All rational numbers (the proper mathematical term for fractions)
can be expressed as a/b where a and b are integers. These can be separated into two
categories depending on whether a is greater or less than b. (There is a third category
in which a=b, but the rational number thus described is simply the number 1. For the
purposes of this argument, the number 1 can be safely ignored.) It is clear that one
category includes all the rational numbers between 0 and 1, such as 5/8, 1/3, 4/17, and
that the other category includes all the rational numbers greater than 1, such as 8/5, 3/1,
17/4. It is also clear that there is an equal number of rational numbers in each
category, since for every a/b there is a corresponding b/a. Therefore...
Conclusion number one: Half the rational numbers come between 0 and 1.
On the other hand, however many rational numbers there are between 0 and 1,
there are the same number between, say, 4 and 5, (Proof? Just add 4 to the first lot
of numbers to get the second lot) or indeed between any two consecutive integers.
So there are, for example, a hundred times as many rational numbers
between 0 and 100 as there are between 0 and 1. If you compare the interval 0 to 1
with the interval 0 to infinity, then out of all possible rational numbers the proportion
between 0 and 1 is 'one divided by infinity' which to all intents and purposes comes to zero.
Conclusion number two: There are no rational numbers between 0 and 1.
This apparent contradiction disappears if you realise that the
first conclusion above is pure nonsense. This is easily demonstrated
by the application of a little logic.
a) There are no rational numbers between 0 and 1 (see conclusion two above).
b) A half is a rational number between 0 and 1.
c) Therefore there is no such thing as a half.
d) Therefore it is nonsense to claim that 'half' the rational numbers come between 0 and 1.
Incidentally, pirates have long known that there were no fractions between zero and one, as shown by their rallying cry, "No quarter!" and even common labourers on building sites have often take time out from their appreciation of the feminine form to express this profound truth - as in the exchange, "Cor, she's a bit of alright, eh?" / "Not half!"
The number of fish caught by the disciples according to John 21:11 was 153. As far as I know this has no theological significance, but it is an interesting number mathematically.
Take any number. Take 12532590168 for example. Now take the cubes of each of its digits and add them together. 1 + 8 + 125 + 27 + 8 + 125 + 729 + 0 + 1 + 216 + 512 = 1752. Now do it again. 1 + 343 + 125 + 8 = 477. And again. 64 + 343 + 343 = 750. And again. 343 + 125 + 0 = 468. Keep going. 64 + 216 + 512 = 792. 343 + 729 + 8 = 1080. 1 + 0 + 512 + 0 = 513. 125 + 1 + 27 = 153.
Once you reach 153, you reach equilibrium. (Try it and see.) Don't ask me for proof, but someone told me this works if you start with any three-digit number. Personally I think it will work for any starting number at all, with the obvious exception of 1.
Correction: I have since discovered that it works for any number which is a multiple of three. (Probably the person who showed me this, over a church Harvest Supper as it happens, had wrongly remembered 'multiple of three' as 'three-digit'.) There exist three other numbers which are the sum of the cubes of their digits, namely 370, 371 and 407. There are also a 'few cycles' such as 55, 250, 133. Perhaps 153 is not quite so uniquely interesting as I'd first thought.
The Problem: Given that helixes, such as screws or DNA, fall into two distinct and separate categories, which we might call clockwise and anti-clockwise, why is it that a spiral staircase can wind clockwise going up and anti-clockwise coming down?
The strange nature of spiral staircases will not be apparent until you understand a feature of the helix - that it looks the same whichever direction you approach it. All DNA and most manufactured screw-threads are 'right-handed', which means that as you move forward along their axes the loops turn clockwise. Left-handed threads are fairly rare and tend to be made for specialist reasons, such as the screw-in light bulbs in New York subway cars, which used to have a left-handed thread to prevent people stealing them for domestic use. This is not easy to imagine and you will need to get hold of a physical object to satisfy yourself that I am telling the truth....
An experiment: Find a right-handed helix from somewhere. An ordinary screw might do, though it is a bit small to see clearly. The corkscrew on your swiss army knife would be better, or you can manufacture your own helix by wrapping a shoelace around a pencil. Look along the top of the helix and notice how the loops turn clockwise, from near left to far right. Now turn the helix around and look along it the other way. What do you see? The loops still turn clockwise, don't they?
So do you get the point? A right-handed helix has loops which turn clockwise whichever direction you view it and a left-handed helix has loops which turn anti-clockwise whichever direction you view it. So how do you explain the peculiar nature of a spiral staircase? When you walk up it turns clockwise. When you come back down it turns anti-clockwise! If you wish to wrap your brain around this paradox, then please accompany me on three thought experiments....
Thought Experiment One: Imagine walking through a rotating tunnel such as you might find in a funhouse in an amusement park. You are in a cylinder with the wall on your left descending and the wall on your right rising. You can see the mouth of the tunnel ahead of you and it is spinning anti-clockwise. Now imagine turning and looking over your shoulder to the entrance of the same tunnel. Which way is it spinning? Clockwise! The spin has not changed direction in reality, but your perception of it has changed.
Thought Experiment Two: Imagine standing inside a round room facing the door. Look down at the circular floor and imagine it to be the face of a clock. The number nine would be over by your left ear, three by your right ear, six by your chin and twelve by your forehead. Now tilt your head back and look at the ceiling. Again imagine a clock face overhead, with nine o'clock over by your left ear, three by your right ear, six by your chin and twelve by your forehead. Look back down at the 'clock' on the floor. The door in front of you is at twelve o'clock. Now look again at the 'clock' on the ceiling. Where is the door? At the six o'clock position!
Thought Experiment Three: Imagine you are standing on the step of a spiral staircase, facing downwards. The step you are on sticks out of the central column at the three o'clock position. In front of you, the next step down is jutting out at the two o'clock position. The step below that is at one o'clock and below that at twelve. The stairs seem to be turning anti-clockwise. Turn around on the same step to face upstairs. You are now at the nine o'clock position on the stair. The next step up is at ten o'clock, then eleven, then twelve. The stairs seem to be turning clockwise. But now tilt your head back and look at the steps above you. Immediately above your head is a step jutting out at nine o'clock. The step just in front of it, and slightly higher, is jutting out at eight o'clock. The one beyond that at seven and the next one at six. The spiral is behaving itself after all and spiralling anti-clockwise upwards, the same as when you were heading downwards.
The Answer: When you walk down a spiral stair you visualise it turning as you should. The confusion comes when you walk up a spiral staircase because you are heading one way (upwards) and looking the other (downwards). In simple terms, you are looking back the way you came and therefore see it rotating in the opposite direction.
POSTSCRIPT: Why did all this occur to me in the first place? Because I came across a trivial question - "Spiral staircases in medieval buildings were always built clockwise. True or false?" - which at first made no sense because I thought it depended which way you were walking. Then it made no sense because I remembered that helixes ought to wind the same way whichever direction you look at them. Then finally, when I had come to terms with the paradox (as outlined above), I decided that the book had provided the wrong answer ("true"). The question-setter was probably thinking of how a spiral staircase seems to turn clockwise as one ascends. My own understanding of medieval staircases (at least in castles) is that they turn anti-clockwise so that a right handed defender above can more easily thrust the pointy end of his sword into a defender coming up from below.
Like it or not, the English language is constantly changing. Words acquire new meanings. In the early stages most educated people wince at the misuse of their native tongue, but eventually common usage makes the new meaning acceptable to all but the most hardened pedants. Here are a few changes which I have found at least mildly interesting:
It seems to me that there is sometimes confusion between the sounds we use in our everyday speech and the use of the 26 letters of the alphabet to represent these sounds. For what it's worth, here is my analysis.
Consonants. In my opinion there are 23 basic consonants. (In the list below it
may help if you think of the sounds in the old childish way so P is pronounced as in super,
F as in sofa and Y as in soya.) First there are seven pairs in which the
first sound is merely a whispered or unvoiced version of the second. Namely:
P (as in pea or pin) and B (as in bee or bin)
T (as in tie or tin) and D (as in dye or din)
F (as in feel or fan) and V (as in veal or van)
S (as in Sue or sip) and Z (as in zoo or zip)
TH (as in thick or thin) and DH (as in this or that)
CH (as in chew or chin) and J (as in Jew or gin)
SH (as in shoe or shin) and ZH (as in regime, measure or French Je t'aime)
Nine further consonants don't seem to fit in pairs:
G (as in go or get)
H (as in hoe or hat)
K (as in key or cat)
L (as in low or let)
M (as in mow or met)
N (as in no or net)
R (as in row or rat)
W (as in woe or wet)
Y (as in you or yet)
Other consonants. We actually use a wider range of consonants than these. For example
There is the Scottish coughing sound at the end of the word loch;
There is the sort of nasal swallow at the end of the word sing;
There is the French nasal sound at the end of the word restaurant;
There is the clacking gottal stop in the middle of the word bottle;
And there are many other subtle variations on our everyday consonants that we need to employ when pronouncing words of exotic origin. But for the purposes of this analysis lets stick to the above twenty-three.
These consonants can be combined together. The word scrambled, for example, has three consonants S-K-R in one batch followed by two pairs M-B and L-D. (Don't forget we are considering sounds rather than spelling, so the L sound comes directly before the D sound. You wouldn't normally pronounce it as if it were two separate words - scram / bled - unless you wanted to sound like a girly poet.) Such combinations do not produce new consonantal sounds, merely two or more basic sounds coming close together.
Vowels. Now these are much trickier to categorise, given that the range of regional dialects render the vowel sounds so differently. As a Yorkshireman I fail to see why my dictionary distinguishes between the u in mud and tough and the u in book and put. On the other hand the dictionary gives exactly the same pronunciation for one and won, when clearly the former ought to rhyme with gone and the latter with gun. But I'll have a go at categorising them, with my own notation which avoids all those confusing bars and dots and accents often used in dictionaries.
Single vowel sounds. There are five pairs in which the second vowel is more or less a
shorter version of the first. For example, try saying teen teen teen several times as fast
as possible and it comes out sounding a bit like tin tin tin.
AH (as in father or carp) and A (as in fatter or cap)
EE (as in sheep or heal) and I (as in ship or hill)
AW (as in port or walk) and O (as in pot or wok)
OO (as in mood or roof) and U (as in mud or rough)
UR (as in early or demur) and UH (as in alike or dimmer)
There is one other short vowel, E (as in bed or pet), which doesn't seem to have a longer version other than AIR, which is a diphthong and I haven't got to those yet. There are two other long vowels, IE (as in ride and bite) and OH (as in rowed and boat)
Diphthongs. Eight vowels are effectively two of the above combined, but such combinations
are more subtle than the aforementioned combinations of consonants, and therefore deserve
some notation of their own.
E and I combine to give EH (as in day or cane)
E and UH combine to give AIR (as in dare or cairn)
IE and UH combine to give IRE (as in dire or higher)
I and UH combine to give EAR (as in deer or here)
U and UH combine to give URE (as in pure or sure)
A and U combine to give OW (as in cow or shout)
OW and UH combine to give OUR (as in power or sour), which seems to me more like a triphthong
O and I combine to give OY (as in coy or soil)
The difference between spelling and sound. We all know that some words which look
the same can be pronounced differently depending on the meaning of the word. For example,
bow can be pronounced B-OW or B-OH. But even the same word with the same meaning can have
a varied pronunciation. For example, the is usually DH-UH (as in Winnie-the-Pooh) but before
a vowel becomes DH-I (as in The End) and when requiring emphasis becomes DH-EE (as in no, I'm
only a Bruce Willis, not the Bruce Willis).
Then again, sounds and letters sometimes just don't match even in common words. You can hear
the sounds of Z, V and I (twice) in the words is, of and women. But the letters chosen to
represent these sounds don't seem to correspond.
To put it another way, don't confuse the letters you see with the sounds you hear. Sometimes the link is apparent but not always.
The Alphabet. So how does our alphabet correspond to the sounds analysed above?
We have five vowels - a; e; i; o; u - which can be used on their own or in a wide range
of combinations to represent all the above 21 vowel sounds. Further analysis in this
direction is too complex for me to tackle.
How are the 23 consonantal sounds represented? Eighteen by single letters - b; d; f; g*; h; j; k; l; m; n; p; r; s; t; v; w; y; z - though the asterisked letter g is also frequently used to represent the J sound as well as the G sound, and is sometimes used to represent ZH. Two consonantal sounds are represented by combinations of two letters - ch; sh - and two by the combination th, which is pronounced either TH or DH depending on context.
If you have kept count you will realise that three letters of the alphabet are unaccounted for. These are c, which is sounded as either K or S, q which represents the combination K-W, and X which represents the combination K-S. In other words, we have three letters which I ekspekt we kould kwite probably live without if nessessary.
Pronouncing the Alphabet. Still with me?
My original musings on this whole subject began with a realisation that not all letters
use the appropriate sounds.
The five vowels are pronounced EH, EE, IE, OH, Y-OO. The first of these is a diphthong, the next three are long vowel sounds and the last is a long vowel sound preceded by a consonant.
Seven letters begin with the appropriate consonant - B-EE, D-EE, P-EE, T-EE, V-EE, J-EH, K-EH - two begin with one of their possible pronunciations - S-EE and J-EE - and six end with the appropriate consonant - E-F, E-L, E-M, E-N, E-S, E-K-S.
This leaves six which have stranger pronuncations.
Z-E-D begins with the appropriate sound, but adds an extraneous D.
K-Y-OO begins with K-Y instead of K-W, perhaps because the letter is pronounced that way in the word queue.
W-IE begins with the wrong consonantal sound.
EH-CH doesn't contain the appropriate sound at all.
AH is simply a long vowel sound without any consonant attached.
D-U-B-UH-L-Y-OO consists of four consonants and three vowels, none of which correspond to the usual sound of the letter.
The Greek Alphabet. Out of curiosity, can we analyse the Greek alphabet in a similar way? I am no expert, so forgive me for any errors, but here goes. The alphabet has 17 consonants and 7 vowels making 24 letters altogether.
How are the sounds represented? 14 of the 23 consonantal sounds are represented by
single letters - beta; gamma; delta; zeta; theta; kappa; lambda; mu; nu; pi; rho;
sigma; tau; phi. Strictly speaking zeta represented the sound D-Z, though it is
pronounced Z at the start of a word. One letter, chi, represents that Scottish
coughing sound at the end of loch which I dismissed earlier. The H sound is not represented
by a letter, but by a single open quote above the vowel at the start of a word.
The vowel iota is sometimes sounded as Y - as in the Greek name pronounced
Y-EH-S-OO-S for example. Another vowel, upsilon, can be sounded as V - as in the word
evangelism, which is derived from a Greek word whose second letter is upsilon.
The remaining six sounds - DH, CH, J, SH, ZH, W - don't seem to be used in Greek.
Two Greek letters, xi and psi, represent a combination of consonants, K-S and P-S.
The vowel sounds are represented by 7 letters - alpha, epsilon, eta, iota, omicron, upsilon, omega. These correspond roughly to our 5 English vowels plus EH (eta) and OH (omega).
How are the letters pronounced? Seven consonants are straightforward - K-S-IE, P-IE,
F-IE, CH-I (as in och aye rather than watch eye), P-S-I, R-OH, T-AW. Two others slip in an
extra consonant almost unnoticed - M-Y-OO and N-Y-OO. Three have the extra syllable 'ta' -
B-EE-T-UH, Z-EE-T-UH, TH-EE-T-UH, as does the vowel EE-T-UH. Another vowel, IH-OH-T-UH, has
the same ending but includes an extra vowel sound. This leaves five consonants and five
vowels with more elaborate pronunciations.
L-A-M-D-UH (with a B sound squeezed in faintly between the M and D)
U-P-S-I-L-O-N (this is that vowel sound as in book which I refuse to distinguish from the one in mud)
OH-M-I-G-UH (or perhaps OH-M-EH-G-UH)
And what do I conclude from this? On the one hand the Greeks seem to have given excessively elaborate names to more of their letters than we English have. Only h and z in our alphabet come close, though w beats them all when it comes to over-the-top names. On the other hand, at least the Greeks have ensured that the pronunciation of each letter begins with the sound that the letter usually represents - which cannot be said for the English alphabet.
One final footnote. I must apologise for the cavalier way I have equated the letter phi with our English f. They may sound the same, but there is a subtle psychological difference between the river Alph as described by Coleridge in his poem Xanadu and the king Alf who burnt the cakes. And I can sympathise with Tolkien (a famous linguist) when someone mis-spelt the name of his wizard. However much they might sound alike, the fact is that Gandalf is right and Gandalph is a horribly distorted abomination.