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.

 Figure 1: Representation of 4 + 3 = 7

Real numbers can also be added. Figure 2 gives a representation of 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 - ba + -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
a + 0 = 0 + a = a Any number plus zero equals the original number. 0 is the additive identity for real and complex numbers.
Additive inverse a + (-a) = 0 The additive inverse of any real or complex number is the negative of that numbers.
Associative property of addition a + ( b + c ) = ( a + b ) + a The order of in which multiple additions of real and complex numbers are performed does not change the result.
Commutative property of addition a + b = b + a It doesn't matter which of two numbers come first in addition of real and complex numbers.
Distributive Property of Multiplication over Addition and Subtraction a ( b + c ) = ab + ac, and a ( b - c ) = ab - ac Multiplication is distributive over addition and subtraction.
Additive property of equality If 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 equality If 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.

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

### How to Add 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.

### References

1. McAdams, David E.. All Math Words Dictionary, addition. 2nd Classroom edition 20150108-4799968. pg 10. Life is a Story Problem LLC. January 8, 2015. Buy the book

### Revision History

4/5/2019: Updated formating of equations. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)