Counting by Piles
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 •a = 1.
4. COMMUTATIVE AXIOM FOR MULTIPLICATION — If 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) = 2x + 2y & 2(x - y) = 2x - 2y
such that a• = 1 or •a = 1.
4. COMMUTATIVE AXIOM FOR MULTIPLICATION — If 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) = 2x + 2y & 2(x - y) = 2x - 2yBelow 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?