Tuesday, 18 February 2020

Fun with numbers - follow up

Statement from my earlier post "The relative length of the height is simply N x 2 'units'."

Bayard's comment: "I can't see from where you derive that."
Easy enough, it's a bog standard quadratic/algebra slog.

You are told, for example, that the height of a right triangle is one seventh of the total perimeter.

Start off by writing "n = 7" and "N = 6" as before.

As we are looking at relative values, not absolute values, you can ascribe any value you like to the height to get the ball rolling.

It's easiest starting with height = 1. Therefore, the total length of base + hypotenuse = 6. The base is shorter than the hypotenuse, so as a first approximation, the ratios are 1 to 3-minus-a-bit to 3-plus-the-same-bit. We express these as 1; 3 - x; and 3 + x.

Applying Pythagoras, 1^2 + (3-x)^2 = (3+x)^2.

Expand those to get 1 + 9 - 6x + x^2 = 9 + 6x + x^2.

Subtract the right hand side from the left hand side, shift the -12x over, change the sign and you end up with 1 = 12x.

So you have height = 1; base = 2 11/12 and hypotenuse = 3 1/12.

Then express everything in terms of 1/12's.

Height = 12/12; base = 35/12 and hypotenuse = 37/12.

We are looking at relative not absolute values, so we can define our "unit" as 1/12 (or "x", which we now know is 1/12), and we express the relative lengths as:

Height = 12 units; base = 35 units; and hypotenuse = 37 units.
Our "unit" must come out as 1/(2N), so the height, which we somewhat arbitrarily set as 1 to start with, must come out as 2N units...

In the above example, x = 1/12 = one unit. The height = 1/(1/12) = 12 units.

The 12 will always be 2N. The "3" in (3 - x ) and in (3 + x) is simply N/2 (it must be, that's how you set up the equation).

When you square (3 - x) you get -6x (= -N, two lots of half of N). When you square (3 + x) you get +6x (=N, two lots of half of N). Subtract 6x from -6x, you get -12x, So the "12" = 2N (two lots of twice half of N).

Whatever n or N is, when you do the workings, you always end up with 2N*x = 1.

That explains why height = 2N units.

The base is 1 unit (or 1*x or 1/12) shorter and the hypotenuse is 1 unit (or 1*x or 1/12) longer than N/2 (again, by definition, that's how you set it up). Once you multiply up by 2N (to get rid of the fractions), the apparent difference is two units (or "2x" or 2/12).
Sorry to ruin the magic for you :-(