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), \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>. | ||
| + | * 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]]. | + | * 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)
.
is equivalent to 
. This sum is usually denoted 
.
, then 
, 
, and 
.
for any complex number 
, is equal to 
, then 
.
.
(See also See also
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