Fact-orials: Associative Property

Saturday, October 6, 2018

Associative Property

Background

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

Answer

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:

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.

Vocabulary used:

Commutative Property - the property that states that a + b = b + a, and a x b = b x a

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations:

Operations with different kinds of numbers: