Why does a negative times a negative equal a positive?

B>Number LineImagine a number line on which you walk. Multiplying x * y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking, and negative step sizes indicate they should go backwards (i.e.

, heel first). So, (-x) * (-y) means to stand on zero, face in the negative direction, and then take x backward steps, each of size y. A Proof Let a and be be any two real numbers.

Consider the number x defined by x = ab + (-a)(b) + (-a)(-b). We can write x = ab + (-a) (b) + (-b) (factor out -a) = ab + (-a)(0) = ab + 0 = ab. Also, x = a + (-a) be + (-a)(-b) (factor out b) = 0 * be + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b).

So we have x = ab and x = (-a)(-b) Hence, by the transitivity of equality, we haveab = (-a)(-b). Because the opposite of a negative is a positive.

Number LineImagine a number line on which you walk. Multiplying x * y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking, and negative step sizes indicate they should go backwards (i.e.

, heel first). So, (-x) * (-y) means to stand on zero, face in the negative direction, and then take x backward steps, each of size y. A Proof Let a and be be any two real numbers.

Consider the number x defined by x = ab + (-a)(b) + (-a)(-b). We can write x = ab + (-a) (b) + (-b) (factor out -a) = ab + (-a)(0) = ab + 0 = ab. Also, x = a + (-a) be + (-a)(-b) (factor out b) = 0 * be + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b).

So we have x = ab and x = (-a)(-b) Hence, by the transitivity of equality, we haveab = (-a)(-b). Because the opposite of a negative is a positive.

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