The Truth of Algebra

This is a piece of fun, playing around with numbers and equations. It need not be taken seriously but has an intriguing result.

We are used to the easy truth:
consider
x + 6 = 3x
6 = 3xx
6 = 2x or 2x = 6
x = 3
simple, but it can miss a hidden mystery .

If we put what we found (x = 3) back into what we started with,
we get
3 + 6 = 3 × 3
which is:
3 + 2 × 3 = 3 × 3      (or 9 = 9)
of course! – This is how our world is,
but something lies beyond this.

When the x enters our world
we can write:
x + 2x = 3x
which gives
3x = 3x
in which x can be any number from minus infinity to plus infinity.

Of course 3 is in x.
x and 3 are now in both our equation and the unknown.
The unknown is infinite and our equation is part of it.

It need not have been three but that seems apt:
an unknown entered into our equation as a number in it, and a piece of fun.

Sue’s Birthday Bunnies

Dedicated to my friends Dick & Sue.

For her birthday little Sue
was given by her Daddy two
little bunnies in a hutch.
She said, ‘I love them! Oh so much!’
She loved them, and they loved Sue,
And they loved each other too.

Bunnies did what bunnies do,
so what a great surprise had Sue
when she peeped inside their door:
her two bunnies now were four.

‘Daddy come! Oh Daddy quick!
(Daddy by the way was Dick)
One and one have just made four!’
Daddy came and Daddy saw
that the present he’d supplied
had gone forth and multiplied.

‘Oh no!’ he said, ‘I gave you two.
Pretty soon we’ll have a zoo!
There first were two, and now two more,
It’s two and two that becomes four.’

Later talking to his neighbour
Dick said, ‘How I had to labour!
‘Sue may be bright and pretty quick,
but no good at arithmetic!’

The neighbour said,
‘Now don’t you fuss,
although I only drive a bus
I study speed and things like that
when in my driver’s seat I’m sat.
A speed of just two miles an hour
If doubled needs a bit more power.
But, and this is hard to scan,
I’ll try and do it if I can,
two miles an hour plus two again,
Is not four m.p.h.’

‘Explain!’

‘The actual sum, as I have found,
Is two miles per hour, plus two miles more
less two divided by the speed of light in miles per hour.
This argument you can’t resist.
I am a Quantum Physicist
not a poet.’

‘I thought you drove a bus!’

Dick’s other neighbour, on his way,
Stopped to pass the time of day.
‘Math and physics show us we
really need philosophy.
One and two and three and four
mean nothing if not joined to more.
They are shorthand, abstract terms,
to count the stars and sticks, and worms.’

‘And bunnies!’ spoke up little Sue.
We really must give her her due.
She knows that numbers are a tool
not bound to any other rule
than Einstein’s relativity
and Heisenberg’s uncertainty.
Schrodinger’s unhappy cat
would surely say, ‘Amen.’ to that.

The Language of the Universe?

In the beginning was the Word?

Many say the language of the universe is mathematics, but the language of mathematics is not necessarily numbers. Numbers are a shorthand for words. In quantum mechanics collapsed waveforms is the term for a relationship between particles and their waveforms, in which particles, or indeed any combination of particles (atoms, molecules, chemical and organic compounds, even you) are seen as the focus or point of action of the energy waves involved. In the same way numbers and equations are like the collapsed waveforms of the huge quantity of words that might otherwise be needed to describe them. It is a good analogy. For many mathematical concepts the number of words would be as infinite as the cosmic extent of particle waveforms.

Pi (π), the circumference of a circle divided by its diameter, is one such mathematical concept. Written as a decimal it extends to an infinite number of decimal places, of which the first thirty two are as below:

3.14159265358979323846264338327950…

Should you have any need to remember this, some time ago I came across a mnemonic for the first dozen or so places. I changed and extended it to thirty-two before getting bored. The number of letters in each word is the number at each decimal place.

Now –
I sing a scale excelling,
in mystic voice and magic spelling,
sublimest strains incarnate.

Art in its measures will reveal
an angel’s song for the carousel,
and in eternal harmonies dwell,
O!

Feel free to add more of your own.

The idea of numbers being a shorthand for words is not a difficult concept, after all without words how could we explain what numbers are to children? But there may be more to them than that. Pi is far more than the simple relationship of a circle to its diameter.

Pi

Are numbers and equations
the collapsed waveforms of words?
And is pi’s definition
the circling of the birds
round and wide above the hills?
or the volume of a drop of water from the rills
rolling down to plop into shining highland ghylls?

Then the circle of the sphere
and the rolling of a tear
when a sobbing child cries, ‘Why!’
and the Earth around the Sun in perihelion,
and the wide, wide width of tears is pi.

The quick birds’ wheeling cry,
and the crying tears of pain,
and the earth around the sun,
and the round drops in the rain,
and the signs of endless sky,
the music of the spheres,
and the circle of the years,
tell us why.

Birds circling round their prey
know the distance from their nest,
and swooping down from sky,
sharp claw and shining eye,
returning straight and high,
the circle and the swoop,
the short returning loop,
and the gather of the storm
round the centre still at rest
say more to you and I
than the radius and the circle
that are pi.