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Repeating Decimals

Tool to find the period of a fraction or a decimal number with repeating decimals. The period is a set of digits that is repeated at infinity in the decimals of the number (usually a rational number or a periodic fraction).

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Repeating Decimals -

Tag(s) : Arithmetics

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# Repeating Decimals

## Fraction Finder

### What are repeating decimal in a fraction? (Definition)

The periodic decimal expansion/development of a rational number or a fraction (numerator over denominator) is the sequence of numbers that are repeated at infinity in the decimal writing of the number.

Example: 1/3 = 0.3333333333 ... The digit 3 is repeated to infinity

Example: 1/27 = 0.037037037037037 ... The digits 037 are repeated to infinity

All fractions do not have a repeating decimal form, some have a terminating decimal form.

### What are terminating decimal in a fraction? (Definition)

A terminating decimal indicates that no sequence of numbers repeats infinitely in the decimal writing of the number.

Example: 4/25 = 0.16 the development is finished and does not continue

Any number that is written in decimal form with a finite number of digits (after the decimal dot) has is a terminating decimal number.

### How to write repeating decimals?

Multiple notations are possible.

The first uses ... points of suspension, but does not define the part that repeats. It is practical but not rigorous and therefore not recommended.

Example: $37/300 = 0.12333333333 ...$

Notation with a bar above the repeated part.

Example: $37/300 = 0.12 \overline{3}$

Notation with a bar below the repeated part.

Example: $37/300 = 0.12 \underline{3}$

Notation between brackets

Example: $37/300 = 0.12 [3]$

NB: For clarity, it is better to write the fraction in an irreducible form.

### How to find decimals from a fraction?

Divide the numerator by the denominator. Lay the Euclidean division by hand or use a calculator.

### How to find the fraction from decimals?

Take $x$ a number, and $n$ the size (the number of digits) of the periodic part of the decimal expansion. To get a fractional writing, solve $x \times 10^n - x$.

Example: $x = 0.1\overline{6} = 0.1666666...$, the repeating portion is $6$, a single digit so $n=1$. Calculate $10^1 \times x = 1.\overline{6} = 1.6666666...$ and solve $10x−x = 9x = 1.\overline{6}−0.1\overline{6}=1.5 \iff 9x=1.5 \iff x=1.5/9 = 15/90 = 1/6$

### What are the most known decimal developments?

The inverses of prime numbers provide long and interesting periodic decimal developments.

Example: $1/3 = 0.333333...$

Example: $1/7 = 0.142857142857...$

### Is there an infinite decimal expansion with a series of digits that never repeats?

Any rational number (any fraction) has a finite developpement or a periodic decimal expansion with a finite number of digits that repeat themselves ad infinitum.

But there are real numbers that are not rational numbers (which are not fractions) which have decimals without repetition

Example: $\pi = 3.14159265 ...$ has no known repetition to date.

Example: Champernowne's constant will never have any repetition, it is a universe number.

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