Sign of product

Sign of a product of two numbers comes down to: $1 \times 1$ , $1 \times -1$ , $-1 \times 1$ , and $-1 \times -1$ . For example, $5 \times -3 = 1 \times 5 \times (-1) \times 3 = (1 \times -1) \cdot (5 \times 3)$ .

Since $1$ is the multiplicative identity, by definition: $a \times 1 = a = 1 \times a$ . So, $1 \times 1 = 1$ , $1 \times -1 = -1$ , and $-1 \times 1 = -1$ . What should $-1 \times -1$ be?

By definition: $-1 + 1 = 0$ . More generally, $-a + a = 0$ .

Since $1$ is the multiplicative identity, $-1 \times -1 = -1 \times 1 \times -1$ . Assuming negative numbers abide by associative law of multiplication: $-1 \times -1 = (-1 \times 1) \times -1$ . Therefore, $(-1 \times -1)$ is a quantity when added to $(-1 \times 1)$ we get 0. Therefore, $-1 \times -1 = 1$ .