What do you do to check whether a number is rational or irrational? In your explanation, use an example of an irrational and a rational number.
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To check whether a number is rational or irrational, you need to understand their definitions.
Rational numbers can be expressed as a fraction of two integers, where the denominator is not zero. For example, 2/3, -4/5, and 0.25 are all rational numbers.
On the other hand, irrational numbers cannot be expressed as fractions and do not terminate or repeat their decimal representation. Common examples of irrational numbers include π (pi), √2 (square root of 2), and e (Euler's number).
To determine if a number is rational or irrational, here are two approaches:
1. Fraction Form: If the number can be expressed as a fraction, it is rational. For instance, if we take the example of 2/3, which is a rational number, you can check by calculating whether the number terminates or repeats when expressed as a decimal. In this case, 2 divided by 3 gives 0.666666... which repeats the digit 6 infinitely. Therefore, 2/3 is a rational number.
2. Irrationality Test: If the number cannot be expressed as a fraction and doesn't terminate or repeat, it is irrational. Consider the square root of 2 (√2) as an example. When expressed as a decimal, √2 is approximately 1.4142135623730950488016887242097... The decimal digits do not terminate or repeat, so √2 is an irrational number.
By following the above methods, you can determine whether a number is rational or irrational.