The Numerator of a Fraction

A hook that survives the exam

Denominator begins with d, for down, and it is also the divisor. Numerator is the other one, up on top. It is a slight hook, but it is enough, and it needs to hold because the words themselves are what instructions are written in.

Multiply the numerator by 3 is unfollowable by a student who has to stop and guess which number is meant, and questions on equivalent fractions, on simplifying and on adding are all phrased this way. Getting the two names automatic is not vocabulary for its own sake; it is what makes the instruction readable.

Change the numerator and you move along the line

Draw a line from 0 to 1 and cut it into eighths. The denominator has fixed the size of the step. The numerator says how many steps to take from zero, so raising it walks you steadily to the right without changing anything about the ruler you are using.

The same line shows what an equivalent fraction is doing. 4/8 and 1/2 land on exactly the same point, so they are one number described with two different sizes of step.

• 1/8 — one step of an eighth from zero
• 3/8 — three steps, still well short of the halfway mark at 4/8
• 7/8 — seven steps, one short of the whole
• 8/8 — eight steps, which lands exactly on 1

The numerator counts, it does not measure

The 3 in 3/8 is not three of anything ordinary. It is three eighths, and its size depends entirely on the number underneath. This is the single idea that explains why comparing 3/8 with 1/2 by looking at the top numbers gives the wrong answer.

Two consequences follow. The numerator may be larger than the denominator, giving a fraction worth more than 1 — that is an improper fraction, and it is a legitimate thing to write. And the numerator may be 0, which makes the whole fraction 0, since zero eighths is nothing at all. It also follows that the numerator on its own tells you nothing about size: 3/8, 3/4 and 3/100 share a numerator of 3 and are three very different quantities.