Rational Numbers and Decimals: Converting Between Forms

Converting Rational Numbers to Decimals 🧮

A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. To convert a rational number to a decimal, perform long division.

Example 1: Converting $\frac{3}{4}$ to a Decimal

Divide 3 by 4:

      0.75
    --------
4 | 3.00
  - 2.8
  ------
    0.20
  - 0.20
  ------
    0.00

Therefore, $\frac{3}{4} = 0.75$.

Example 2: Converting $\frac{1}{3}$ to a Decimal (Repeating Decimal)

Divide 1 by 3:

      0.333...
    --------
3 | 1.000
  - 0.9
  ------
    0.10
  - 0.09
  ------
    0.010
  - 0.009
  ------
    0.001...

The division continues indefinitely, resulting in a repeating decimal. We write $\frac{1}{3} = 0.\overline{3}$.

Converting Decimals to Rational Numbers 🔄

To convert a decimal to a rational number, follow these steps:

1. Write the decimal as a fraction: Place the decimal number over a power of 10. The power of 10 depends on the number of decimal places.
2. Simplify the fraction: Reduce the fraction to its simplest form.

Example 1: Converting 0.625 to a Rational Number

1. Write 0.625 as a fraction: $\frac{625}{1000}$
2. Simplify the fraction: $\frac{625}{1000} = \frac{5}{8}$

Therefore, $0.625 = \frac{5}{8}$.

Example 2: Converting a Repeating Decimal to a Rational Number ♾️

Converting repeating decimals requires a bit more algebra. Let's convert $0.\overline{4}$ to a fraction.

1. Let $x = 0.\overline{4}$
2. Multiply by 10: $10x = 4.\overline{4}$
3. Subtract the first equation from the second: $10x - x = 4.\overline{4} - 0.\overline{4}$ which simplifies to $9x = 4$
4. Solve for $x$: $x = \frac{4}{9}$

Therefore, $0.\overline{4} = \frac{4}{9}$.

General Method for Repeating Decimals

• For a repeating decimal of the form $0.\overline{a_1a_2...a_n}$, let $x = 0.\overline{a_1a_2...a_n}$.
• Multiply by $10^n$: $10^n x = a_1a_2...a_n.\overline{a_1a_2...a_n}$.
• Subtract $x$ from $10^n x$: $(10^n - 1)x = a_1a_2...a_n$.
• Solve for $x$: $x = \frac{a_1a_2...a_n}{10^n - 1}$.

Example: Convert $0.\overline{123}$ to a fraction.

1. $x = 0.\overline{123}$
2. $1000x = 123.\overline{123}$
3. $999x = 123$
4. $x = \frac{123}{999} = \frac{41}{333}$

Therefore, $0.\overline{123} = \frac{41}{333}$.

Summary 📝

• Rational numbers can be converted to decimals through division.
• Terminating decimals can be directly converted to fractions by placing them over a power of 10 and simplifying.
• Repeating decimals require an algebraic approach to convert them into fractions.