Some Simple Sums
1. Wrestling with Numbers
Maths is notorious for inducing mental
panic – and yet it is supposed to SIMPLIFY thinking. Suppose that there are n
people in this room, to stop fights breaking out, let’s all shake hands. Let t be the time to shake hands, then if
everyone shakes hands with everyone else, then each of n people must shake hands with n-1
others, so the total time, big T, taken by this orgy of peace-making
equals n(n-1)t – or does it?
Let’s run a test for a small value of n. Suppose n=2, then the formula yields 2t,
but we know only one handshake is required. Aha, our formula has counted every
handshake from the point of view of every person- every shake has been
double-counted, so the correct formula is n(n-1)/2. Let’s tabulate this
for various n, Café Philosophiques
do attract a variable attendance -actually, rather more people than I have
included below, with our normal attendance of between 20 and 30, we would be a
long time shaking.
|
A Constant
(Unity)
|
Number of people: n/1
|
No. of shakes n(n-1)/1x2
|
No. of three- hugs
n(n-1)(n-2)/1x2x3
|
|
1
|
0
|
0
|
0
|
|
1
|
1
|
0
|
0
|
|
1
|
2
|
1
|
0
|
|
1
|
3
|
3
|
1
|
|
1
|
4
|
6
|
4
|
|
1
|
5
|
10
|
10
|
|
1
|
6
|
15
|
20
|
|
1
|
7
|
21
|
35
|
|
1
|
8
|
28
|
56
|
|
1
|
9
|
36
|
84
|
|
1
|
10
|
45
|
120
|
|
1
|
11
|
55
|
165
|
|
1
|
12
|
66
|
220
|
You notice I have added some other
columns. A New-age three-hug
requires at least three people, and to work out the number of different
three-hugs we have to divide by 3 times
2, the number of different orders in which the three people can be chosen
in order to eliminate double-counting and get the right answer. When counting just
the number of people (column 2) I have put in a “divide by 1”, as there is no double counting. I have also added a
column of constant unity to the left and finally also a row for zero people.
These modifications are there to bring out the beautiful simplicity of this
table. You can now see that you don’t need to bother with multiplying figures
together. As you go to the right, the figure in each cell of the table can be
found by simply adding the figure in the cell above to the figure in the cell
above and to the left. This means that the figure in each cell is also the sum of the series of ALL the numbers in the column to the left up
to and including the cell above and to the left. If you know some algebra, you
recognise that the first column tabulates value of a constant, the second of a linear
function, the second of a function including a square of n ( a quadratic function),
the third a cubic function, and that
we could go on adding as many columns to the right as we liked in order to
tabulate values of the function n(n-1)(n-2)……..(n-r+1)/ 1x2x3…xr, which
is the number of r-hugs and is a polynomial involving n to the power r,
where r can be as large as we like. We call 1x2x3x4…..xr, r factorial, or r! for
short, as factorials crop up all over the place in combinatorial mathematics
(which is what we have been talking about), probability theory and statistics.
For me, this has been a classical and beautiful mathematical exercise,
embarking on a potentially endless journey into more and more general results,
with wider and wider implications, starting from just one simple idea – that of
shaking hands.
Does this sort of thing worry you? Well,
you are in good company. We can trace mathematical thinking back to, for
instance, Babylonian student excercises still preserved on 4000-year-old inscribed and
baked tablets. Already in those days, the master was setting examples that can
still terrify us today, such as the brain-twister below.
I found a stone but did not weigh it. I
weighed out six times its weight and added 2 gin, then added one third of one
seventh of this weight multiplied by 24. The total weight was finally 1 man-na.
What was the original weight of the stone?
Answer, 4 and1/3 man-na .This works out correctly if 1 man-na equals 60
gin.
Incidentally, we see just how long ago
and far away the basis of our conventional division of hours into 60 minutes
and minutes into 60 seconds, both of time and of angle, was established.
An Egyptian problem, found on a Papyrus
about 3600 years old, seems simpler:
If 10 hekat of fat is given out for a
year, what is the amount used in a day?
(c.f.
if 104 black bin-bags are given out for a year, how many are used in a week?)
The answer however is not so simple,
being expressed as:
1/64
hekat and 3+2/3+1/10+1/2190 ro.
(I hekat = 320 ro)
Here we see that, with the exception of 2/3, and maybe also 3/4, the Egyptian notation and probably
the Egyptian mind could not deal with fractions other than “One share of
however many”. A fraction like 56/73
(to which the above addition of shares is equal) was beyond writing down and
probably beyond thinking about.
Problems with notation have taken much of
the following three and a half thousand years to deal with. Modern fractions
like 56 73rdsonly
gradually permeated Europe from Arabia and Italy during the Middle Ages. Present-day algebraic
notation, x‘s and y’s and all that, came into use
gradually between the fifteenth and seventeenth centuries; British calculus is
said to have been held up for more than a century by Newton’s now largely
abandoned notation – patriotism inhibited the adoption of Leibniz’s more
adaptable dy/dx expressions. Maybe there are improvements still to be made.
Our own personal struggles with algebra and mathematics in general can perhaps
be excused when you consider how many centuries even those
cleverest-of-the-clever leading mathematicians took to get their notation, and
their corresponding thought processes, straightened out.
Note that the above-mentioned three
and a half thousand years, say 3511
years, to be precise, could have been expressed as: three millennia plus five
centuries plus one decade plus one year. It is revealing to quote a Derbyshire
sale catalogue dating from 1920 for lands belonging to His Grace the Duke of
Rutland: Here is a typical lot description (shortened):
Yeld Wood Farm, Woodlands & Cottage
situate close to the Village of Baslow…,
containing an area of about 82 Acres 3 Roods and 22 Perches.
As you will know (?) there are forty perches to a rood and four roods to
an acre which was the area that could be ploughed in a day, 4840 square yards. A rood can be well visualised as the
typical area of a mediaeval strip (or perhaps half a strip), one furlong long by one pole wide, or 220 yards by 5 ½ yards. A perch is simply a square rod (or pole), 5 ½
yards by 5 ½ yards.
So, you see how
the prospective buyer can actually visualise the land area involved, whereas
giving it as 82.87265, say 82.88, acres, or 82 and 71/80 acres,
requires the farmer to visualise 0.88
of an acre; probably he’d rather have it in roods and perches. An ancient
Egyptian might render the area for sale as 82
½ ¼ 1/8 1/80 acres.
All this takes
me back to primary school, I used to be able to add, subtract and even multiply
in acres, roods and perches, and in miles, furlongs, chains, rods, yards, feet
and inches. Measurement systems like this are meant to avoid the need to think
in terms of fractions or decimals. Almost any quantity can be expressed as
integral (whole number) multiples of units you have a habitual feel for. Same
with old money: one pounds, seven shillings and six pence ha’penny. Nowadays,
since decimalisation, we still have the various sized coins reflecting our physical need to hand over change in “roods
and perches”, to speak metaphorically, namely the 50p, the20p, the10p, 5p, 2p
and 1p coins, but we no longer (or, I hope, do not YET) have common names for these coins and may get quite
confused trying to convert, say, 83p into a practical palmful of change.
We are not confused, say the mathematicians! They,
or maybe We, have surrendered the primitive need to visualise the sizes
of the quantities we deal with in exchange for the extreme simplicity of
manipulating numbers in the decimal system. Everything is in multiples or
divisors by ten. 82.87265 acres may be hard to imagine, but it can be
multiplied by a hundred by a simple double shift of the decimal point. Other
multiplications, additions and subtractions take a bit longer than this but are
completely straightforward. In contrast, imagine trying to work out how
many 4 ounce bags of sweets can be made up from a day’s production of 2 tons, 7
hundredweights, 3 stones, 5 pounds and 12 ounces, which latter is
expressed entirely in units I still have a real feel for. 2.372 tonnes = 2372 Kilograms
= 23,720 100gramme bags is so
much easier!
This handy
decimal system goes back at least to the India of a couple of thousand years ago. For
the next stage of the discussion, we need to simplify it a bit more.
10, 100, 1000, 10000, 100000, 1000000,….
is getting hard
to write, let alone distinguish just how many 0’s there are. So generalising on
100 being 10 squared, written 10², we can write the sequence as:
10,10,²10³,104,105,106, … and so on, these being the sort of
numbers that appear in successive columns of our handshake and hug table.
Similarly, the
sequence of numbers less than 1:
0.1, 0.01, 0.001, 0.0001, 0.00001,….
Can be written 10‾1 (that is one over ten),
10-2 (one over ten squared),
10-3, and so on.
(For
mathematicians to deal with numbers less than 1 in these elegant decimal and
power expressions took a lot longer in the adoption than dealing with the
larger-than -one numbers. Finally, to join these sequences of powers together
we are forced to adopt the convention that the number 1, unity itself, = 10° , even though multiplying a number by itself
zero times seems meaningless)
Now, at last we are in a position to start
measuring the Universe.
2.
Getting the Measure of the Universe
In reaching out
to see and grasp the great and the little, we start with a handy measure of our own size, one
metre – a stride, an arm’s length, a child’s height. – this is our unity, our 10°.
Lets start going
up in scale in powers of ten, or, as practical scientists say “orders of
magnitude”
Order 1 10metres; the width of the bookshop
Order2 100
metres; a sprint, how far a shout
will carry, or a missile be thrown
Order 3 1
kilometre; a walk to the station, a waving friend visible
Order 4 10
Km; the distance to the nearest
town, a long run
Order 5 100
Km; a journey to the regional capital, to court, to prison, to the seaside.
Now we are beginning to reach the edge of the pre-industrial ordinary person’s
experience
Order 6 1000
Km; travelling to the national capita city, to another country, or on a
pilgrimage
An experience of
only a few courtiers, merchants, churchmen, armies
Order 7 10,000
Km; The voyage of Christopher Columbus, Marco Polo’s travels
We are now
getting to the edge of the ordinary person’s experience, but not beyond the
ingenious measurements of the Classical
Greek geometers, who obtained very good estimates for the size of the
Earth.
Order 8 100,000
Km; A girdle around the Earth.
Order 9 1
million Km; The distance to the Moon; beyond the ordinary mortal, but again
not beyond the ingenious Greeks, whose curiosity and method is humbling. They
visualised the distance in “stades”, a foot-race distance of about 200 yards.
Order 11 100
million Km; Of the order of the distance to the sun. Even this the Greeks tried to measure, and their
estimate was “correct” to within an order of magnitude; it was more accurately known
by the 17th.Century. A digression, a triumph in science may long remain useless
in practice!
Order 12 1
billion Km; The distance to Jupiter. Only the invention of the “Gallilean”
telescope made this possible to conceive .Galileo first observed Jupiter’s 4
largest moons in 1610. They are
easily seen with binoculars and, as early as 1676, small delays in their eclipsing by the planet were used to
obtain a good stab at the speed of light, unimaginably high at 186,000 miles (
Anglo-Saxon motorway units) per second.
Order 16 1
Light Year, or a third of a Parsec. The nearest star is about 4 light-years
away. Note the introduction of new units to try to help us visualise the
immensities. The distance to a star (not
a planet) was first measured in 1838,
using parallax, the slight change in direction from opposite sides of the
Earth’s orbit. A parsec is the distance at which the parallax of a star,
subtended by the RADIUS of the Earth’s orbit, is one second of arc
Order 21
Diameter of the Milky Way, our galaxy (100,000 light years) Advances in the measurement and understanding
of starlight spectrums and in so-called Cepheid variable stars made
measurements like this possible by about 1915.
A digression. To
quote from the internet, how to bring
the Universe down to size:
There are an estimated 150 globular clusters that swarm around our
galaxy. Each of them contains 100,000 to 1,000,000 stars in a spherical
region ONLY a few hundred light-years in
diameter. Order 22 1 million light years, is the approximate distance to the nearest other galaxy, The Great Nebula in Andromeda, M 31. Controversy about whether “galaxies”, those fuzzy objects, were gas clouds, perhaps forming stars, in the Milky Way, or other collections of stars at a great distance, was finally settled only in 1923 (the approximate birth-date of physicist Freeman Dyson) with the aid of the 100 inch Mount Wilson telescope– Hubble identified individual variable stars in nebula M 31.
Order 25 1 billion light years. In 1929, Edwin Hubble proposed his celebrated “expanding universe” theory. The dimmer the supernovae, the further away the galaxy and the bigger the red shift, explained by its receding from us. A billion light years was uintil recently about the limit for observations of this type on galaxies.
Order 26 Ten billion light years. The distance to the edge of the observable universe is currently estimated as about 16 billion light years, giving a visible diameter of twice this.
So, the largest number we can come up with, relating the size of the observable Universe to a stretchy human pace-length is some 3 X 1026.
Though the Greeks’ imagination reached out to their estimate for the Sun’s distance, of order 10, it was not until the 17th.Century that planetary distances became accepted, order 11 to 12, not until the 19th.Century that the true remoteness of the fixed stars was revealed, order 16 and not until the lifetime of the parents of many at this meeting that the true scale of the observable Universe, orders 22 to 26, was understood and accepted.
Surely we must question whether any existential philosophy more than 200 years old can have more than inspirational or allegorical significance?
3.
Smaller and ever-smaller
Time now to turn from telescopy to microscopy
and go down in scale to the smaller and smaller, starting again from our
“zeroth” order of I metre.Order Minus 1 10cms. A “handy” size, the scale of a handspan, a fist, a stone, a sheet of writing paper, a jug of milk.
Order Minus 2 1 cm. A finger’s-breadth, a flower, handwriting, an easily snapped twig, a pebble
Order Minus 3 1 mm. Getting hard to see. Grit, a seed, a pin-head, your nails needing cutting
Order Minus 4 0.1 mm. About as small as can be seen or imagined to be visible to the naked eye. Small seeds, sand-grains, eye of a needle. From mediaeval times, magnifiable to a more comfortable scale by single-lens “reading glasses” (as in Umberto Eco’s monastically set Name of the Rose)
Order Minus 5 0.01mm., 10 micro-metres. Silt or “soil” particles; they don’t float but do smear. Pollen grains – may blow about but can and need to settle. The cells of animal and plant tissues are often in this range; first described by Robert Hooke, c. 1670, from his microscopic observations
Order Minus 6 1 micrometre. Dust. As we know, you can’t see it till it settles. In the 17th. Century, the double-lens microscope allowing X20 to X200 magnification brought this scale into view.
Order Minus 7 0.1 micrometres or 100 nanometres. The wavelength of visible light is in the range 400-700 Nm. and this limits what could be distinguished using the best optical microscopes by late Victorian times.
Order Minus 9 1 nanometre, a billionth of a metre; about the diameter of a sugar molecule. The actual existence of “molecules” became accepted during the 19th. Century, but the direct investigation of their structure only began c. 1920 with X-ray crystallography, X-rays having a wavelength comparable to molecular sizes.
Order Minus 10 1 Angstrom, a tenth of a nanometre, 100 picometres; typical effective size and separation of atoms.
Order Minus 12 1 picometre , a billionth of a millimetre. The wavelength of the electrons used in microscopy is about 5 picometres. This limits the electron microscope, developed in the 1930’s.
ORDER minus 13 100 Femtometres This is where “High Energy Physics” takes over; larger and larger linear and circular accelerators:-, particularly since about 1960, CERN and the infant Large Hadron Collider
Order Minus 15 1 femtometre roughly the radius (whatever that means) of a proton or electron. The existence of the electron was deduced about 1900, but protons and neutrons not until their tracks could be followed in cloud or bubble chambers from about 1930
Order Minus 18 1 attometre or nano-nanometre. About the feasible limit of High Energy Physics and correspondingly the scale of elementary forces and particles studied.
Order Minus 45 The Planck Length. An entirely hypothetical and especially hard to understand concept. May perhaps be thought of as the ultimate limit of the precision with which a particle’s position could be ascertained in the quantum theory. The energy of the probing particle/wave would be such that a black hole would be formed, so no measurement would result (!?!)
I have gone down to the very (to the 27th. Power) silly Planck Length just because I want to give the Universe a chance to resist the power of the human mind!
So, the extension of man’s ability to look at very small things has gone from order 7 to order 18 in little more than 100 years. As with the very large, surely our outlook should have changed radically with such an expansion in our ability to observe the sub-sub-microscopic entities of which everyday objects and ourselves are made up. Certainly, we are filled with wonder by television programmes, articles and books, but most of even us “educated classes” experience little outside the everyday scale in our everyday lives. We are mostly scientifically ill-informed and even more inexperienced ; we probably do not own or rarely use a microscope or a telescope, let alone an x-ray diffractometer or a linear accelerator! It is very easy, still, for us to live in an unquestioning mental world akin to that of the “ancients” in which only a few visionaries posed fundamental questions. How many of us have ever thought of estimating the moon’s distance by timing the length of a lunar eclipse, as did the Greek natural philosophers.
4.
But how BIG is the Universe, actually?
"Space is big. Really big. You just
won't believe how vastly hugely mind-bogglingly big it is. I mean, you may
think it's a long way down the road to the chemist, but that's just peanuts to space.”
Douglas Adams: The Hitch-hiker’s Guide to the Galaxy
4πR³, where R is the radius in Planck Lengths, 1.6πX10 to power71x3, let’s approximate a bit, after all my calculations may not be that precise, 10 to the power 324 is near enough.
5.
God’s Numbering System
But the very hairs
of your head are all numbered. Matthew Ch. 10
Verse 30
God is supposed to have
no problem numbering, that is describing and knowing, every point in the World,
now known to be so much bigger than the Evangelist could have thought, and
implied in the quotation.
The readers of the Bible
are supposed to be awe-struck by this degree of omniscience. Not so Archimedes, who
explains, in the 3rd. century B.C, in a paper addressed to a King
Gelon, that his numbering system is capable of dealing easily with the number
of grains of sand that might be needed to fill the Universe. Our numbering
system is also perfectly capable of dealing with the scale of things, as we
have already seen.
Another way to look at
this is to perform a card trick. Imagine I am holding three perfectly normal packs of 52 cards, excluding troublesome
jokers, 156 cards which to simplify things can be considered all to be
different, each pack having a different design on the back. I assure you that
this pack has not be prepared or tampered with in any unfair way, but the cards
are of course in one particular order, with one letter per card, I’ve written
out a short passage from Shakespeare. Now, watch carefully.
Dave fans,
shuffles, fumbles and drops the whole pack on the floor. What a mess!
Oh *******!! How am I ever to sort them out again? Well,
if I work through all the possible orders to find the right one – the 152-letter
message from Shakespeare, I will need not merely all the time in the world, but
much more than that!
There are just two ways
of ordering two cards, six of ordering three, twenty-four of ordering four, in
short 1x2x3x4x……x156 of ordering
them all – its that FACTORIAL again, 156!
This number is rather large. I haven’t had time to calculate it, but I’m told
that 70! Is approximately 10 100, so 100! is going to be at least 10150 and somewhere around
the three-pack mark the number of ways of ordering the cards is going to exceed
10324 , the
number of “Planck points” in the visible universe. So there on the floor is all
that God needs to set up a one-to-one correlation with number all the hairs on
the head of all of space.
Now there is a very
theoretical minimum conceivable time interval called the Planck Time, about 10 -43
seconds.( this is about one ten thousandth. of the time
it takes light to cross the diameter of an atom) The present age of the
Universe is a mere 1018 seconds (and counting,
slowly) which comes to 1061 Planck Times. So, there
is no way those playing cards could be got into the right order to rediscover
Shakespeare’s message in a time equal to the lifetime of the Universe so far.
We can also see that God
would not need many more playing cards to number not merely each point in the
Universe, but each point at each instant in the life of the Universe, each
point in space time, with plenty of room left over to describe what is going on
at each point, whether empty, or associated with a particle, or with the scale
and direction of each possible force field, and finish by giving this point in
space-time a fanciful name, perhaps inspired by Peak Rock-climb or Lead-mine
nomenclature! Don’t Sneeze Now Arete or
Second-Cousin’s
Fortune. There are even more names available that orderings of playing
cards, which leads us to:
6.
Monkeys Typing Shakespeare
Shall
I compare thee to a Summer’s Day,
Thou
art more beautiful and more temperate.
Rough
winds do shake the darling buds of May
And
summer’s lease hath all too short a date.
At
this point I had intended to embark on some intricate calculations of the time
it would take the proverbial “monkeys” to type even this and the other eight
lines of one supreme sonnet of Shakespeare, but I think you have got my drift
by now. It's going to be an absurdly long time, though we could calculate a decent estimate! Even were the monkeys able to employ, not typewriters, but the still
proverbial “Quantum Computer”, they would have no chance of discovering even an
early draft by the Bard within the lifetime of the Universe. This is, I
suppose, a commonplace observation, but it is less commonplace to ask;
“How then DID the
sonnets of Shakespeare ever GET written
– starting from the blankish, even if anthropophilic slate of the early
Universe”
particularly
as the Universe got off to such a
laggardly start in the race against the monkeys to see who could write Shakespeare’s works first. Nearly ten billion years were given over just
to forming giant stars, letting them manufacture heavy elements and then blow
themselves up, so that the scattered materials could condense into a second
generation solar system, our sun and planets, some solid and iron- and silicon-rich.
Another half-billion years at least
were required for things to solidify a bit on planet Earth and for Jupiter to
vacuum up most of the dangerous impacting meteors. Another mere hundred million years or two sufficed for life to appear, but
more than three billion years were
used up before it crept out of the sea. Another 300 million years were needed to evolve mammals and 298 million years to evolve the first
self-consciously intelligent species. And during all this time, the monkeys can be imagined typing
randomly away, by now they are well in
the lead – they’ve got as far as several very beautiful lines of a risqué
sonnet. Even the last two million years
before the present, about a ten-thousandth of the lifetime of the Universe,
have largely been employed in developing language from scratch, honing all our
subtle passions, emotions and abstract intelligence and in developing the art
of story-telling and aural tradition. Only in the last 50,000 or less years have written alphabets allowed remembered
culture to develop and be passed on. Urban
living, with all its special crafts, including those of playwright and
poet, seems to extend back no further than 10,000
years before the present, and this period is essentially that which has
allowed literature, philosophy and science to be recorded so that the likes of
Shakespeare and Newton could rejoice in “standing upon the shoulders of giants”
to achieve there own dazzling in- and out-sights.
So
we allowed the typing Monkeys a very long start indeed, but we still got there
first! This shows the true scale of the wonder of human thought. Douglas Adams
once again got here first – his super-computer Deep Thought, faced with
discovering the QUESTION to which 46 is the ANSWER to The Riddle of
Life, the Universe and Everything announces that it needs to design:
“A computer which can calculate the Question to the Ultimate Answer,
a computer of such infinite and subtle complexity that organic life itself
shall form part of its operational matrix…. And it shall be called..the
Earth”
7.
The Interstellar Pen is mightier than the Sword
So much for the scale of the Universe, we can
hack it! But can we affect it, or even explore it - hardly at all. You might think of scientists and engineers working on Earth to be analogous to a gathering of pub philosophers gathered in an English inn scheduled for closure! But that is another story - SETI, the Search for Extraterrestrial Intelligence - and this article has gone on too long already!
