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!
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!
