Distributive property for integers

The idea

Integer multiplication distributes over addition: for any integers a, b and c, a × (b + c) = (a × b) + (a × c).

Add first, or multiply first?

What is 4 × (2 + (−3))? One way: add inside the bracket first, then multiply. Another way: find 4 × 2 and 4 × (−3), then add them.

Will the two ways give the same answer?

• Yes, the same
• No, different answers
• Only when all the numbers are positive

One row, many rows

One row holds the tokens of b and then the tokens of c. A rows of it make the big rectangle. Cut the rectangle into its b part and its c part. Change a, b and c. Do the two ways ever differ?

Both ways fill the same rectangle

4 × (2 + (−3)) = 4 × (−1) = −4. And 4 × 2 + 4 × (−3) = 8 + (−12) = −4. Multiplying by 4 puts the row in 4 times. The whole rectangle and its two parts hold the same tokens.

For all integers a, b and c, a × (b + c) = (a × b) + (a × c). Integer multiplication distributes over addition.

• Both ways | Bracket first | Multiply first
• 5 × (4 + (−2)) | 5 × 2 = 10 | 20 + (−10) = 10
• (−2) × (4 + (−3)) | (−2) × 1 = −2 | (−8) + 6 = −2
• 4 × (2 + (−3)) | 4 × (−1) = −4 | 8 + (−12) = −4

Notes

For all integers a, b and c, a × (b + c) = (a × b) + (a × c). Integer multiplication distributes over addition.

Check yourself

Find 5 × (4 + (−2)).

Answer: 10

5 × (4 + (−2)) = 5 × 2 = 10. Also 5 × 4 + 5 × (−2) = 20 + (−10) = 10.

Find (−3) × (2 + (−6)).

Answer: 12

(−3) × (2 + (−6)) = (−3) × (−4) = 12. Also (−3) × 2 + (−3) × (−6) = (−6) + 18 = 12.

Which is equal to 4 × (2 + (−3))?

Find 6 × (−13) + 6 × 13. Take the 6 out first.

Answer: 0

6 × (−13) + 6 × 13 = 6 × ((−13) + 13) = 6 × 0 = 0. The two parts are inverses: −78 and 78.

• 4 × 2 + (−3). The 4 must multiply the (−3) too.
• 4 + 2 × (−3). That multiplies only the 2 and the (−3), not by 4. Here it gives −2, not −4.
• 4 × 2 + 4 × (−3) — correct. Yes! Multiply each number in the bracket by 4, then add: 8 + (−12) = −4.