# Mixing Things Together

- Author:
- Jaime Dianzon

- Topic:
- Addition, Algebra, Real Numbers

## How do you mix things together?

## OBJECTIVE: To learn about the axioms of addition

In a previous lesson, you learned about the AXIOMS OF EQUALITY.

*Keeping Your Balance:*https://www.geogebra.org/m/v5jyuhxm In this lesson, you're going to learn about the AXIOMS OF ADDITION. The following axioms define the rules for adding real numbers. 1. CLOSURE AXIOM FOR ADDITION — If*a*and*b*are real numbers, then*a*+*b*is equal to a unique real number, i.e., if two real numbers are added together, the sum is a real number. 2. IDENTITY AXIOM FOR ADDITION — For any real number*a*,*a*+ 0*= a*or 0 +*a*=*a*. Zero ( 0 ) is the identity element for addition. 3. ADDITIVE INVERSE FOR ADDITION — Every real number*a*has an additive inverse*-a*such that*a + (-a) =*0 or*(-a) + a =*0. 4. COMMUTATIVE AXIOM FOR ADDITION — If*a*and*b*are real numbers, then*a*+*b = b*+*a*, i.e., the order in which two or more numbers are added together does not affect the sum. 5. ASSOCIATIVE AXIOM FOR ADDITION — If*a*,*b*, and*c*are real numbers, then*(a*+*b)*+*c = a*+*(b*+*c)*; i.e., the grouping of three or more numbers for addition does not affect the sum. Algebraic Examples: 1. CLOSURE AXIOM FOR ADDITION — 5 + 7 = 12 ⟶ Since 5 and 7 are real numbers, 12 is also a real number. 2. IDENTITY AXIOM FOR ADDITION — 10 + 0 = 10 & 0 + (-12) = -12 3. ADDITIVE INVERSE FOR ADDITION — 5 + (-5) = 0 & (-7) + 7 = 0 4. COMMUTATIVE AXIOM FOR ADDITION — 2 + 3 = 5 ⟷ 3 + 2 = 5 5. ASSOCIATIVE AXIOM FOR ADDITION — (6 + 7) + 13 = 13 + 13 = 26 ⟷ 6 + (7 + 13) = 6 + 20 = 26Below is a set of problems involving axioms of addition.

## Axioms of Addition

## ANSWER BOX:

Check your answers below.

## In this lesson, you learned about the axioms of addition.

In the next lesson, you'll learn about the axioms of multiplication. Did you ENJOY today's lesson?