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).
Repeating Decimals - dCode
Tag(s) : Arithmetics
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Repeating Decimals
Recurring Decimal Detection A/B
Terminating Decimal Detection
Fraction Finder
Answers to Questions (FAQ)
What are repeating decimals 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 decimals 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\dots $
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\dots $, the repeating portion is $ 6 $, a single digit so $ n=1 $. Calculate $ 10^1 \times x = 1.\overline{6} = 1.6666666\dots $ 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\dots $
Example: $ 1/7 = 0.142857142857\dots $
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\dots $ has no known repetition to date.
Example: Champernowne's constant will never have any repetition, it is a universe number.
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- Recurring Decimal Detection A/B
- Terminating Decimal Detection
- Fraction Finder
- What are repeating decimals in a fraction? (Definition)
- What are terminating decimals in a fraction? (Definition)
- How to write repeating decimals?
- How to find decimals from a fraction?
- How to find the fraction from decimals?
- What are the most known decimal developments?
- Is there an infinite decimal expansion with a series of digits that never repeats?
