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Mathematics · Chapter 2: Real Numbers
The Real Number Line
Every real number on the number line: ordering, locating √2, inequalities with open and filled dots, and absolute value.
Key termRememberExampleWatch out
01Concept
Draw a straight line, mark a point 0 (the origin), choose a length for 1 unit, and mark the units both ways. That is the number line. Positive numbers go to the right, negative numbers to the left.
Here is the big idea: every real number has exactly one point on the line, and every point on the line is exactly one real number. That is why it is called the real number line.
Placing rational numbers
- ½ is halfway between 0 and 1.
- −2.5 is halfway between −3 and −2.
- 7/3 = 2⅓: divide the gap from 2 to 3 into three equal parts and take the first mark.
Placing √2 exactly
We cannot write √2 as a fraction, but we can still locate it exactly using Pythagoras' theorem and a compass.
Order on the number line
A number to the right is greater than a number to its left. −1 > −5 because −1 is to the right of −5.
| Symbol | Meaning | Example |
|---|---|---|
| < | less than | −3 < 2 |
| > | greater than | √2 > 1.4 |
| ≤ | less than or equal to | x ≤ 5 |
| ≥ | greater than or equal to | marks ≥ 33 to pass |
Showing inequalities on the line
Distance from zero: absolute value
The absolute value |x| is the distance of x from 0 on the number line. Distance is never negative, so |5| = 5 and |−5| = 5.
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Real World Examples
Thermometer
A thermometer is a number line standing up: 0 °C in the middle, positive temperatures above and negative below.
Lift buttons
In a Karachi shopping mall, the lift may show B2, B1, G, 1, 2: basement floors work like negative numbers.
Passing marks
"Marks ≥ 33" to pass a board exam is an inequality you can draw on a number line.
04Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Arrange in ascending order: −2, √3, 1.7, −2.5, 0.Show answer
Answer
−2.5, −2, 0, 1.7, √3 (since √3 ≈ 1.732).
Q2.Explain how to show √2 on the number line.Show answer
Answer
- Draw a square of side 1 unit on the number line, with one corner at 0.
- Its diagonal is √(1² + 1²) = √2.
- Put the compass point at 0 and open it to the far end of the diagonal.
- Draw an arc down to the number line. It cuts the line at √2.
Q3.Find |−7| and |3 − 10|.Show answer
Answer
|−7| = 7; |3 − 10| = |−7| = 7.
Q4.Draw the inequality −2 < x ≤ 4 on a number line (describe it).Show answer
Answer
An open dot at −2, a filled dot at 4, and a line joining them.
Q5.Name two numbers between 1/3 and 1/2.Show answer
Answer
For example 0.4 and 0.45. (1/3 ≈ 0.333, 1/2 = 0.5)
Long questions
Q6.What is meant by the real number line? Explain why it has no gaps.Show answer
Answer
The real number line is a line on which every real number has exactly one point, and every point stands for exactly one real number.
It has no gaps because between any two real numbers there are infinitely many more, rational and irrational, so no point is left without a number.