Question

prove that product of two integers is also an integer

Answer 1

Answer:

Step-by-step explanation:

Given: A and B are integers; every positive integer is a sum of ones; a sum of integers is an integer; 0 is an integer; the integers reflect about 0

Prove: AB is an integer

 

1. First, we will prove the case where at least one of A or B is a positive integer.

2. By the commutative property of multiplication, we may assume B is a positive integer.

3. Because B is a positive integer, and because every positive integer is a sum of ones, B is a sum of ones.

4. Therefore, we can substitute this sum of ones in for B to obtain AB = A(1 + 1 + ... + 1).

5. By the distributive property and the identity property of multiplication, A(1 + 1 + ... + 1) = A + A + ... + A.

6. A + A + ... + A is a sum of integers, and we know that a sum of integers is an integer.

7. Then we know that AB is an integer.

8. Next, we consider the case where neither A nor B is a positive integer, but at least one of them is 0.

9. By the commutative property of multiplication, we can assume B is 0 without loss of generality.

10. Then AB = A(0) = 0. Because AB = 0, and because 0 is an integer, we know AB is an integer.

11. Finally, we consider the only case left, where A and B are both negative integers.

12. By the associative, commutative, and identity properties of multiplication, AB = (1)AB = (-1)(-1)AB = (-A)(-B).

13. Because the integers reflect about 0, we are able to write (-A)(-B) = FG, for positive integers F and G.

14. Then, AB = FG. This case is then reduced to the first, which has already been proven.