The Number Line — Mathematics Knowledge System

Overview

The number line is a straight line on which every point corresponds to a real number, with numbers increasing from left to right. It gives a visual, geometric picture of order (which number is bigger), distance (how far apart two numbers are), and operations like addition (moving right) and subtraction (moving left).

Formal Definition

Definition

The number line represents the real numbers ℝ as points on a line with a fixed origin (0), a unit length, and a direction of increase. The distance between two numbers a and b is given by the absolute value of their difference.

d(a,b) = |a - b|

Distance on the number line

Properties

Order

a < b \iff a \text{ lies to the left of } b

Distance

d(a,b) = |a-b|

Worked Examples

Take the absolute value of the difference.
$d(-3,5) = |-3-5| = |-8| = 8$

Answer: 8

Practice Problems

Difficulty 1/10

What is the distance between -7 and -2 on the number line?

Common Mistakes

Common Mistake

Computing distance as a - b without taking the absolute value, giving a negative 'distance.'

Distance is always non-negative: d(a,b) = |a - b|, regardless of which number is larger.

Summary

The number line represents real numbers as points on a line, ordered left to right.
Distance between two points a and b is |a - b|.
It gives a visual way to understand order, magnitude, and arithmetic operations.

References

WebsiteWikipedia — Number line