📌 Integers: Rational Numbers to Decimals

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 how to convert rational numbers to terminating and repeating decimals using long division.

Rational Numbers to Decimals

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Learning Guide

3. Converting Rational Numbers to Decimals

To convert any rational number

\frac{p}{q}

to a decimal, divide the numerator

p

by the denominator

q

using long division.

Terminating Decimal: The long division reaches a remainder of $0$ .
Example: $\frac{3}{8} = 3 \div 8 = 0.375$ .
Repeating Decimal & How to Find It:
When dividing $p \div q$ , track the remainders after each step. Once a remainder repeats, the subsequent digits in the quotient repeat in the exact same order forever!

Step-by-Step Example: Finding $\frac{5}{11}$ as a Repeating Decimal:
1. Divide $5.0$ by $11$ : $50 \div 11 = \mathbf{4}$ with a remainder of $6$ ( $50 - 44 = 6$ ).
2. Bring down $0 \rightarrow 60$ : $60 \div 11 = \mathbf{5}$ with a remainder of $\mathbf{5}$ ( $60 - 55 = 5$ ).
3. Identify Repeating Remainder: The remainder $5$ is identical to our starting number $5$ ! Continuing long division will yield remainders of $6, 5, 6, 5\dots$ forever.
4. Write with Overline Notation: The quotient digits $45$ repeat indefinitely:
$\frac{5}{11} = 0.4545\dots = 0.\overline{45}$

💡 Pro-Tip / Shortcut (Prime Factors of Denominator):
For any simplified fraction $\frac{p}{q}$ :
If the prime factors of denominator $q$ contain only $2$ 's and/or $5$ 's (e.g. $8 = 2^3$ , $20 = 2^2 \times 5$ ), it will always be a terminating decimal.
If denominator $q$ has any other prime factors (like $3, 7, 11$ ), it will always be a repeating decimal (e.g. $\frac{5}{11} = 0.\overline{45}$ ).

📊 Long Division Diagram

Tracking remainders in long division: when remainder 5 repeats, quotient 45 repeats forever ($0.\overline{45}$).

Using SealMath: Typing Repeating Decimals

To write a repeating decimal in the math field:

Bar / Overline notation: Type \overline over repeating digits (e.g. 0.\overline{45} or 0.1\overline{6}).
Expanded Decimal: Type repeating digits (e.g. 0.4545 or 0.4545...).
Fraction form: Enter the exact fraction (e.g. 5/11 or \frac{5}{11}).

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.)