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 = 26
Below 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?