Difference between revisions of "Every rational number has an irreducible representation"
(Created page with "'''Every rational number has an irreducible representation''' is a claim in number theory that all rational numbers must have a representation where the numerator and denominator cannot be further reduced. '''Proof''' Suppose one has a rational number m/n that is reducible. We can assume n≥1 (divide the numerator and denominator by −1 otherwise). Take an integer k≥2 which divides both of m and n, and so m′=m/k and n′=n/k are smaller integers satisfying m...") |
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'''Every rational number has an irreducible representation''' is a claim in [[number theory]] that all rational numbers must have a representation where the numerator and denominator cannot be further reduced. | '''Every rational number has an irreducible representation''' is a claim in [[number theory]] that all rational numbers must have a representation where the numerator and denominator cannot be further reduced. | ||
A simple proof may suppose one has a rational number m/n that is reducible. We can assume n≥1 (divide the numerator and denominator by −1 otherwise). | |||
Take an integer k≥2 which divides both of m and n, and so m′=m/k and n′=n/k are smaller integers satisfying m′n′=mn. | Take an integer k≥2 which divides both of m and n, and so m′=m/k and n′=n/k are smaller integers satisfying m′n′=mn. | ||
Revision as of 04:41, 23 January 2022
Every rational number has an irreducible representation is a claim in number theory that all rational numbers must have a representation where the numerator and denominator cannot be further reduced.
A simple proof may suppose one has a rational number m/n that is reducible. We can assume n≥1 (divide the numerator and denominator by −1 otherwise).
Take an integer k≥2 which divides both of m and n, and so m′=m/k and n′=n/k are smaller integers satisfying m′n′=mn.
Repeat this process as long as the fraction remains reducible.
One cannot repeat forever, because the sequence of denominators is a sequence of positive integers that gets smaller and smaller, and no such sequence can continue infinitely (essentially one is using mathematical induction in this step).
When the process stops, one has an irreducible fraction representing the same rational number
| Statement of the claim | The square root of 2 is irrational |
| Level of certainty | Proven |
| Nature | Theoretical |
| Counterclaim | |
| Dependent on | |
| Dependency of |
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