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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]].
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'''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 ==
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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), \cdots, f(b)</math>, where <math>f</math> is a [[function]], is denoted <math>\sum_{i=a}^bf(i)</math>. (See also [[Sigma notation]])
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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>.
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* 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>.
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* Inverse: The sum of a number and its [[additive inverse]], <math>a+(-a)</math>, is equal to [[Zero (constant)|zero]].
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* 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 $3+2=5$.

Notation

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

Properties

  • Commutativity: The sum $a+b$ is equivalent to $b+a$.
  • Associativity: The sum $(a+b)+c$ is equivalent to $a+(b+c)$. This sum is usually denoted $a+b+c$.
  • Distributivity: $a(b+c)=ab+ac$
  • Closure: If $a$ and $b$ are both elements of $\mathbb{R}$, then $a+b$ is an element of $\mathbb{R}$. This is also the case with $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{C}$.
  • Identity: $a+0=a$ for any complex number $a$.
  • Inverse: The sum of a number and its additive inverse, $a+(-a)$, is equal to zero.
  • Equality: If $a=b$, then $a+c=b+c$.
  • If $a$ is real and $b$ is positive, $a+b>a$.
  • The sum of a number and its Complex conjugate is a real number.
  • $a+(-b)=a-b$ (See also Subtraction)

See also

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