📌 Integers: Multiplication & Division

Q: How do you know if a rational number becomes a terminating or repeating decimal?

Perform long division ($p \div q$) and track the remainders: if long division reaches a remainder of $0$, it is a **terminating decimal** (e.g., $\frac{3}{8} = 0.375$). If a remainder repeats, the quotient digits repeat indefinitely (**repeating decimal**, e.g., $\frac{5}{11} = 0.\overline{45}$). *(Shortcut: A simplified fraction terminates if its denominator $q$ contains only $2$'s and/or $5$'s as prime factors.)*

Learn rules of signs for multiplying and dividing positive and negative integers.

Multiplication & Division

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Learning Guide

1. Rules of Signs

Rules of Signs for Multiplication:
◦ $\text{positive} \times \text{positive} = \text{positive}$ ( $(+) \times (+) = +$ )
◦ $\text{positive} \times \text{negative} = \text{negative}$ ( $(+) \times (-) = -$ )
◦ $\text{negative} \times \text{negative} = \text{positive}$ ( $(-) \times (-) = +$ )
Derivation via Distributive Property:
Why does $(-1)(-1) = 1$ ? Since $1 + (-1) = 0$ , multiplying by $(-1)$ gives:
$(-1)(1 + (-1)) = (-1)(1) + (-1)(-1)$
$0 = -1 + (-1)(-1)$
$\implies (-1)(-1) = 1$
Rules of Signs for Division:◦ $\dfrac{\text{positive}}{\text{positive}} = \text{positive}$ ( $(+) \div (+) = +$ )◦ $\dfrac{\text{negative}}{\text{negative}} = \text{positive}$ ( $(-) \div (-) = +$ )◦ $\dfrac{\text{positive}}{\text{negative}} = \text{negative}$ ( $(+) \div (-) = -$ )◦ $\dfrac{\text{negative}}{\text{positive}} = \text{negative}$ ( $(-) \div (+) = -$ )

Rules of Signs Summary

Multiplying or dividing numbers with the SAME sign yields POSITIVE (+). Different signs yield NEGATIVE (-).

2. Division of Integers & Fraction Equivalent Forms

Division by Zero: Division by zero is undefined ( $\frac{p}{0}$ is undefined).
Rational Numbers: Every quotient of integers $\frac{p}{q}$ (where $q \neq 0$ ) is a rational number.
Equivalent Forms of Negative Fractions: The negative sign can be placed on the numerator, denominator, or in front of the fraction:
$-\frac{p}{q} = \frac{-p}{q} = \frac{p}{-q}$

Example: $\frac{-12}{4} = \frac{12}{-4} = -\frac{12}{4} = -3$ .

💡 Mastering Multiplication & Division Pro-Tip

When multiplying or dividing signed numbers: Numbers with the same sign always yield a POSITIVE (+) result, while numbers with different signs yield a NEGATIVE (-) result. Also, division by zero is always undefined.

Learning Topics

Frequently Asked Questions

What is an additive inverse?

An additive inverse of a number is another number that, when added to it, yields zero. For example, the additive inverse of 7 is -7 because 7 + (-7) = 0.

Why does multiplying two negative numbers result in a positive number?

Multiplying two negative numbers yields a positive result because multiplying by a negative number flips the sign on the number line. Reversing a negative direction twice turns it back into a positive direction (e.g., $(-1) \times (-1) = 1$ ).

How do you know if a rational number becomes a terminating or repeating decimal?

Perform long division ( $p \div q$ ) and track the remainders: if long division reaches a remainder of $0$ , it is a terminating decimal (e.g., $\frac{3}{8} = 0.375$ ). If a remainder repeats, the quotient digits repeat indefinitely (repeating decimal, e.g., $\frac{5}{11} = 0.\overline{45}$ ). (Shortcut: A simplified fraction terminates if its denominator $q$ contains only $2$ 's and/or $5$ 's as prime factors.)