b=c.b=c\text{.}
  • Compatibility with order: if a<b,a<b\text{,} then a+c<b+c.a+c<b+c\text{.}

Examples

  • 2+3=52 + 3 = 5
  • 0+n=n0 + n = n for any natural number n.n\text{.}
  • 4+(−4)=0.4+(-4)=0\text{.}

Prerequisites

Sources

Keywords: summation, summand Symbols: integers, natural numbers, rational numbers, real numbers

Last updated: 20 October 2026 03:02