Order does not matter

With whole numbers, swapping the two factors leaves the product alone. Does that still hold when the numbers may be negative? Test some pairs.

3 × (−4) = −12, and (−4) × 3 = −12. The same. (−15) × (−8) = 120, and (−8) × (−15) = 120. The same. Work out (−30) × 12 and 12 × (−30) and you will find they match too.

Will this ALWAYS happen? Rather than testing for ever, split the product into its two parts — magnitude and sign — and check each.

The MAGNITUDE first. The magnitude of the product depends only on the magnitudes of the two numbers being multiplied. And we already know that the product of two POSITIVE integers does not change when they are swapped. So the magnitude cannot change.

Now the SIGN. If both numbers are positive, the product is positive before and after swapping. If both are negative, the product is positive before and after swapping. So in those cases the sign does not change.

And if one is positive and the other negative? The product is negative before swapping, and after swapping there is still one of each, so it is negative again. The sign does not change here either.

Neither the magnitude nor the sign changes, so the product does not change — whatever the signs may be. Multiplication is commutative for integers: for any two integers a and b, a × b = b × a.