Lesson 4: Multiplying and Dividing Rational Numbers

Property

Multiplying any number by $-1$ changes its sign and reflects its position across zero on the number line: $(-1) \times a = -a$. The product of two negative numbers is positive because $(-a) \times (-b) = (-1)(a) \times (-1)(b) = (-1)(-1) \times (a)(b) = 1 \times (a)(b) = ab$.

Examples

Property

Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as $(-1) \cdot (-1) = 1$ and the rules for multiplying signed numbers. The product of two numbers with the same sign is positive, and if the numbers have opposite signs, it is negative.

Examples

• For same signs, like $(-6) \times (-7)$, both numbers are negative, so the result is positive. $6 \times 7 = 42$, so $(-6) \times (-7) = 42$.
• For different signs, like $9 \times (-5)$, one is positive and one is negative, so the result is negative. $9 \times 5 = 45$, so $9 \times (-5) = -45$.
• For a series of multiplications like $(-3) \times (2) \times (-5)$, work step-by-step. First, $(-3) \times 2 = -6$. Then, $(-6) \times (-5) = 30$.

Explanation

A simple rule for signs: if the signs of the two numbers are the same, the product is positive. If the signs are different, the product is negative. Think of a 'double negative' becoming a positive.

Property

For any numbers $n$ and $a$, the properties of operations extend to positive and negative numbers. This can be summarized by the equation (which works for either $n$ or $a$ being positive or negative):

Examples

• Using $(-n) \times a$: We can calculate $(-4) \times 8 = -32$.
• Using $n \times (-a)$: We see that $4 \times (-8) = -32$, which is the same result.
• Using $-(n \times a)$: We can also calculate $-(4 \times 8) = -(32) = -32$.

Explanation

When multiplying a positive and a negative number, the result is always negative. This rule shows that it doesn't matter which of the two numbers is negative; you can move the negative sign around and the answer stays the same.