Is It Rational?

One of the classifications into which mathematicians put numbers is that of rational numbers. Here, the word ‘rational’ has nothing to do with whether they are based on reason or logic, but rather on the fact that rational numbers are numbers that can be expressed as a ratio. So, the fundamental definition of a rational number is that a rational number is one that can be expressed as the ratio (or fraction) of two integer numbers.

The integer numbers, 1, 2, 3, 4, etc., are rational numbers because the can be expressed as the ratios 1/1, 2/1, 3/1, 4/1, and so on.

But rational number are not limited to the natural integer numbers. Decimal numbers like 1.5, 0.75, and 5.375 are also rational numbers because they can be expressed as the ratios (fractions) 3/2, 3/4, and 43/8, respectively.

Numbers that cannot be expressed as the ratio of two integers are called, as you might expect, irrational numbers. The value of the square root of 2 (), for example, is an irrational number because it cannot be expressed as the ratio of two integers. The value of the square root of 2 is where the ‘...’ means that the exact value extends for an infinite number of digits. The exact value of the square root of 2 cannot be expressed as the ratio of two integers, and therefore it is an irrational number.

The decimal value of the rational number 22/7 is 3.1428571... where again the ‘...’ means that the exact value extends for an infinite number of digits to the right. Note, then, that whether the exact value of a decimal number requires an infinite number of digits to express is not the determining factor about whether the number is rational or not. The decimal equivalent of 22/7 (3.1428571...) is a rational number because it is the ratio of two integers even though its exact value requires an infinite number of digits to express. Thus, the value of the square root of 2 is irrational because its exact value cannot be expressed as the ratio of two integers, not because it requires an infinite number of digits to express exactly.

Can you think of other numbers that you encounter in your daily activities that are rational or irrational numbers?

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