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An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).

What are the applications and uses of unreducible fractions?

A classical one is the proof that square root of 2 is irrational (https://en.wikipedia.org/wiki/Irreducible_fraction#Applications).

Are there other uses of it in Basic School (High School or earlier) besides to express a fraction in simplified terms as answer to a math problem?

Related question: when the concept should be first presented to students? K4? K5? Only in High School? Why?

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    $\begingroup$ Could you please clarify what you mean by Basic School? $\endgroup$ – J W Jun 2 at 15:08
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    $\begingroup$ Even a first-grader would agree that cutting a pizza in three pieces and having one is quicker and less messy than cutting it into six pieces and having two. $\endgroup$ – Rusty Core Jun 2 at 18:08
  • $\begingroup$ @JW, Basic School here means High School or earlier. $\endgroup$ – Humberto José Bortolossi Jun 3 at 15:41
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Lots of different fractions all represent the same rational number. So if someone hands you two fractions, it can be difficult to tell if they represent the same number or not. For instance, you might need to do some work to determine if the fractions $\frac{18399}{30665}$ and $\frac{20991}{34985}$ represent the same number.

However, all rational numbers have only one reduced fraction representation! So you can detemine equality of rational numbers by finding the reduced fraction representation. In the example above, both fractions reduce to $\frac{3}{5}$ so we can be sure they are equal. Even more interesting is that we can be sure two rational numbers are not equal if they do not have the same reduced fraction representation. For instance, we can instantly tell that $\frac{23}{11}$ and $\frac{27}{13}$ are not equal because they are both reduced, but not identical as fractions.

So the most basic application of reduced fractions is in making rational number equality easy to detect. This is often why instructors request their students to report their answer as a reduced fraction: it makes the answer easy to check! Otherwise, you would need to either reduce the fraction to check the equality, or use the definition of rational number equality to decide equality ($a/b = c/d \leftrightarrow ad=bc$).

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    $\begingroup$ Another practical consequence is calculations on a computer, which has limited number of bits to represent numbers. Smaller absolute value (in particular, of nominator and denominator) is better. $\endgroup$ – Rusty Core Jun 2 at 18:17
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    $\begingroup$ Yes, while I was working on my examples, I noticed that Python thinks that $18399/30655.0=0.6001957266351329$. $\endgroup$ – Steven Gubkin Jun 2 at 20:28

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