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Irreducible Fractions

Tool to reduce fractions in lowest terms. A Fraction in Lowest Terms (Irreducible Fraction) is a reduced fraction in which the numerator and the denominator are coprime (they do not share common factors)

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Irreducible Fractions -

Tag(s) : Arithmetics, Symbolic Computation

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# Irreducible Fractions

## Simplify Fractions in Lowest Terms

Use the character / for the fraction bar

## Decimal to Fraction in Lowest Terms Converter

### What is a fraction in lowest terms? (Definition)

Any fraction can be written in different ways while keeping its value.

Example: $\frac{50}{100} = \frac{5}{10} = \frac{2}{4} = \frac{1}{2} = 0.5$

A fraction in lowest form (irreducible fraction) is a fraction whose denominateur (the divisor, the number under the fraction bar) is the smallest possible integer. NB: the numerator (the dividend number above the fraction bar) must also be an integer.

Giving the result as an irreducible fraction is usually the preferred format for writing a fraction, as it is its simplest form.

Example: $\frac{1}{2}$ is fraction in lowest form while $\frac{2}{4}$ is not a fraction in lowest form.

### How to make a fraction in lowest terms?

To simplify a fraction $a / b$ or $\frac{a}{b}$ composed of a numerator $a$ and a denominator $b$, find the greatest common divisor (GCD) of the numbers $a$ and $b$. The irreducible fraction is obtained by dividing the numerator and the denominator by the calculated GCD.

Example: The fraction $12/10$ has $12$ for numerator and $10$ for denominator. Calculate that $GCD(12,10) = 2$ and divide both the numerator $12/2 = 6$ and the denominator $10/2 = 5$, so the corresponding irreducible fraction is $6/5$

dCode offers tools to calculate the GCD via, for example, Euclid's algorithm.

### How to calculate and give the result under the lowest terms form?

Use the above calculator form: enter the expressions/fractions and the simplifier will use formal calculations in order to keep variables and find the irreducible form of the division (simplification of the fraction in lowest terms).

### How to make a fraction from a decimal number?

If the number has a limited decimal development then it only needs to be multiplied by the right power of 10, then simplify the fraction and solve the equation.

Example: The number $0.14$ is equivalent to $0.14/1$, multiply by $10/10 (=1)$ until having no comma: $0.14/1 = 1.4/10 = 14/100$ then simplify $14/100 = 7/50$

If the number has a non finite decimal expansion then it is necessary to locate the repeating portion of the number after the repeating decimal point.

Example: The number $0.166666666\dots$ where the $6$ is repeated

If $x$ is the decimal number, and $n$ the size (number of digits) of the smallest repeated portion. To obtain a fraction, multiply $x$ by $10^n$ and then subtract $x$.

Example: $x = 0.1666666\dots$, the smallest repeated portion is $6$, which has a single digit so that $n = 1$. Then compute $10^1 \times x = 1.6666666\dots$ and $10x-x$. $$10x-x = 9x = 1.666666\dots - 0.1666666\dots = 1.5 \\ \iff 9x = 1.5 \\ \Rightarrow x = 1.5/9 = 15/90 = 1/6$$ so $1/6 = 0.1666666\dots$

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Irreducible Fractions on dCode.fr [online website], retrieved on 2023-02-08, https://www.dcode.fr/irreducible-fraction

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