Why Draw Numbers as a Line? A Picture That Does Real Work
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Representing numbers as positions along a line is so familiar it looks like a convention, and it encodes substantive claims about what numbers are. It also appears to correspond to something the brain does anyway.
What the picture claims
Placing numbers on a line asserts several things at once. It asserts an order, so that any two numbers can be compared and one is to the left of the other. It asserts that the spacing is meaningful, so equal differences occupy equal distances, which distinguishes it from a mere ranking. It asserts completeness in the sense that there are no gaps, since every point on the line corresponds to a number, which is a substantial claim requiring the irrational numbers to fill the spaces the fractions leave. And it makes operations geometric, with addition becoming movement, subtraction becoming movement in the opposite direction and multiplication becoming stretching, which is what makes the picture useful rather than decorative.
What it makes easy
Several ideas that are awkward symbolically become obvious on a line:
- •Negative numbers, which are simply positions on the other side of zero
- •That subtracting a negative moves right, which follows from the geometry rather than being a rule
- •Absolute value as distance from zero, regardless of direction
- •Inequalities as regions of the line
- •Fractions as points between whole numbers rather than as pieces of something
- •Why the irrationals must exist, since a diagonal's length is a point the fractions do not reach
The mental version
Experimental work suggests something line-like operates in the mind independently of instruction. People are faster to judge which of two numbers is larger when they are far apart than when they are close, which is what a spatial representation would predict and not what looking up a fact would. People respond faster to small numbers with the left hand and large numbers with the right in several tasks, which suggests an association between number and space, and the direction of that association varies with reading direction across cultures. Young children asked to place numbers on a line space small ones far apart and large ones close together, in a pattern resembling a logarithmic scale, and shift towards even spacing as they gain experience, which several researchers interpret as a native representation being overwritten by education.
How it got extended
The line as drawn today includes numbers that took a long time to be accepted, and the resistance is instructive. Negative numbers were treated as absurd by many European mathematicians well into the seventeenth century, described as fictitious and rejected as solutions to equations, while Chinese and Indian mathematicians had used them far earlier. Zero as a number rather than a placeholder arrived through Indian mathematics and was resisted in Europe. Irrational numbers troubled Greek mathematics enough that the discovery that a square's diagonal cannot be expressed as a ratio was reportedly disturbing. In each case the objection was that the supposed number did not correspond to a quantity of anything, and in each case acceptance followed from the arithmetic working rather than from anybody resolving the philosophical worry.
Where the line stops being enough
The representation has limits and knowing them prevents confusion later. Complex numbers cannot be placed on a line at all and require a plane, with the second dimension representing multiples of the square root of minus one, which is why they appear mysterious to anybody committed to the line and straightforward to anybody who accepts the plane. Infinity is not a point on the line, and treating it as one produces most of the classic errors about it. Different sizes of infinity cannot be represented spatially at all. And the line suggests that between any two numbers there is always another, which is true, alongside the fact that the rationals are countable and the reals are not, which the picture gives no way of seeing.
The takeaway
Placing numbers along a line asserts order, meaningful spacing and completeness, and turns addition into movement and multiplication into stretching. It makes negative numbers and irrationals obvious. Experiments suggest something spatial operates in the mind anyway, with children spacing small numbers widely and shifting towards even spacing as they learn.