Lesson 11: Multiplying and Dividing Real Numbers

New Concept

Algebra is built on consistent rules called properties. The properties of real numbers apply to all real numbers, rational and irrational.

What’s next

Next, you'll see these properties in action with worked examples on multiplying, dividing, and raising signed numbers to a power.

Property

The product of two numbers with the same sign is a positive number. The product of two numbers with opposite signs is a negative number.

Examples

$(-7)(-5) = 35$
$9 \cdot (-4) = -36$
$(1.5)(4) = 6$

Explanation

Multiplying signed numbers is simple! If two numbers have the same sign (both positive or both negative), their product is always positive—they're on the same team! If they have different signs (one positive, one negative), their product is negative because they are working against each other. Just multiply the numbers and then check the signs!

Property

The quotient of two numbers with the same sign is a positive number. The quotient of two numbers with opposite signs is a negative number.

Examples

$-24 \div (-3) = 8$
$\frac{50}{-10} = -5$
$3.6 \div 0.9 = 4$

Explanation

Division follows the exact same sign rules as multiplication! Think of them as best friends. If the signs of the two numbers match (positive/positive or negative/negative), the answer is positive. If the signs are different, the answer is negative. So, just do the division as usual, then apply the same sign rule you already learned for multiplying!

Property

An even exponent on a negative base gives a positive result, while an odd exponent gives a negative result. Note the difference: $(-a)^n$ is not the same as $-a^n$.

Examples

$(-5)^2 = (-5)(-5) = 25$
$(-5)^3 = (-5)(-5)(-5) = -125$
$-5^2 = -(5 \cdot 5) = -25$

Explanation

Parentheses are VIP passes! In $(-3)^4$, the negative is included in the repeated multiplication, resulting in a positive. But in $-3^4$, the negative is an outsider, applied only after the power is calculated, making the final answer negative. Always check if the negative sign is inside the parentheses to see if it gets to join the party!

Property

To divide by a fraction, multiply by its reciprocal. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$. The rules for signed numbers still apply.

Examples

$-\frac{2}{3} \div (-\frac{3}{4}) = -\frac{2}{3} \cdot (-\frac{4}{3}) = \frac{8}{9}$
$\frac{1}{5} \div (-\frac{2}{7}) = \frac{1}{5} \cdot (-\frac{7}{2}) = -\frac{7}{10}$

Explanation

Don't let fraction division scare you! Just use the 'keep, change, flip' trick. Keep the first fraction the same, change division to multiplication, and flip the second fraction upside down to get its reciprocal. Then, multiply the fractions and use your sign rules to find the final answer. It turns a tricky problem into a simple multiplication!