Rational and Irrational Numbers

Hello, and welcome to this video about rational and irrational numbers, which are key components of the real number system.

Today, we’re going to explore these two subsets of real numbers that you encounter every day, often without even realizing it. Rational numbers are the familiar ones you use in daily life, like 10 dollars or ¾ of a cup. Then, there are the irrational numbers—less obvious but equally fascinating, like the square root of 2 or the constant \(\pi\).

What are Real Numbers?

This Venn diagram is a visual representation of how real numbers are classified.

You can see that rational numbers include natural numbers, whole numbers, and integers. Natural numbers comprise the smallest subset, which is also known as the set of “counting” numbers. These are all positive, non-decimal values starting at one. Whole numbers encompass all natural numbers, with the addition of zero. Integers are whole numbers and their additive inverses (negatives).

What are Rational Numbers?

Rational numbers include all of the sets seen here in addition to some values in between.

An easy way to remember this is that the word ratio is in the name of this classification. All numbers included in the rational number set can be written as a ratio of integers.

Rational numbers are any numbers that can be written as \(\frac{a}{b}\), as long as \(a\) and \(b\) are integers and \(b≠0\).

For example, the integer 3 could be represented as the fractions \(\frac{3}{1}\), \(\frac{6}{2}\), or even \(\frac{-24}{-8}\).

The integer 0 could be represented as the fractions \(\frac{0}{3}\), \(\frac{0}{-2}\), or \(\frac{0}{123}\).

Fractions can also be written as decimals. For example, \(\frac{13}{100}\) is equal to 0.13 because the 3 is in the hundredths decimal place and the 1 is in the tenths decimal place. This is an example of a terminating decimal.

Other decimals have repeating patterns. These are considered rational because they can be expressed as a fraction. For example, the repeating decimal \(2.1 \overline7\) represents the digits 2.17171717… and so on. This can be represented as the fraction \(\frac{215}{99}\).

Irrational Number Examples

It is important to note that not all decimals are repeating. Some decimals have an infinite number of non-repeating digits. These types of real numbers cannot be expressed as a ratio of integers and are therefore classified as irrational.

While there are an infinite number of irrational numbers in the real number system, those most commonly used in mathematics are the square roots of non-perfect squares, like the \(\sqrt2\), for example, and the constants \(\pi\) and Euler’s number (\(e\)). The notation for irrational numbers allows for efficiency in mathematical applications.

Let’s take a look at some example problems involving rational and irrational numbers.

Determine if the following numbers are rational or irrational and explain your reasoning.

1. \(-2\)
2. \(0\)
3. \(\sqrt 7\)
4. \(\frac{1}{3}\)

(A) is rational because -2 is an integer, which is a subset of rational numbers. (B) is rational because 0 is a whole number, which is also a subset of rational numbers. (C) is irrational because the square root of 7 is approximately equal to 2.6457513, but is an infinitely non-repeating decimal. (D) is rational because \(\frac{1}{3}\) is equal to \(0.\overline3\). Any repeating decimal or number that can be written as a fraction of integers is a rational number.

Here’s another example: List 3 rational numbers between 3 and 4. For this example, let’s stick with decimals.

We can keep it simple and do 3.25, 3.5, and 3.75, but in reality any terminating decimal like 3.58 or 3.987 would work in this scenario. Remember, any number between 3 and 4 that can be written as a fraction would also be correct.

All right, that’s all for this review. Thanks for watching, and happy studying!

Frequently Asked Questions

Rational and Irrational Number Practice Questions