Lesson 16: Irrational Numbers

New Concept

Mathematics is built upon a complete system of numbers, called the real numbers, which includes all rational and irrational values on the number line.

What’s next

Our journey begins by exploring the building blocks of this system. Next, you'll discover a fascinating type of number: the irrational numbers.

Property

An irrational number is a number that cannot be expressed as a fraction. The square root of any counting number that is not a perfect square is an irrational number.

Examples

• Which number is irrational? A: $\sqrt{25}=5$, B: $\sqrt{36}=6$, C: $\sqrt{42}$, D: $\sqrt{49}=7$. The answer is C, as 42 is not a perfect square.
• The number $\pi \approx 3.14159...$ is a famous irrational number because its digits go on forever with no repeating pattern.
• The side length of a square with an area of $3 \text{ cm}^2$ is $\sqrt{3}$ cm, which is an irrational number.

Explanation

Think of irrational numbers as the wild ones of the number world! Unlike their rational cousins, their decimal forms go on forever without ever repeating a pattern. You can't pin them down as a simple fraction, which is why your calculator screen fills up with digits when you try to find the value of $\sqrt{2}$!

Property

Rational numbers and irrational numbers together form the set of real numbers. All of the real numbers can be represented by points on a number line, and all of the points on a number line represent real numbers.

Examples

• The set $\{ -5, 0, \frac{3}{4}, \sqrt{2}, 2.5, \pi \}$ contains a mix of rational and irrational numbers, all of which are real numbers.
• Arrange these real numbers in order from least to greatest: $1, \sqrt{3}, \frac{1}{2}, 0, 2$. The correct order is $0, \frac{1}{2}, 1, \sqrt{3}, 2$.
• The number $-2$ is a real number; it is also a member of the rational numbers and the integers.

Explanation

Imagine the number line is a giant party. The rational numbers are the organized guests who stand in neat lines (like fractions), while the irrational numbers fill in all the weird gaps in between. Together, they make up the entire crowd, leaving no empty spaces. This complete set, with everyone included, is the real numbers!

Property

To determine which two counting numbers a square root lies between, find the perfect squares immediately below and above the number inside the root.

Examples

• To estimate $\sqrt{108}$, you can see that $10^2=100$ and $11^2=121$. Therefore, $\sqrt{108}$ is between the counting numbers 10 and 11.
• Which two counting numbers is $\sqrt{30}$ between? Since $5^2 = 25$ and $6^2 = 36$, $\sqrt{30}$ must be between 5 and 6.
• To find which number is between 3 and 4 (A: $\sqrt{8}$, B: $\sqrt{9}$, C: $\sqrt{15}$), the answer is C, since $3=\sqrt{9}$ and $4=\sqrt{16}$.

Explanation

Think of it as a number sandwich! To locate a tricky square root like $\sqrt{108}$, you just need to find the perfect square 'bread slices' on either side. Since you know that $\sqrt{100}=10$ and $\sqrt{121}=11$, you can be sure that $\sqrt{108}$ is happily squished right between 10 and 11. It's that easy!