WaywardHouse

5  Numbers and the number line

Placing any number exactly

A bank balance of −£40 is not the same as no money — it is a real position, 40 units to the left of zero. An altitude of −85 metres is the Dead Sea shore. A temperature of −18°C is your freezer.

Numbers below zero are not a trick. They are the language of anything that can be in deficit, in debt, below a reference point. The number line is the map that shows where every number lives — and why the rules of arithmetic have the shape they do.

5.1 What the notation is saying

Draw a horizontal line. Mark a point in the middle and call it 0. Every positive number lives to the right; every negative number lives to the left. The further from zero, the larger the absolute value.

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

Absolute value \(|x|\) strips the direction: \(|{-3}| = 3\) and \(|3| = 3\). It answers “how far from zero?” — not “which side?”

The inequality symbols \(<\) and \(>\) describe position on the line: \(-5 < -1\) because \(-5\) sits further left. This trips people up — \(-5\) feels “bigger” in magnitude, but it is smaller as a number.

Fractions and decimals also live on this line. \(\frac{1}{2}\) sits halfway between 0 and 1. \(-0.7\) sits between \(-1\) and \(0\). The line is continuous: there is a location for every number, however precise.

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

5.2 The method

Placing a number on the line

  1. Determine the sign: positive (right of zero) or negative (left) — this tells you which direction to go.
  2. Determine the magnitude: how far from zero — this tells you how far to go.
  3. Locate it. Every number has exactly one position.

Ordering numbers

To order a list of numbers, locate each one on the line. Left is smaller, right is larger. For negative numbers: \(-10 < -2 < 0 < 3 < 7\).

Absolute value

The formula below looks slightly odd. Here is what it is doing: if \(x\) is already positive, its distance from zero is just \(x\). If \(x\) is negative, its distance from zero is the positive version of it — which is \(-x\). Negating a negative gives a positive.

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

So \(|{-7}| = -(-7) = 7\). We will prove in Chapter 4 why \(-(-7) = 7\).

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.

5.3 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.

As signed numbers: +3, −2, +1, −4.

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

\[-4 < -2 < 1 < 3\]

Worst to best: lost by 4, lost by 2, won by 1, won by 3.

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?

Vertical distance = \(|2450 - (-340)|\).

That subtraction involves two negatives, so pause: subtracting a negative is the same as adding.

\(|2450 - (-340)| = |2450 + 340| = 2790\) m.

The absolute value ensures we get a distance (positive), not a directed displacement.

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: 12, 3, 7, 19, 1.

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

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: 0. Receive £15: position +15. Repay debt −8: position +7. Spend £4: position +3.

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

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.

5.4 Where this goes

The number line is the foundation every later chapter assumes. Integers and signed arithmetic (Chapter 4) takes addition, subtraction, multiplication, and division into negative territory — all of which are just movements along this line. Fractions and decimals (Chapter 5) fill in the gaps between the integers.

Further on, the line becomes an axis. In Volume 2, the horizontal axis of a graph is the number line, and a linear equation traces a path along it. In Volume 5, limits are defined precisely as behaviour near a point on the line. The model you are building here does not change — it accumulates meaning.

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.

5.5 Exercises

Each problem has a clean answer. The interesting part is setting up which numbers go where before you compute.

  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?