We looked at the ordering of terms in the Commutative Property post, asking if a operator b was the same thing as b operator a. So what happens if we group things in different ways? That is, if we have (a operator b) same operator c, is it the same as a operator (b same operator c)?
Question
Evaluate the following to see if they are true:
- 3 + (2 + 4) = (3 + 2) + 4
- 3 - ( 2 - 4) = (3 - 2) - 4
- 3 x (2 x 4) = (3 x 2) x 4
- 3 ÷ (2 ÷ 4) = (3 ÷ 2) ÷ 4
Addition and Multiplication are Associative and so the first and third questions above are true in that they equal each other. The subtraction and division questions are false - they don't equal each other.Analysis
Let's work each question to test to see if they are true or not:
Addition
3 + (2 + 4) = (3 + 2) + 4
3 + 6 = 5 + 4
9 = 9 ✅
And in fact this is true regardless of the numbers we put in for a, b, and c.
Addition is associative.
Subtraction
3 - (2 - 4) = (3 - 2) - 4
3 - (-2) = 1 - 4
5 = -3 X
While we can choose certain values for a, b, and c to make the expression true, it's not always true (like we just saw here).
Subtraction is not associative.
Multiplication
3 x (2 x 4) = (3 x 2) x 4
3 x 8 = 6 x 4
24 = 24 ✅
And in fact this is true regardless of the numbers we put in for a, b, and c.
Multiplication is associative.
Division
3 ÷ (2 ÷ 4) = (3 ÷ 2) ÷ 4
I'll rearrange how we write these to reflect the groupings:
XWhile we can choose certain values for a, b, and c to make the expression true, it's not always true (like we just saw here).
Subtraction is not associative.
Vocabulary used:
- Commutative Property - the property that states that a + b = b + a, and a x b = b x a
Where might you have come from?
Fact-orials Index
Operations:
- Addition
- Subtraction
- Multiplication
- Division
- Grouping Terms, Parentheses
- Order of Operations (PEMDAS)
Operations with different kinds of numbers:
Properties:
Where might we go?Properties:
Operations with different kinds of numbers:
Properties:
