Lesson 2: Square Roots of Non-square Integers

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include $\pi$ and the square roots of numbers that are not perfect squares.

Examples

• The number $\pi$ is a famous irrational number, beginning with $3.14159...$ and continuing infinitely without repetition.
• The square root of 3, $\sqrt{3}$, is irrational because 3 is not a perfect square. Its decimal form is $1.7320508...$.
• A decimal that continues without a pattern, such as $67.121231234...$, is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.

Property

To estimate a square root between two consecutive whole numbers, first locate the number between two consecutive perfect squares.
Its square root will then lie between the square roots of those perfect squares.

Examples

• To estimate $\sqrt{20}$, we know that 16 and 25 are the closest perfect squares. Since $16 < 20 < 25$, we can say that $4 < \sqrt{20} < 5$.
• To estimate $\sqrt{90}$, we look for perfect squares near 90. Since $81 < 90 < 100$, the estimate is $9 < \sqrt{90} < 10$.
• To estimate $\sqrt{150}$, we see that $144 < 150 < 169$. Therefore, we know that $12 < \sqrt{150} < 13$.

Explanation

This method works because as numbers get bigger, their square roots also get bigger. By finding the two perfect square neighbors for your number, you can trap its square root between two whole numbers, giving you a great estimate.