Converting Decimal Expansion of Rational Number to Fraction

Theorem

Let $x \in \Q$ be a rational number.

Let the decimal expansion of $x$ be:

$0 \cdotp a_1 a_2 \ldots a_m \dot b_1 b_2 \ldots \dot b_n$

where $a_i: i \in \set {1, 2, \ldots, m}$ and $b_j: j \in \set {1, 2, \ldots, n}$ be the digits in the base $10$ expansion of $x$.

Then $x$ can be expressed as the following fraction:

$x = \dfrac {a_1 a_2 \ldots a_m b_1 b_2 \ldots b_n - a_1 a_2 \ldots a_m} {10^m \times \paren {10^n - 1} }$

Proof

First we note that:

\(\ds 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n\)

\(=\)

\(\ds b_1 b_2 \ldots b_n \cdotp \dot b_1 b_2 \ldots \dot b_n\)

\(\ds \leadsto \ \ \)

\(\ds 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times 10^n - 0 \cdotp \dot b_1 b_2 \ldots \dot b_n\)

\(=\)

\(\ds b_1 b_2 \ldots b_n \cdotp \dot b_1 b_2 \ldots \dot b_n - 0 \cdotp \dot b_1 b_2 \ldots \dot b_n\)

subtracting $0 \cdotp \dot b_1 b_2 \ldots \dot b_n$ from both sides

\(\ds \leadsto \ \ \)

\(\ds 0 \cdotp \dot b_1 b_2 \ldots \dot b_n \times \paren{10^n - 1}\)

\(=\)

\(\ds b_1 b_2 \ldots b_n\)

\(\ds \leadsto \ \ \)

\(\ds 0 \cdotp \dot b_1 b_2 \ldots \dot b_n\)

\(=\)

\(\ds \dfrac {b_1 b_2 \ldots b_n} {\paren {10^n - 1} }\)

\(\ds \leadsto \ \ \)

\(\ds 0 \cdotp a_1 a_2 \ldots a_m \dot b_1 b_2 \ldots \dot b_n\)

\(=\)

\(\ds \dfrac {a_1 a_2 \ldots a_m} {10^m} + \dfrac 1 {10^m} \times \dfrac {b_1 b_2 \ldots b_n} {\paren {10^n - 1} }\)

Definition of Decimal Expansion

\(\ds \)

\(=\)

\(\ds \dfrac {a_1 a_2 \ldots a_m} {10^m} \times \dfrac {\paren {10^n - 1} } {\paren {10^n - 1} } + \dfrac 1 {10^m} \times \dfrac {b_1 b_2 \ldots b_n} {\paren {10^n - 1} }\)

multiplying top and bottom by $10^n - 1$

\(\ds \)

\(=\)

\(\ds \dfrac {a_1 a_2 \ldots a_m \overbrace{000 \ldots 000}^n - a_1 a_2 \ldots a_m } {10^m \paren {10^n - 1} } + \dfrac {b_1 b_2 \ldots b_n} {10^m \paren {10^n - 1} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {a_1 a_2 \ldots a_m b_1 b_2 \ldots b_n - a_1 a_2 \ldots a_m }{10^m \times \paren {10^n - 1 } }\)

$\blacksquare$

Examples

Decimal Number $0 \cdotp 7 \dot 1 82 \dot 8$

The number:

$0. 7\dot 1 82 \dot 8$

can be expressed as a fraction as:

\(\ds 0. 7 \dot 1 82 \dot 8\)

\(=\)

\(\ds \dfrac {71828 - 7} {10^1 \paren {10^4 - 1} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {71821} {99990}\)

Decimal Number $0 \cdotp 14 \dot 1 \dot 6$

The number:

$0. 14 \dot 1 \dot 6$

can be expressed as a fraction as:

\(\ds 0. 14 \dot 1 \dot 6\)

\(=\)

\(\ds \dfrac {1416 - 14} {10^2 \paren {10^2 - 1} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {1402} {9900}\)