What Are Rational And Irrational Numbers?

By Manjit Singh Atwal
Rational and irrational numbers are taught in grade nine math. Students are learning rational numbers already since grade six, but irrational numbers are introduced in grade nine (in most schools). When I start teaching grade nine students, they look very confused about these two kinds of numbers. Let's take on these both types one by one.

Rational numbers:

In the real number system, rational numbers are the fractions (mainly). Any number that can be written in the form "p/q", where "p" and "q" are both integers and "q" is not equal to zero, is called the rational number. There is a letter symbol of "Q" to denote these.

For example; 2/3 and -2/3 are both the examples of rational number.

But they are not limited to fractions only. All the terminating (ending) decimals and repeating decimals are the in the this category. For example; 2.5, - 2.5, 5.009 and repeating decimals like 0.3333... and 2.666.... fall under the symbol Q.

Also, all the integers can be changed to fractions by making one as their denominator; hence all the integers such as - 5, - 4, - 3, 0, 1, 2, 3 and so on fall in this category.

Therefore rational numbers contain a variety of numbers in them. Below there are more example of rational numbers.

0, 1, -1, 2, -2, 0.56, 3.125, 3/6,-5/2, 3.22222...., 0.99999....

Irrational Numbers:

These are defined as the non-repeating and non-terminating decimals. In other words, if a decimal is not ending and numbers after decimals are not in a pattern that number is a rational number. These kinds of numbers are obtained when square root of a number (which is not a perfect square) is calculated.

For example; 3.013004751224... is an irrational number. Look at the pattern after the decimal are not in a pattern and no body can predict what is coming after last digit "4" and also this is a non-terminating decimal.

If we find the square root of number "2" using the calculator, we find a decimal which is an irrational number. Similarly the square root of number "3" falls in the same category. But be careful in case of perfect squares such as "4", as the square root of four is "2" which is a natural number and hence a rational number but not an irrational number because four is a perfect square. Similarly all other perfect squares like, 16, 25, 36, 49, 64, 81, 100, 121, 144 and so on should be cared about their category.

Similarly the square root of next perfect square "9" is "3" which is not an irrational.

I always ask my students to remember the irrational number as they are non-ending and non-repeating decimals and everything else is rational numbers.

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