Lesson 03 · Cutting the unit
A fraction isn’t a slice of pizza and it isn’t the symbol you write down. It’s a point on the number line — one you build by cutting the unit into equal parts and laying the new piece end to end, exactly the way you laid down the unit to make the whole numbers.
“A fraction is not a symbol but, rather, a point on the number line.”— Hung-Hsi Wu
Definition · H. Wu
The fractions are the points on the number line defined in the following manner: Fixing a whole number n > 0, we divide the unit segment into n parts of equal length. Then the first division point to the right of 0 will be denoted by 1n. The multiples of 1n then form an equi-spaced sequence associated with n. The totality of all the points in these sequences as n runs through 1, 2, 3, … is by definition the collection of all the fractions. — Understanding Numbers in Elementary School Mathematics, §12.2 (quoted with permission)
Watch what the picture actually says. To build 23 you cut the unit segment [0, 1] into 3 equal parts, take the first division point as your new unit 13, and lay 2 copies of it end to end. Where you land is the fraction. No pizza, no rule to memorize — just a point you constructed.
In Lesson 01 you made the whole numbers by fixing the unit 1 and laying down its multiples: 1, 2, 3… Here, you first divide the unit segment into n equal parts to get the new piece 13 — and then its multiples are to fractions what the multiples of 1 are to whole numbers. The division is the new step; the multiplying is the old one.
“Fractions are… points constructed on the number line in a specific way.”— Hung-Hsi Wu
And because each copy is a length of 1n, stamping is just the concatenation you met in Lesson 02 — lengths laid end to end and measured. The fraction is the total length you reach.
“mn is the length of the concatenation of m segments each of length 1n.”— Hung-Hsi Wu, fractions as length
Because a fraction is a point, a single point can wear many names. The point 1 is also 22, 33, 44; the point 2 is 42 and 63. Concatenate the copies in the tool until you land exactly on 1 or 2 and you’ll see it. Set the parts to 1 and the fractions become the whole numbers again: m1 = m. The whole numbers never left; they were fractions all along.
We choose to cut the unit into n equal parts, and we count copies to the right of 0 — the same direction and same unit we fixed back in Lesson 01.
ForcedOnce the unit is cut into n equal parts, the piece 1n is pinned down, and so is every mn — each is one exact point, reachable in exactly one way.
Not every point on the line is a fraction. Cut and concatenate all you like and you will still miss some points entirely — the length of the diagonal of a unit square, for instance, is a real point on the line that no mn ever lands on. Fractions are a specific, constructed family of points, which is exactly why pinning down what they are matters before doing anything with them. That precise starting point is the whole idea behind correctmath.io.