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Good Math by Mark C. Chu-Carroll

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What Are Irrational Numbers?

Let’s start with the irrational numbers. These are numbers that aren’t integers and also aren’t a ratio of any two integers. You can’t write them as a normal fraction. If you write them as a continued fraction (which we’ll describe in 11, Continued Fractions), then they go on forever. If you write them in decimal form, they go on forever without repeating. They’re called irrational because they can’t be written as ratios. Many people have claimed that they’re irrational because they don’t make sense, but that’s just a rationalization after the fact.

They do make sense, but they are uncomfortable and ugly to many mathematicians. The existence of irrational numbers means that there are numbers that you cannot write ...

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