Fractions on a number line

The number line already has marks for 1, 2, 3 and so on. Now we mark fractions on it. The distance from 0 to 1 is one unit. If we divide that unit into 2 equal parts, each part is 1/2 unit long, so a line from 0 to the first mark is 1/2 unit long.

If a unit is divided into 5 equal parts, the marks between 0 and 1 are 1/5, 2/5, 3/5 and 4/5, in this order. The fifth mark is 5/5, which is the same as 1. A line from 0 to the third mark is 3/5 unit long: 3 times 1/5.

We can keep counting past 1 in the same fractional units, just as you keep counting steps. With fifths, the marks after 1 are 6/5, 7/5, 8/5 and so on, and 10/5 lands on 2. With halves, the marks go 1/2, 2/2 (which is 1), 3/2, 4/2 (which is 2), and so on. A line from 0 to 3/2 is longer than 1 unit.

How many fractions lie between 0 and 1? We can split the unit into 10 equal parts, or 100, or 1000, or even more, and every split gives new fractions. So there are endlessly many fractions between 0 and 1.