Difference between revisions of "Addition"
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− | '''Addition''' is the mathematical [[operation]] | + | '''Addition''' is the mathematical [[operation]] represented by the <math>+</math> sign which combines two quantities. The result of addition is called 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 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]]) | 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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* [[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 [[ | + | * Inverse: The sum of a number and its [[additive inverse]], <math>a+(-a)</math>, is equal to [[0|zero]]. |
* Equality: If <math>a=b</math>, then <math>a+c=b+c</math>. | * 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>. | ||
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* <math>a+(-b)=a-b</math> (See also [[Subtraction]]) | * <math>a+(-b)=a-b</math> (See also [[Subtraction]]) | ||
− | == See also == | + | ==See also== |
− | * [[Arithmetic]] | + | *[[Arithmetic]] |
− | * [[Number theory]] | + | *[[Number theory]] |
− | * [[Subtraction]] | + | *[[Subtraction]] |
− | * [[ | + | *[[Hyperoperation]] |
− | + | *[[Counting]] | |
− | * [[Counting]] | ||
− | |||
[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Operation]] | [[Category:Operation]] | ||
+ | {{stub}} |
Latest revision as of 10:24, 15 February 2025
Addition is the mathematical operation 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.