Addition

Pronunciation: /əˈdɪ ʃən/ Explain

Addition is joining two or more quantities together to make a sum. Figure 1 gives an example of integer addition. In the addition statement 4 + 3 = 7, 4 and 3 are addends and 7 is the sum. Addends are what is being added together, and the sum is the result of the addition. The symbol '+' is called the addition sign.

4 dots and 3 dots and 7 dots representing 4 + 3 = 7
Figure 1: Representation of 4 + 3 = 7

Real numbers can also be added. Figure 2 gives a representation of 2 + 1 = 3.

Number line showing 2 + 1 = 3
Figure 2: Representation of 2 + 1 = 3

Manipulative 3 represents the addition of real numbers. Click on the blue points and drag them to change the figure.

Click on the points A and B near the top to change the figure.

The blue and red arrows on the number line show A added to B. The sum A+B is below them in purple.
Manipulative 1 - Addition Created with GeoGebra.

Addition is also defined for many types of math entities such as vectors and matrices.

Subtraction

Mathematicians usually define subtraction as adding the additive inverse of a value. Subtraction is defined this way so that subtraction of vectors and matrices and other math entities makes more sense. Stated mathematically, a - b ≡ a + -b. A difference is the result of subtracting one number from another. For example, in the equation 7 - 4 = 3, 3 is the difference.

A example of this is 5 - 4 = 5 + -4 = 1

Properties of Addition of Real and Complex Numbers

Property NameExampleDescription
Additive property of zero
Additive identity
a + 0 = 0 + a = aAny number plus zero equals the original number. 0 is the additive identity for real and complex numbers.
Additive inversea + (-a) = 0The additive inverse of any real or complex number is the negative of that numbers.
Associative property of additiona + ( b + c ) = ( a + b ) + cThe order of in which multiple additions of real and complex numbers are performed does not change the result.
Commutative property of additiona + b = b + aIt doesn't matter which of two numbers come first in addition of real and complex numbers.
Distributive Property of Multiplication over Addition and Subtractiona ( b + c ) = ab + ac, a ( b - c ) = ab - acMultiplication is distributive over addition and subtraction.
Additive property of equalityIf a = b then a + c = b + c.The additive property of equality states that any number can be added to both sides of an equation without changing the truth value of the equation.
Subtractive property of equalityIf a = b then a - c = b - c.The subtractive property of equality states that any number can be subtracted from both sides of an equation without changing the truth value of the equation.
Table 1: Properties of Addition

Addition Facts

Addition facts are two operands and the result of adding those two operands. The following table gives the addition facts for 0 through 10.

+012345678910
0012345678910
11234567891011
223456789101112
3345678910111213
44567891011121314
556789101112131415
6678910111213141516
77891011121314151617
889101112131415161718
9910111213141516171819
101011121314151617181920

Adding complex numbers

To add two complex numbers, add the corresponding parts. Given two complex numbers a + bi and c + di, (a + bi) + (c + di) = (a + c) + (b + d)i. Example: (3 - 2i) + (-1 + 3i) = (3 + (-1)) + (-2 + 3)i = 2 + 1i = 2 + i.

Educator Resources

Cite this article as:

McAdams, David E. Addition. 6/18/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/addition.html.

Image Credits

Revision History

6/12/2018: Removed broken links, changed Geogebra links to work with Geogebra 5, updated license, implemented new markup. (McAdams, David E.)
10/12/2010: Added "Adding complex numbers", additive identity, additive inverse, and additive property of equality. (McAdams, David E.)
1/4/2010: Added "References". (McAdams, David E.)
3/3/2009: Added discussion of difference to the section on subtraction. (McAdams, David E.)
9/16/2008: Changed figure 3 to a manipulative. (McAdams, David E.)
6/7/2008: Corrected spelling errors. (McAdams, David E.)
5/14/2008: Initial version. (McAdams, David E.)

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