Decimal Representation Formula

The Formula

0.d_1 d_2 d_3 \ldots = \frac{d_1}{10} + \frac{d_2}{100} + \frac{d_3}{1000} + \cdots

When to use: Just like 234 = 200 + 30 + 4, we have 2.34 = 2 + 0.3 + 0.04.

Quick Example

0.75 = \frac{7}{10} + \frac{5}{100} = \frac{75}{100} = \frac{3}{4}

Notation

A decimal point separates the whole-number part from the fractional part; digits to the right represent 10^{-1}, 10^{-2}, 10^{-3}, \ldots

What This Formula Means

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

Just like 234 = 200 + 30 + 4, we have 2.34 = 2 + 0.3 + 0.04.

Formal View

0.d_1 d_2 d_3 \ldots = \sum_{k=1}^{\infty} d_k \cdot 10^{-k} where each d_k \in \{0,1,\ldots,9\}. A decimal terminates iff the fraction \frac{p}{q} in lowest terms has q = 2^a \cdot 5^b.

Worked Examples

Example 1

easy
Convert \dfrac{5}{8} to a decimal by long division, and determine whether it terminates or repeats.

Solution

  1. 1
    Divide 5 \div 8: 8 goes into 50 six times (48), remainder 2. So far: 0.6.
  2. 2
    8 goes into 20 twice (16), remainder 4. Decimal: 0.62.
  3. 3
    8 goes into 40 five times (40), remainder 0. Decimal: 0.625.
  4. 4
    Remainder is 0, so the decimal terminates: \dfrac{5}{8} = 0.625.

Answer

\dfrac{5}{8} = 0.625 (terminating decimal)
A fraction in lowest terms terminates as a decimal if and only if its denominator has no prime factors other than 2 and 5. Here 8 = 2^3, so the decimal terminates.

Example 2

medium
Convert 0.\overline{142857} to a fraction in simplest form.

Common Mistakes

  • Thinking 0.125 > 0.5 because 125 has more digits — compare digit by digit from the left: 0.1 < 0.5
  • Reading 0.40 as larger than 0.4 — trailing zeros after the decimal do not change the value
  • Placing the decimal point incorrectly when converting fractions — \frac{1}{4} = 0.25, not 0.14 or 0.41

Why This Formula Matters

Decimals make fractions compatible with place-value computation.

Frequently Asked Questions

What is the Decimal Representation formula?

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

How do you use the Decimal Representation formula?

Just like 234 = 200 + 30 + 4, we have 2.34 = 2 + 0.3 + 0.04.

What do the symbols mean in the Decimal Representation formula?

A decimal point separates the whole-number part from the fractional part; digits to the right represent 10^{-1}, 10^{-2}, 10^{-3}, \ldots

Why is the Decimal Representation formula important in Math?

Decimals make fractions compatible with place-value computation.

What do students get wrong about Decimal Representation?

More digits after decimal doesn't mean larger (0.5 > 0.125).

What should I learn before the Decimal Representation formula?

Before studying the Decimal Representation formula, you should understand: place value, fractions.

← Back to Decimal Representation See All Examples Practice Problems