What is the root of the word irrational
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A ratio is effectively a fraction of the form m/n where m and n are integers. The fraction is therefore a rational number. So it follows that an irrational number is a number that cannot be expressed as such a ratio.

Rational in other contexts means reasonable, logical, sensible, commonsense,... So irrational implies the opposite: unreasonable, illogical, nonsensical, ...

It may have been assumed at one time that all numbers could be expressed as ratios or whole numbers. That's a reasonable assumption. But then numbers that didn't conform to this expression were discovered, numbers like √2, etc. This family of numbers had to be given a name, so what better than irrational?

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