Real numbers on the number line

Real numbers

Put the rational numbers and the irrational numbers together. We get the real numbers, written $R$. Every real number is a point on the number line, and every point on the number line is a real number. There are no gaps.

Rational numbers between two numbers

Between any two different real numbers, there are infinitely many rational numbers, and infinitely many irrational numbers. To find a rational number between $a$ and $b$, we can take the average:

Example: between $\dfrac{1}{3}$ and $\dfrac{1}{2}$, the average is $\dfrac{1}{2}\left(\dfrac{1}{3} + \dfrac{1}{2}\right) = \dfrac{1}{2} \times \dfrac{5}{6} = \dfrac{5}{12}$. We can do the same between the new numbers, again and again, so there is no end.

Locating the square root of 2

1. Mark $0$ and $1$ on the number line.
2. At the point $1$, draw a line of length $1$, perpendicular to the number line.
3. Join $0$ to the top of this line. By the Pythagoras theorem, this length is $\sqrt{1^2 + 1^2} = \sqrt{2}$.
4. With a compass at $0$, draw an arc with this radius, until it meets the number line.

The point where the arc meets the line is exactly $\sqrt{2} \approx 1.414$.

The square root spiral

We can do the same for other numbers. At the end of the hypotenuse $\sqrt{2}$, draw a line of length $1$, perpendicular to it. The new hypotenuse is $\sqrt{(\sqrt{2})^2 + 1} = \sqrt{3}$. Continue again, and we get $\sqrt{4}$, $\sqrt{5}$, and so on.

At $n = 3$ the red side is $\sqrt{4} = 2$, a whole number. Which other values of $n$ give a whole number?

Between which whole numbers?

To place $\sqrt{40}$, find the perfect squares on both sides. $6^2 = 36$ and $7^2 = 49$. So $6 < \sqrt{40} < 7$. Since $40$ is nearer to $36$ than to $49$, the root is nearer to $6$. In fact $\sqrt{40} \approx 6.32$.

Practise with new numbers

Try it in Python

This program finds the lengths of the hypotenuses in the spiral.

Remember these

• Rational and irrational numbers together make up the real numbers.
• Every real number is a point on the number line, and the other way round.
• Between two real numbers, there are infinitely many rational and irrational numbers.
• A right triangle with sides of $1$ and $1$ gives $\sqrt{2}$. A spiral gives $\sqrt{3}$, $\sqrt{4}$ and so on.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find a rational number between $2$ and $3$.

2. Between which two whole numbers is $\sqrt{30}$?

3. How long is the long side of a right triangle with sides of $1$ and $2$?

4. Find two rational numbers between $\dfrac{1}{2}$ and $1$.