Number line

The number line is younger than it looks. Greek mathematics treated numbers and lengths as different kinds of thing, and medieval European arithmetic had no use for negative numbers at all, so there was nothing to put to the left of zero. The first clear description of the picture we now draw is in John Wallis's Treatise of Algebra, published in London in 1685. To make sense of negative quantities, Wallis imagined a man walking forward and back along a line: advance five yards and retreat eight, and he stands three yards behind where he started. The negative number was a place on the line, on the other side of the start.

Once numbers had positions, the line became the axis of coordinate geometry, the ruler against which a curve is measured. In the nineteenth century it became something more exact. Richard Dedekind, in a short book of 1872, defined the real numbers so that the line has no gaps: every point on it, including ones like √2 that no fraction reaches, is a number.

In schools the line is a teaching tool for the step between counting and calculating. Adding is a jump to the right and subtracting a jump to the left, so a child can see why 3 − 5 lands below zero. Dutch researchers at the Freudenthal Institute promoted the "empty number line", with only the marks a problem needs, as a way of doing mental arithmetic on paper. The Common Core standards in the United States ask third-graders to understand a fraction as a number on the number line — a quarter is the first of four equal parts between 0 and 1 — which is exactly what the split marks here are for.

Key properties

• Numbers increase to the right and decrease to the left; the arrowheads mean the line carries on in both directions.
• The distance between two numbers on the line is the size of their difference, |a − b|, whichever order they come in.
• Every negative number sits as far left of zero as its positive partner sits to the right: −3 is the reflection of 3 in 0.
• Between any two different fractions there is always another, for example their average, so the fractions never run out between two marks.
• Not every point is a fraction: the diagonal of a unit square, swung down onto the line with a compass, marks √2, which is irrational.
• Splitting each whole step into four equal parts makes 2/4 land on the same mark as 1/2, which is the idea of equivalent fractions in one picture.

Where they turn up

• Rulers, tape measures and thermometers are number lines with units attached, and a thermometer is where most people first meet numbers below zero.
• Timelines in history books, where the years before the common era run to the left like negative numbers.
• The x- and y-axes of every graph are two number lines crossing at zero.
• Primary-school classrooms, where a long number line along the wall is used for counting on, counting back and skip counting.
• Fraction lessons, where a line split into equal parts shows that 1/2, 2/4 and 4/8 are the same number.