4  Numbers and the number line

Placing any number exactly

A bank balance of −$40 is not the same as having no money. It tells you that you are 40 dollars below zero. An altitude of −85 metres means 85 metres below sea level. A temperature of −18°C means 18 degrees below zero.

In each case, the number does two jobs at once. It tells you which side of zero you are on, and it tells you how far from zero you are. The number line gives us a clear picture of both ideas. In this chapter, we use that picture to place numbers, compare them, and understand what their symbols mean.

4.1 What this chapter helps you do

Symbols to keep handy

These are the bits of notation you'll see a lot. If a line of symbols feels like a fence, read it out loud once, then keep going.

  • −: negative, or the opposite direction from positive

  • |x|: the absolute value of x — how far from zero, ignoring direction

Here is the main move this chapter is making, in plain terms. You do not need to be fast. You just need to keep the thread.

  • Coming in: You can count. You know that 7 is more than 3, and that you can add, subtract, multiply, and divide whole numbers.

  • Leaving with: Every number — whole, fractional, negative, irrational — has exactly one location on a line. Understanding arithmetic means understanding how those locations relate to each other and to zero.

4.2 What the notation is saying

Start with something simple. Imagine a straight road with a marker at the centre. Label that marker 0. Now think of every number as a place on that road.

Numbers greater than zero go to the right. Numbers less than zero go to the left. A number’s position tells you its value.

\cdots \quad -4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4 \quad \cdots

This picture is called the number line. It helps us compare numbers by location. A number further left is smaller. A number further right is larger.

For example, -5 < -1 because -5 is further left on the line. This means that even though 5 is a greater distance from zero than 1, -5 is still the smaller number.

Fractions and decimals belong on the same line. For example, \frac{1}{2} sits halfway between 0 and 1, and -0.7 sits between -1 and 0. This means the number line is not just for whole numbers. It has a place for every number we will use.

Now we can introduce one more idea: distance from zero. Sometimes we care about where a number is. Sometimes we care only about how far it is from zero.

The symbol |x| means the absolute value of x. It tells us the distance from zero and ignores direction.

For example, |{-3}| = 3 and |3| = 3.

This means that both -3 and 3 are the same distance from zero, even though they are on opposite sides.

The symbols < and > are called inequality symbols. They compare positions on the line. The symbol < means “is less than,” and the symbol > means “is greater than.”

So when we write -5 < -1, we are saying that -5 is to the left of -1. This means -5 is the smaller number.

Use the slider below to place any number on the line. The arrow shows its distance from zero — that distance is its absolute value.

4.3 The method

Placing a number on the line

Suppose you want to place a number exactly. There is a simple method.

  1. Determine the sign. If the number is positive, move right from zero. If the number is negative, move left from zero.
  2. Determine the magnitude. The magnitude is the size of the number without its direction, so it tells you how far to move.
  3. Mark the position. Each number has one exact place on the line.

This means that sign gives direction, and magnitude gives distance.

Ordering numbers

To order numbers, place them on the line and read from left to right. Left means smaller. Right means larger.

For example:

-10 < -2 < 0 < 3 < 7

This means -10 is the smallest number in the list, and 7 is the largest.

Absolute value

We have already described absolute value as distance from zero. The formula below says the same thing in a precise way.

|x| = x if x \geq 0; |x| = -x if x < 0.

Here, x means any number. The symbol \geq 0 means “greater than or equal to zero.” So if x is already zero or positive, its absolute value is just x. If x is negative, we use -x to turn it into the matching positive distance.

For example,

|{-7}| = -(-7) = 7

This means that the number -7 is 7 units from zero.

We will justify ideas like -(-7) = 7 more fully in Chapter 4, when we study signed arithmetic in detail.

Why this works

“Negative” means “opposite direction from positive” — not “less real” or “doesn’t exist.” Absolute value removes direction because sometimes you only care about size (distance, error) not which way. When you multiply two negatives to get a positive, it is because reversing direction twice returns you to where you started.

4.4 Worked examples

Example 1 — Sport. A football team’s goal difference across four matches: won by 3, lost by 2, won by 1, lost by 4. Express these as signed numbers and order them from worst to best.

First translate the results into signed numbers:

+3, −2, +1, −4

Place each on the number line and read left to right:

-4 < -2 < 1 < 3

So the order from worst to best is:

lost by 4, lost by 2, won by 1, won by 3.

This means more negative results are worse, and more positive results are better.

Example 2 — Geography. A group records these elevations relative to sea level: mountain peak +2,450 m; valley floor −60 m; coastal town +8 m; lake bed −340 m. What is the vertical distance from the lake bed to the peak?

We want the distance between two positions on the number line: 2450 and −340.

Set up the calculation:

|2450 - (-340)|

Now simplify step by step. Subtracting a negative is the same as adding the matching positive:

|2450 - (-340)| = |2450 + 340| = 2790

So the vertical distance is 2790 m.

The absolute value bars show that we want a distance, which must be positive.

Example 3 — Weather. Five cities report overnight temperatures: −12°C, +3°C, −7°C, −19°C, +1°C. Which temperature is furthest from zero? Which is closest?

Find the absolute value of each temperature:

|{-12}| = 12,\quad |3| = 3,\quad |{-7}| = 7,\quad |{-19}| = 19,\quad |1| = 1

Furthest from zero: −19°C (absolute value 19). Closest to zero: +1°C (absolute value 1).

This means absolute value helps us compare size without worrying about direction.

Example 4 — Money. Your weekly pocket money is $15. You owe a friend $8 from last week. You spend $4 on a snack. Express your running total as signed numbers from the start of the week.

Start at 0.

Receive $15: move to +15.

Repay the $8 debt: move 8 units left to +7.

Spend $4: move 4 more units left to +3.

Each transaction is a movement along the number line. You end up at +3, meaning $3 remaining.

This means gains move you right and costs move you left.

The interactive below lets you explore addition and subtraction as movement along the line. Set a starting value A, choose an operation, and adjust B to see where you land.

4.5 Where this goes

This chapter gives us the picture that the rest of the volume uses. We began by placing numbers relative to zero. Next, we will use the same idea in more situations.

In Integers and signed arithmetic (Chapter 4), addition and subtraction become movements left and right on this line. In Fractions and decimals (Chapter 5), we fill in the spaces between whole numbers more carefully.

Later in the series, the number line becomes an axis in a graph. So this is not a one-time picture. It is a model we will keep using and extending.

Where this shows up

  • A weather forecaster reading −18°C is placing a temperature on the number line with zero at freezing.
  • A diver tracking depth uses a signed number with zero at the surface.
  • A games scoreboard showing −5 (penalty points) is using the same idea.
  • An app showing your bank balance in red is showing you a negative number.

Same structure. Different labels.

4.6 Exercises

Each problem has a clean answer. The interesting part is setting up which numbers go where before you compute. Sketching a quick number line will help.

  1. A diver descends from the surface to −28 m, then swims to a coral shelf at −15 m, then ascends to −5 m before surfacing. List these depths in order from deepest to shallowest.

  2. Four cities report overnight temperatures: Edmonton −23°C, Vancouver +4°C, Winnipeg −31°C, Toronto −8°C. Order from coldest to warmest. What is the difference in temperature between the warmest and coldest cities?

  3. A football team’s weekly goal differences are: Mon −1, Tue +2, Wed −3, Thu +4, Fri −1. On which day was the result furthest from zero? (Use absolute values to compare.)

  4. Five friends pool money for a trip. Their balances in the group fund are: +$12, −$5, +$3, −$18, +$8. Order from least to most. Who owes the most?

  5. Place these numbers on a number line sketch and order them from least to greatest: -1.5, 2, 0, -3, 0.7, -0.5.

  6. A submarine is at −180 m. A fish swims at −45 m. A buoy floats at +2 m. What is the distance between the submarine and the buoy? Between the fish and the submarine?