Additive Property of Equality

Pronunciation: /ˈæd.ɪ.tɪv ˈprɒ.pər.ti ʌv ɪˈkwɒl.ɪ.ti/ Explain

The additive property of equality states that we can add the same expression to both sides of an equation and the new equation is still has the same truth value. Stated algebraically: For real or complex numbers a, b, and c, if a = b then a + c = b + c. This means one can add any value to both sides of an equation without changing the truth value of the equation.

The additive property of equality applies to any relation on a set which:

• is closed with respect to addition; and
• is reflexive.

Proof of Additive Property of Equality

Step	Equation	Description
1	Let a, b and c be members of a set S and '=' be a relationship on that set. Set S is closed with respect to addition.	Define the mathematical objects needed for the proof.
2	Let a = b.	Establish the criterion.
3	Let d = a + c, d ∈ S.	Define member 'd' of the set.
4	Then d = b + c.	Use the substitution property of equality to substitute b for a.
5	d = d.	Use the reflexive property of equality to establish d = d.
6	a + c = d.	Use the substitution property of equality to substitute a + c in for d.
7	a + c = b + c. QED.	Use the substitution property of equality to substitute b + c in for d.

Cite this article as:

McAdams, David E. Additive Property of Equality. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/a/additivepropofequality.html.

Revision History

4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
4/11/2019: Changed wording to clarify proof. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/18/2018: Clarified wording. (McAdams, David E.)
6/12/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
12/24/2009: Fixed typographical error in step 7 of proof. (McAdams, David E.)
2/12/2009: Added limitation for real numbers. (McAdams, David E.)
10/5/2008: Added proof, discussion for systems other than real numbers. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)