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In real life one can do very well with nothing but rational numbers. Real numbers that are not rational can be approximated with whatever degree of accuracy is needed, which is exactly what computers and calculators do .

Real numbers are essential to mathematical theory but unless you are a mathematician or need to understand higher mathematics in a rigorous way, you can do very well with only a vague , intuitive understanding of real numbers. In fact, it was not until 1870 ,or so, that mathematicians devised a satisfactory definition of real numbers.

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