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
Fraction Finder
Answers to Questions (FAQ)
What are repeating decimals in a fraction? (Definition)
The repeating decimal expansion of a rational number (that is, a fraction $ \frac{p}{q} $ where $ p $ and $ q $ are integers, and $ q \neq 0 $) is a decimal representation in which a sequence of digits repeats infinitely (sometimes called repetend).
Example: $ \frac{1}{3} = 0.\overline{3} = 0.3333\dots $ (the digit 3 repeats indefinitely)
Example: $ \frac{1}{27} = 0.\overline{037} = 0.037037037\dots $ (the sequence 037 repeats indefinitely)
If a rational number does not have a repeating decimal expansion, then it has a finite/terminating decimal expansion.
What are terminating decimals in a fraction? (Definition)
A finite decimal expansion is a decimal representation that stops after a certain number of digits after the decimal point.
Example: $ \frac{4}{25} = 0.16 $ the expansion is finite and does not continue.
A fraction $ \frac{p}{q} $ has a finite decimal expansion if and only if the denominator $ q $, once simplified, contains in its prime factorization only 2 and/or 5.
Example: $ \frac{1}{8} = 0.125 $ because $ 8 = 2^3 $, but $ \frac{1}{6} = 0.1\overline{6} $ because $ 6 = 2 \times 3 $ (the factor 3 causes the repeating pattern)
Why do some fractions have a finite expansion and others an infinite period?
The nature of the expansion depends on the denominator of the simplified fraction.
If the prime factors decomposition of the denominator contains only the prime factors 2 and 5, the expansion is finite.
Otherwise, it is periodic.
How to write repeating decimals?
Several notations exist to represent the repeated part of a repeating decimal expansion:
— With ellipses (non-rigorous notation):
Example: $ \frac{37}{300} = 0.12333333\dots $
— With a bar above (standard notation):
Example: $ \frac{37}{300} = 0.12\overline{3} $
— With a bar below (alternative notation):
Example: $ \frac{37}{300} = 0.12\underline{3} $
-In square brackets (occasional notation):
Example: $ \frac{37}{300} = 0.12[3] $
For clarity, write the fraction in its simplest form before expanding its decimal writing.
How to find decimals from a fraction?
Perform the division of the numerator by the denominator. Set up the Euclidean division by hand or use the dCode calculator.
To determine the decimal expansion of a fraction $ \frac{p}{q} $, perform the Euclidean division of the numerator $ p $ by the denominator $ q $.
As soon as a remainder that has already been encountered reappears, the sequence of digits begins to repeat again: the period begins.
Example: $ \frac{1}{7} = 0.142857142857\dots $ (the remainder repeats after 6 steps, so the period is 142857)
How to find the fraction from decimals?
Method for converting a repeating decimal $ x $ to a fraction:
— Identify the number of digits $ n $ in the period (the repeating part)
Example: For $ x = 0.1\overline{6} $, the period is $ \overline{6} $, so $ n = 1 $
— Calculate $ 10^n \times x - x $. This subtraction cancels the repeating part, leaving a finite number.
Example: Calculate $ 10x - x = 9x = 1.\overline{6} - 0.1\overline{6} = 1.5 $
— Solve the equation for $ x $
Example: $ x = \frac{1.5}{9} = \frac{15}{90} = \frac{1}{6} $
How do you determine the length of the period of a decimal expansion?
For an irreducible fraction $ \frac{p}{q} $ whose denominator is not a multiple of 2 or 5, the length of the period is the smallest integer $ n $ such that $ 10^n - 1 $ is divisible by $ q $ (in other words, $ 10^n \equiv 1 \mod{q} $)
Example: For $ \frac{1}{7} $, look for the smallest $ n $ such that $ 10^n - 1 $ is a multiple of 7: $ 10^6 - 1 = 999999 $ is divisible by 7, so the period has 6 digits: $ 142857 $
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
- Fraction Finder
- What are repeating decimals in a fraction? (Definition)
- What are terminating decimals in a fraction? (Definition)
- Why do some fractions have a finite expansion and others an infinite period?
- How to write repeating decimals?
- How to find decimals from a fraction?
- How to find the fraction from decimals?
- How do you determine the length of the period of a decimal expansion?
- What are the most known decimal developments?
- Is there an infinite decimal expansion with a series of digits that never repeats?
