# Equal

Pronunciation: /ˈi.kwəl/ Explain

Two values are equal if they have the same value. For example, a = 15, states that the variable a has the same value as 15. The symbol '=' is called the equal sign /ˈikwəl saɪn/ and is used to show that the expression on the left is equal to the expression on the right. The equation x + 2 = 5 states that when 2 is added to an unknown value x, the sum is equal to 5.

### Properties of Equality for Real Numbers

#### Operations

NameProperty
Addition property of equality If a = b then a + c = b + c
Subtraction property of equality If a = b then a - c = b - c
Multiplication property of equality If a = b then a · c = b · c
Division property of equality If a = b and c ≠ 0 then a ÷ c = b ÷ c
Table 1: Operative properties of equality

#### Reflexive Property of Equality

The reflexive property of equality states that: a = a. This means that every value is equal to itself.

#### Symmetric Property of Equality

The symmetric property of equality states that: If a = b then b = a. A mnemonic for this property is the meaning of symmetric. A object has binary symmetry if both sides are the same. if a = b, a is the same as b, so b = a.

#### Transitive Property of Equality

The transitive property of equality states; if a = b and b = c, then a = c. For example, if y = x+2 and y = x - 4, then x + 2 = z - 4. A mnemonic for this property is taken from the root word 'transit'. A transit is a mode of movement. For example, a mass transit system is a system to move masses of people. Since equality is transitive, given a = b and b = c the value of a 'transits' or moves to c.

#### Equivalence Relation

An equality that is reflexive, symmetrical and transitive is called an equivalence relation.

#### Substitution Property

The substitution property of equality states that; if two values are equal, then one can be substituted for the other in equations.

• if a = b, then for a function F(x), F(a) = F(b);
• if a = b, then a + c = b + c;
• if a = b, then a - c = b - c;
• if a = b, then ac = bc;
• if a = b and c is not zero, then a ÷ c = b ÷ c.

#### Trichotomy Property of Real Numbers

For any two real numbers a and b, exactly one of the following is true:
a < b
a = b
a > b

### References

1. McAdams, David E.. All Math Words Dictionary, equal. 2nd Classroom edition 20150108-4799968. pg 70. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Equivalence Relation. allmathwords.org. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/e/equivalencerelation.html.

McAdams, David E. Equal. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/equal.html.

### Revision History

4/20/2019: Updated expressions and equations to match new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
6/2/2011: Added equal sign. (McAdams, David E.)