Rational and Irrational Numbers

It’s finally the end of the school year, and your class is having a huge end of the year party - with free pizza 🍕! You’re waiting on a slice of pizza, and your teacher turns to you and asks, “Would you like 0.3674643... slices of pizza?”

Wait WHAT? How could the teacher possibly cut a slice that size? You can’t be given this amount of pizza because your teacher offered you an amount that is irrational.

Just like people, we have rational and irrational numbers. So, how exactly do we know when a number is rational or irrational?

What is a rational number?

Rational numbers can be written as a fraction of integers.When we divide these fractions we should end up with an integer, a terminating decimal, or a repeating decimal. Examples of rational numbers include: $- 2, 0, 416, \frac{1}{3}, 0.25, 1.676767 . . ., 9, 12, 25$

Is 0 a rational number?

0 is a rational number. It can be written as a fraction of integers where the numerator is 0 and the denominator is any non-zero integer (ex: $\frac{0}{4}$ ).

Be careful though! It’s important to remember that any number divided by zero is undefined (neither rational nor irrational). So it’s important to remember that $\frac{0}{0} \neq = 0$ .

What is an irrational number?

On the other hand, irrational numbers are a whole different story 🙄. Irrational means NOT rational, so irrational numbers cannot be written as a fraction of integers. These numbers include:

Never-ending NON-repeating decimals
Square roots of imperfect squares
Pi ( $π$ ) or numbers that contain pi when simplified. Most of us think of pi as just 3.14, but it actually goes on forever without a pattern!

Helpful Definitions

Term	Definition
What is an integer?	An integer is a whole number that can be positive, negative, or zero. Ex: ( $. . .,$ $- 3,$ $- 2,$ $- 1,$ $0,$ $1,$ $2,$ $3,$ $. . .$ )
What is a terminating decimal?	A terminating decimal is a decimal number that stops. Ex: $0.123,$ $0.25,$ $0.875$
What is a repeating decimal?	A repeating decimal is a decimal number with a digit, or group of digits, that repeats forever. A horizontal line is usually placed over the digit(s) that repeat. Ex: $\frac{1}{3} =$ $0.333 . . . =$ $0 . \overline{3}$
What is a perfect square?	A perfect square is a whole number that can be expressed as the product of two equal integers. Ex: $25 =$ $5 \times 5 =$ $5^{2}$
What is a numerator?	The numerator is the top value in a fraction. Ex: in $\frac{3}{4}$ , 3 is the numerator
What is a denominator?	The denominator is the bottom value in a fraction. Ex: in $\frac{3}{4}$ , 4 is the denominator

When thinking about the difference between rational and irrational numbers, think of characters like Chris and Andy from Parks and Rec. Chris is like a rational number: he’s straightforward, predictable, and doesn’t do anything out of the ordinary. Andy is like an irrational number: he’s crazy, all over the place, and you never know what you’re going to get! 🤪