Lesson 1.8: The Real Numbers

New Concept

This lesson introduces the full spectrum of real numbers. You will learn to simplify square roots, classify numbers as rational or irrational, and accurately place fractions and decimals on the number line, building a complete map of our number system.

What’s next

Now, let's put these ideas into action. You'll start with interactive cards on simplifying square roots and then move on to classifying different number types.

Property

If $n^2 = m$, then $m$ is the square of $n$. When a number $n$ is multiplied by itself, we write $n^2$. The result is a positive number, whether $n$ is positive or negative. For example, $8^2 = 64$ and $(-8)^2 = 64$. Numbers that are squares of counting numbers are called perfect square numbers.

Examples

• The square of 9 is $9^2 = 9 \times 9 = 81$.

• The square of -6 is $(-6)^2 = (-6) \times (-6) = 36$.

Property

If $n^2 = m$, then $n$ is a square root of $m$. The radical sign, $\sqrt{m}$, denotes the positive square root, called the principal square root.

Square Root Notation
$\sqrt{m}$ is read 'the square root of $m$'
If $m = n^2$, then $\sqrt{m} = n$, for $n \ge 0$.
The square root of $m$, $\sqrt{m}$, is the positive number whose square is $m$.

Examples

• To find $\sqrt{49}$, we ask 'what positive number squared equals 49?'. Since $7^2 = 49$, $\sqrt{49} = 7$.

Property

A rational number is a number that can be written as the ratio of two integers, $\frac{a}{b}$ where $b \neq 0$. Its decimal form either stops or repeats. An irrational number cannot be written as a ratio of two integers, and its decimal form never stops and never repeats. Together, rational and irrational numbers make up the real numbers.

Examples

• The numbers $5$, $-\frac{3}{8}$, and $2.75$ are rational because they can be written as $\frac{5}{1}$, $-\frac{3}{8}$, and $\frac{275}{100}$. The number $0.333...$ is also rational because its decimal repeats.

• The number $\sqrt{50}$ is irrational because 50 is not a perfect square, so its decimal form goes on forever without repeating.

Property

To locate a fraction on the number line: First, locate the integers. For a proper fraction like $\frac{a}{b}$, divide the interval between 0 and 1 (or 0 and -1) into $b$ equal parts and plot the point at the $a$-th mark. For an improper fraction, convert it to a mixed number first to find which two integers it lies between.

Examples

• To locate $\frac{3}{5}$, divide the space between 0 and 1 into 5 equal parts and mark the third part from 0.

• To locate $-\frac{1}{4}$, divide the space between 0 and -1 into 4 equal parts and mark the first part to the left of 0.

Property

To locate a decimal on the number line, use its place value. The decimal $0.4$ is equivalent to $\frac{4}{10}$, so it is located between 0 and 1. Divide the interval between 0 and 1 into 10 equal parts (tenths) and mark the fourth one. For a decimal like $-0.74$, which is $-\frac{74}{100}$, you would divide the interval between 0 and -1 into 100 parts.

Examples

• To locate $0.8$ on the number line, divide the segment from 0 to 1 into 10 equal parts and mark the 8th tick mark.

• To locate $-1.5$, find the point exactly halfway between -1 and -2.