Two is Still Not a Number With any Square Root

Mini-Series On Classical Mathematics:
	(10-2)*(10-2)=64 As History of Mathematics ... Sigh ... Absolute and Relative Roman Numerals Two is Still Not a Number With any Square Root
Now, for the essay:

2 as a number is not the square of any other number. Number answering of course the question "how many" and none other.

2m² having the form of a square has however the side 1 m 414 mm.

If you want to express that fact as 1.⁴¹⁴ or 1.⁴¹⁴²¹⁴ or 1.^4142135624 being the square root of two and being a number, well no one is stopping you from misnaming things, but none of these are numbers. They are proportions. And in this expression, unlike the expression using the root sign over a two, or the expression "sqrt(2)" they are necessarily approximations.

And since these approximations are incommensurable with the measure of the diagonal - already Aristotle knew that a diagonal of either a square or a circle is incommensurable with the side, you will never get any approximation that is exact.

Beyond 1.⁴¹⁴ I used the calculator function on the computer. Does this mean I trust "science" declaring 1.^{414 ....} to be a number and declaring it to have that greatness?

No.

I trust the technology put into the calculator an algorithm for "finding square roots" - and the calculator knows nothing whether it's of a number or of a proportion, it just does it. Exactly as, with less skill available for the button push and more necessitated in manipulator, the beads of an abacus.

Now, the algorithms for finding square roots would be the same whether binary or decimal or duocimal (I tried that in order to find the side of an area 2 square feet). They come from the instances where square numbers are concerned and the algorithm has a nice finish with a zero remaining at the bottom. They are then generalised to cases where that is not the case, like two.

So, do you really need to be able to find the square root of two with so many decimals? Not for architecture at any rate.

It is easier to make a square 1m by 1m, stretch a thread to measure the diagonal, and use the thread to make a square thread length by thread length. That square is then a square of 2m².

Just imagine there are the idiots who will at any costs denigrate what the Greeks knew and what the Christians inherited about Mathematics.

They will even tell you the Greeks tried to find an exact ratio - in the sense of a proportion between two commensurables - for the diagonal of a square or diameter of a circle, and that they even when failing did not realise they were dealing with what is misnamed "irrational numbers", when the correct name irrational ratios, was in fact known by Ancient Mathematics.

Some do it out of a spite against common sense:

Remember the number line. We all thought there were no numbers below one until we were taught the number line.

Which is indeed a line, has indeed both a positive and a negative side and is indeed continuous, but which is not constituted properly speaking of numbers.

All squares that have sides are far more frequent than the square numbers that have numerical roots as in strictly whole numbers as square roots.

Some squares only will even have surfaces with a whole number of square metres. Only some of those in turn have a whole number of square metres which is also a square number in arithmetic.

What we do when we measure is not to count, but to proportionate. 2m² = 2/1 * 1m². And, unlike 2 as a number, 2/1 is a proportion.

Hans-Georg Lundahl
Public Information Library
Georges Pompidou, Paris
St Margaret of Scotland
16-XI-2012