A number line is generally a piece of straight line with a starting point, labeled 0, and equally spaced points labeled 1 and 2 and 3 and 4 and so on till the paper runs out.

The value of a number is the distance from the zero point to the numbered point, in units of the equal spacing.

It is really much easier to draw one of these !

### Addition

Two parallel number lines, same scale.

Notice that the zero points do not have to be in the same vertical line.

### Subtraction

To get the symbolic form 7 – 2 = 5 we start with 0 on the target line (now the upper line) and join it to the 2 on the subtrahend line. (arrow down) (needs a nicer word here)

Then from the 7 on the subtrahend line we produce the line from 7 parallel to the 0 to 2 line. Then “arrow up” to the target line

Magic ! The result is 5 on the target line.

I like the picture, but the subtraction words are a mess.

### Multiplication

We need two number lines, but since multiplication is “proportional” they will now be crossing, and the common point is labeled 0.

Also, the labels are “target” and “multiplier” and each line has its own scale.

### Bonus: Nomograms, with lines.

The first is a simple calculator, with A + B = Sum

The second one calculates parallel resistances

The multiplication is a mind-catching one. I wonder how long it takes people to be convinced that the angle of the second line isn’t relevant to how it works.

It’s not only the angle, it’s the arbitrary (but equal) spacing of the points on the multiplier line as well. I have more to say on this later.