Why Did I Learn This Only Recently?

It arose during a radio discussion on the nature of the Paradox - first examined by Classical Greek philosophers.

Given any true square of integer side length, its diagonal can never be of integer length.

Conversely, if the diagonal of a square is of integer length, the side length can never be an integer.

How? By Pythagoras' The square of the length D of the diagonal of a rectangle is the sum of the squares of two of the sides's Lengths.

So for a true square of side L,

D = √(2.L^2)

= √2.√L

So D = 1.414 L, to 3 decimal places. Irrespective of L or the units.

Despite having realised that equation a long time ago in connection with the maximum size of a square on the end of a shaft (e.g. for a spanner), I had not twigged the general non-integer law!

It works for a rectangle but only if the sides are in the ratio 3:4. The diagonal is then 5. This has for centuries, perhaps two or three millenia, been a very useful, practical rule for setting out large right-angles such as in building work. Still is.

............

Here's another quirk I did learn some time ago when I found that given a Circle of Diameter D,
its area A = D^2( π /4).

You can derive that easily from the standard formula A = π R^2.

Now look at a standard calculator keyboard.

The top corner keys form a neat clockwise square, 7854.

What is quarter pi? The same digits but with a decimal point: 0.7854.

Pure fluke, but making using a calculator physically quicker and easier if you know only the circle's diameter!

For A then = [0.7854 D^2]

..........

An example of Paradox that was quoted in the programme is that of the athlete and the tortoise having a sprint race.

The athlete gives the tortoise a generous head-start, as he is obviously far faster. Yet apparently never overtakes the animal, only draws ever more closely level with it. Why?

Suppose the head-start is 100 metres. When the human has reached the 100m post the tortoise, plodding away with all his might, is now 10m ahead.

So the athlete carries on, reaches the 110m point... the tortoise is 1m ahead...

At 111m, 100mm ahead...

Etc. A diminishing Series; in that example, dividing by 10 each time.

This comes into the sum of the series [ 1 + 1/2 + 1/4 + 1/8 + 1/...] Each term half the preceding. The answer approaches 2, of course, but never is 2, and never exceeds 2.

And forms the principle of the Infinitesimal Calculus, discovered independently but roughly contemporansously by Leibnitz and Newton. Differentiation uses a law of ever-closer values approaching but never reaching 0 (you cannot divide into or by 0) to identify a point on what if plotted would form a curve. Integration arithmetically sums the areas of an infinite series of infinitely narrow "slices" bounded by the curve, to determine the total bounded area.

I never understood Calculus when it was part of my Maths syllabus at school, for the GCE Ordinary-Level Mathematics course.

Then years later, a geology-club lecture about river gradients presented us with a simple formula, of one value over the difference between two others. Something made me use calculus notation when I copied it.

Eureka! This was a very basic Differentiation to find a spot gradient!

Here on something measureable, a river, which has long stretches of steadily diminishing gradient!

.....

Logarithms? I became used to using base-10 logs and slide-rules as arithmetical tools, at school before the advent of the electronic calculator. I still could, with a bit of revision. Understanding what logarithms are though, is a different matter. They are important in many scientific and engineering laws, but it was not until I had to use them decades later, at work, that I twigged what they are!

.......

The one thing I have still to grasp fully and should have done years ago, also in my work, is the concept of the Newton, the standard unit of force used in all sorts of engineering.

I know Pressure is (Force/Unit-area), and that its official unit is the Pascal, = 1N/m^2). So is so tiny it is a right silly little so-and-so for any day-to-day pressure use like tyres. It is the unit of Force, the Newton itself, I find a bit awkward.

There is a practical example in Ninalanyon's blog, where he describes trying to make an apple-press using a car-jack. He quotes the values in N/m^2 though, not Pa. Perhaps he likes the slight alliteration with his nick-name!

For ordinary use the Bar (= 1 X 10^5 Pa) is allowed. That is near-enough mean sea-level atmospheric pressure at 20ºC; and near enough for most purposes, the pressure of a 10m head of water.

For sound measurements the Pascal is a million times too big!

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JoyfulSilence · 51-55, M

Maths is awesome.

About the square stuff. It is because the square root of two is irrational (rational = ratio of integers). So if both the diagonal and sides are rational lengths, then one can put sqrt 2 on one side of the equation and a rational number on the other, so sqrt 2 is rational, a contradiction. So the diagonal and side lengths cannot both be rational. Or integers, too.

ArishMell · 70-79, M

@JoyfulSilence Ooh! Thankyou for the deeper explanation!

There have been occasions when I wanted to use the formula for practical purposes, but had not spotted the underlying law.

JoyfulSilence · 51-55, M

@ArishMell

One can also use a proof by contradiction to show the square root of 2 is irrational.

Assume sqrt(2) = a/b where a and b are positive integers, and the greatest common factor of a and b is 1. That is, a/b is "reduced" to its simplest form (we cannot divide by any common factor to get lower integers).

Square both sides:
2 = a^2 / b^2
Rearrange:
a^2 = 2 * b^2

So a^2 is even. Then a must be even

(since if a were odd then a=2*n+1 for some integer n so a^2 = 4*n^2 + 4*n + 1 which is odd).

Since a is even, then
a = 2n for some integer n. So
4*n^2 = a^2 = 2 * b^2
Divide by 2:
2*n^2 = b^2

So b^2 is even, hence b is even (by the same argument we made above for a).

Hence both a and b are even. But this means a/b is not in reduced form since they have common factor 2. So we have a contradiction. Hence the initial assumption that sqrt 2 is rational is false, hence sqrt 2 is irrational.

ArishMell · 70-79, M

@JoyfulSilence I must admit that really started to stretch my abilities with algebra - hence showing why I took so long, and other ways to see them, to understand the things I first listed.

It is an elegant explanation. I suppose like many people I have tended always to think of mathematics as quite rigid, immune from anything not absolutely logical; so the notion of using paradoxes and contradictions to see how some of it works is not exactly instinctive!

Maybe it's how we were taught it - as a series of bland rules and instructions often with no real-world purpose. Do this, then that, then... and it works. Errr, what works? How does it work? Since when has A called itself B?

It was using fairly basic maths at work or in my interests that helped me see that property of the square, or what logarithms and differentiation do. I need a concrete anchor for them, rather than abstract concepts, to understand them.

Indeed, it was only fairly recently that I began to see Algebra for what it is at a functional level:

Firstly as a shorthand way to write a string of arithmetic instructions for particular tasks, despite having often used formulae of various types.

Secondly as a shorthand way to describe arithmetic generally; and from that realised you can work it back to analyse the algebra itself. Change all those mysterious letters into simple numbers like 2 and 3, and it illustrates what the algebra is doing, or shows if you have operated it correctly.

(A course I took did include a brief look at Dimensional Analysis, as the means to make sure a physics or engineering equation mixing real values like length and mass, will work correctly; or at a higher level, to determine the result's units.)

On the other hand when I took a standard maths course, as an adult-education student but ending in taking the national school-level examination, I was totally flummoxed by a topic entirely new to me.

This was Matrices: boxes of small numbers to add or multiply in particular orders for no particular purposes; and given fancy but meaningless names. All so utterly abstract, with no given link to any other mathematics let alone any practical use, that I failed completely to understand them.

Luckily, though surprisingly, I don't think there was a Matrix question in the exam. - which I passed with high marks because the rest of the course had been revision for me.

I discovered matrices are an old mathematical concept with centuries of history, but which has only fairly recently found anything useful to do... in special technical applications such as Finite Element Analysis in critical engineering designing, certain scientific fields, and top-level areas of computer programming. So nor do I understand why they are introduced in an ordinary school maths syllabus.

JoyfulSilence · 51-55, M

@ArishMell

Proof by contradiction is a pretty standard logical method, and quite useful for complicated things where a clear path is not evident.

For the following, let a "statement" be a variable that is either true (T) or false (F).

Then to show that proof by contradiction works, suppose we have a statement A and any statement B. We want to show A is T.

So we assume A is F, then show B is both T and F, a contradiction. Hence our assumption (A is F) is wrong, so A is T.

Saying this method is valid is the same as saying the statement
"A"
is logically the same as the statement:
"C",
where C is
"(not A) => (B and not B)"

We can case it out:
First, note that
"B and not B" is clearly always false. B is either true or false but not both at the same time!

Here is a truth table of all cases:
A___B___C__________
T___T___(not T) => F
T___F___(not T) => F
F___T___(not F) => F
F___F___(not F) => F

which becomes
A___B___C__________
T___T___F => F
T___F___F => F
F___T___T => F
F___F___T => F

which becomes
A___B___C__________
T___T___T
T___F___T
F___T___F
F___F___F

So A is logically the same as C.

As for matrices, is a convenient way to represent the algebra/arithmetic of a system of m linear equations with n unknowns
x1,...,xn

a(1,1)*x1 + ...+ a(1,n)*xn = b1
....
a(m,1)*x1+ ... + a(m,n)*xn= bm

The a's form an m by n matrix A.
The x's and b's form column vectors with n rows.

In matrix form this is:
A*x = b.

If you were given A and x first, and used the rules of "matrix multiplication" to find A*x, you would get the left hand side of all the equations above.

But this no accident. Matrix multiplication is explicitly defined to make this happen.

Often when you get things in matrix form algebra and proof can be easier. There are some theorems and principles of matrices and matrix algebra that can be used to solve problems, which are not readily apparent if you have to deal with all the expanded equations.

So much of maths, statistics, physics, etc., is expressed in terms of matrix algebra.

It is just a tool to represent complex systems of normal arithmetic in more compact form. Yet when it comes to using them a computer still needs to break it up and do all the normal arithmetic.