Understanding Irrational Numbers

In mathematics, an irrational number is any real number that cannot be written as a simple fraction of two integers (like $\frac{a}{b}$).

When you write an irrational number as a decimal, its digits go on forever without ever forming a repeating pattern.

Common Examples

• Pi ($\pi$): The ratio of a circle's circumference to its diameter. It is approximately $3.14159...$, and it never ends and never repeats.
• Imperfect Square Roots: Numbers like $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ are irrational because there is no rational number that can be multiplied by itself to produce $2, 3,$ or $5$.

It is important to note that not all square roots are irrational. For example, $\sqrt{4}$ is a rational number because $4$ is a perfect square, meaning $\sqrt{4} = 2$, which can easily be written as the fraction $\frac{2}{1}$.

Estimating Irrational Numbers

Even though irrational numbers have endless decimals, you can still estimate their value and figure out exactly where they belong on a number line.

Example: Approximate $\sqrt{10}$

1. Find the closest perfect squares below and above $10$. These are $9$ and $16$.
2. Take the square root of those numbers: $\sqrt{9} = 3$ and $\sqrt{16} = 4$.
3. Because $10$ is between $9$ and $16$, $\sqrt{10}$ must be a decimal between $3$ and $4$.
4. Since $10$ is much closer to $9$ than $16$, $\sqrt{10}$ is slightly more than $3$ (it is actually about $3.16$).

The Real Number System

Together, rational numbers (fractions, terminating decimals, repeating decimals, and integers) and irrational numbers make up the real numbers. Every single point on an infinitely long number line represents a unique real number, whether it is rational or irrational.

Practice Problems

1. Is $\sqrt{5}$ rational or irrational? It is irrational. Because $5$ is not a perfect square, its square root will have a decimal that goes on forever without repeating.

2. Explain why $\sqrt{4}$ is rational but $\sqrt{5}$ is not. $\sqrt{4}$ equals exactly $2$, which can be written as the simple fraction $\frac{2}{1}$, making it rational. $\sqrt{5}$ cannot be simplified to a whole number or a fraction of two integers, so it is irrational.