# Counting by Piles

- Author:
- Jaime Dianzon

- Topic:
- Algebra, Multiplication, Real Numbers

## How do you count by piles?

## OBJECTIVE: To learn about the axioms of multiplication

In previous lessons, you learned about AXIOMS OF EQUALITY and AXIOMS OF ADDITION.

*Keeping Your Balance:*https://www.geogebra.org/m/v5jyuhxm*Mixing Things Together:*https://www.geogebra.org/m/fmktaxpa In this lesson, you're going to learn about the AXIOMS OF MULTIPLICATION. The following axioms define the rules for multiplying real numbers. 1. CLOSURE AXIOM FOR MULTIPLICATION — If*a*and*b*are real numbers, then*a•**b*is equal to a unique real number, i.e., if two real numbers are multiplied together, the product is a real number. 2. IDENTITY AXIOM FOR MULTIPLICATION — For any real number*a*,*a•*1*= a*or 1*•**a*=*a*. One ( 1 ) is the identity element for multiplication. 3. MULTIPLICATIVE INVERSE FOR MULTIPLICATION — Every real number*a*has a multiplicative inverse_{}such that*a•*1 or_{}=_{}*1. 4. COMMUTATIVE AXIOM FOR MULTIPLICATION — If**•a*=*a*and*b*are real numbers, then*a•**b = b•**a*, i.e., the order in which two or more numbers are multiplied together does not affect the product. 5. ASSOCIATIVE AXIOM FOR MULTIPLICATION — 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 multiplication does not affect the product. 6. MULTIPLICATION BY ZERO — If*a*is any real number, then*a**•*0 = 0 or 0*•a*= 0. 7. DISTRIBUTIVE AXIOM OF MULTIPLICATION — If*a*,*b*, and*c*are real numbers, then*a(b + c**)**= a•**b + a•**c*or*a(b - c**)**= a•**b - a•**c**;*i.e.,*a*is OVER ADDITION AND SUBTRACTION distributed across the sum or difference of*b*and*c.*Algebraic Examples: 1. CLOSURE AXIOM FOR MULTIPLICATION — 5*•*7 = 35 ⟶ Since 5 and 7 are real numbers, 35 is also a real number. 2. IDENTITY AXIOM FOR MULTIPLICATION — 10*•*1 = 10 & 1*•*(-12) = -12 3. MULTIPLICATIVE INVERSE FOR MULTIPLICATION — 5*•*= 1 & (-7)_{}*•*= 1 4. COMMUTATIVE AXIOM FOR MULTIPLICATION — 2_{}*•*3 = 6 ⟷ 3*•*2 = 6 5. ASSOCIATIVE AXIOM FOR MULTIPLICATION — (2*•*3)*•*4 = 6*•*4 = 24 ⟷ 2*•*(3*•*4) = 2*•*12 = 24 6. MULTIPLICATION BY ZERO — 8*•*0 = 0 & 0*•*8 = 0 7. DISTRIBUTIVE AXIOM OF MULTIPLICATION OVER ADDITION AND SUBTRACTION — 2(*x*+*y*) = 2*x*+ 2*y*& 2(*x*-*y*) = 2*x*- 2*y*Below is a set of problems involving axioms of multiplication.

## Axioms of Multiplication

## ANSWER BOX:

Check your answers below.

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

In future lessons, you'll learn about order of operations for real numbers. Did you ENJOY today's lesson?