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How can you test whether the square root of a number will be rational or not?
Is this even possible?
I need this because I need to work out whether to display a number as a surd or not in a maths app I'm developing at the moment.
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Identifying Irrational NumbersIf a square root is not a perfect square, then it is considered an irrational number. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating).
A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. But an irrational number cannot be written in the form of simple fractions. ⅔ is an example of a rational number whereas √2 is an irrational number.
√2, √3, √5, and so on are some examples of irrational numbers as they cannot be expressed in form of p ⁄ q.
For integer inputs, only the square roots of the square numbers are rationals. So your problem boils down to find if your number is a square number. Compare the question: What's a good algorithm to determine if an input is a perfect square?.
If you have rational numbers as inputs (that is, a number given as the ratio between two integer numbers), check that both divisor and dividend are perfect squares.
For floating-point values, there is probably no solution because you can't check if a number is rational with the truncated decimal representation.
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From wikipedia: The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares.
So you need to find a rational approxmiation for your input number. So far the only algorithm I've nailed down that does this task is written in Saturn Assembler for the HP48 series of calculators.
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