# Difference between revisions of "Addition"

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− | '''Addition''' is the mathematical [[operation]] which combines two quantities. The result of addition is called [[sum|a sum]]. | + | '''Addition''' is the mathematical [[operation]] (it is represented by the <math>+</math> sign) which combines two quantities. The result of addition is called [[sum|a sum]]. For example, the sum of 3 and 2 is 5 because <math>3+2=5</math>. |

== Notation == | == Notation == | ||

− | The sum of two numbers <math>a</math> and <math>b</math> is denoted <math>a+b</math>, which is read "a plus b." The sum of <math>f(a), f(a+1), f(a+2), f(a+3), | + | The sum of two numbers <math>a</math> and <math>b</math> is denoted <math>a+b</math>, which is read "a plus b." The two numbers being added together, or <math>a</math> and <math>b</math>, are called addends. The sum of <math>f(a), f(a+1), f(a+2), f(a+3), \cdots, f(b)</math>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]]) |

==Properties== | ==Properties== | ||

* Commutativity: The sum <math>a+b</math> is equivalent to <math>b+a</math>. | * Commutativity: The sum <math>a+b</math> is equivalent to <math>b+a</math>. | ||

* Associativity: The sum <math>(a+b)+c</math> is equivalent to <math>a+(b+c)</math>. This sum is usually denoted <math>a+b+c</math>. | * Associativity: The sum <math>(a+b)+c</math> is equivalent to <math>a+(b+c)</math>. This sum is usually denoted <math>a+b+c</math>. | ||

+ | * Distributivity: <math>a(b+c)=ab+ac</math> | ||

* [[Closure]]: If <math>a</math> and <math>b</math> are both elements of <math>\mathbb{R}</math>, then <math>a+b</math> is an element of <math>\mathbb{R}</math>. This is also the case with <math>\mathbb{N}</math>, <math>\mathbb{Z}</math>, and <math>\mathbb{C}</math>. | * [[Closure]]: If <math>a</math> and <math>b</math> are both elements of <math>\mathbb{R}</math>, then <math>a+b</math> is an element of <math>\mathbb{R}</math>. This is also the case with <math>\mathbb{N}</math>, <math>\mathbb{Z}</math>, and <math>\mathbb{C}</math>. | ||

* Identity: <math>a+0=a</math> for any complex number <math>a</math>. | * Identity: <math>a+0=a</math> for any complex number <math>a</math>. | ||

+ | * Inverse: The sum of a number and its [[additive inverse]], <math>a+(-a)</math>, is equal to [[Zero (constant)|zero]]. | ||

+ | * Equality: If <math>a=b</math>, then <math>a+c=b+c</math>. | ||

* If <math>a</math> is real and <math>b</math> is positive, <math>a+b>a</math>. | * If <math>a</math> is real and <math>b</math> is positive, <math>a+b>a</math>. | ||

* The sum of a number and its [[Complex conjugate]] is a real number. | * The sum of a number and its [[Complex conjugate]] is a real number. |

## Latest revision as of 19:34, 4 July 2019

**Addition** is the mathematical operation (it is represented by the sign) which combines two quantities. The result of addition is called a sum. For example, the sum of 3 and 2 is 5 because .

## Notation

The sum of two numbers and is denoted , which is read "a plus b." The two numbers being added together, or and , are called addends. The sum of , where is a function, is denoted . (See also Sigma notation)

## Properties

- Commutativity: The sum is equivalent to .
- Associativity: The sum is equivalent to . This sum is usually denoted .
- Distributivity:
- Closure: If and are both elements of , then is an element of . This is also the case with , , and .
- Identity: for any complex number .
- Inverse: The sum of a number and its additive inverse, , is equal to zero.
- Equality: If , then .
- If is real and is positive, .
- The sum of a number and its Complex conjugate is a real number.
- (See also Subtraction)

## See also

*This article is a stub. Help us out by expanding it.*