Order Doesn't Matter: Commutativity
Swap the terms. Watch the sum stay the same.
Equation
y = (x + 3) - (3 + x)
Graph
Table
| x | y |
|---|---|
| -5 | 0 |
| -4 | 0 |
| -3 | 0 |
| -2 | 0 |
| -1 | 0 |
| 0 | 0 |
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | 0 |
| 5 | 0 |
What this lesson covers
What you do
You shape the function y = (x + a) - (a + x) and watch the graph answer.
Challenges to clear
- Slide a to -3. Even with a negative number, x + a and a + x stay EQUAL — the difference is 0 for every x in the table.
- Commutativity does not care about signs. Slide a to −4 — x + a and a + x are still the same number.
- And it survives multiplication: slide b to 7. The difference stays flat on 0 for every x in the table, which is what 'always true' looks like.
Check yourself
Does commutativity work for subtraction? Try calculating 5-3 and 3-5.
Does commutativity work for division? Try calculating 6/2 and 2/6.
- Yes, both equal 2
- Yes, both equal -2
- No, 5-3=2 but 3-5=-2 — correct
- No, subtraction is undefined
- Yes, both equal 3
- Yes, both equal 1/3
- No, 6/2=3 but 2/6=1/3 — correct
- No, division is undefined
Think about it
- If a = -2 and x = 5, what is (5 + (-2)) - ((-2) + 5)?