If you have any comments or suggestions for More or Less then click here to find an e-mail form. We read and value all your e-mails but we cannot promise a reply.
We would also like to know about your encounters with numbers, whether mystifying, strange or even beautiful.
And we hope you will join us in keeping a watchful eye on the way numbers are used and reported.
These are a selection of e-mails in response to some of the topics covered in the last series of More or Less.
I have synaesthesia in that I see words, names, days and months in colour, and I also see them in a certain visible sequence and pattern.
I see days of the week stretching out in front of me, with the weekends slightly raised, and I am in the pattern according to what day it is.
The year has a strange asymmetric curve, and numbers and percentages and centuries have their own pattern. If I think about it, they too have colours.
I see numbers as colours. Single digit numbers then go on to make up more complex numbers using their 'signature' colour.
For example, one is white and nine is royal blue, therefore 19 is white and royal blue.
I also see years and the days of the week in a very similar way to the patterns Rachel described in the programme.
However mine have colours, for example 11 = yellow, 23 = red, Saturday = red and so on.
It was not always helpful at school to have to locate these on my mind map - but I don't really understand how else people find numbers.
I also "see" numbers, which coil up towards the right.
They are coloured: one, two and three are pale yellow, four is black, five is salmon pink, six is aqua blue, seven is black and so on.
After about 20 the decades have different colours.
Centuries also coil round, coming from low down on my left hand side. It helps me to remember the dates.
Musical keys have colours - e.g. D flat is rather the same colour as number six. Key of G is similar to number five.
I had presumed most people saw numbers, not only in a pattern slowly zigzagging away, but also in colour.
Synaesthesia is useful in remembering dates, people's names, learning spellings etc, but has not helped me with maths!
I see the days of the week as small rectangles arranged from left to right beginning with Saturday and Sunday and extending to the right ending with Saturday and Sunday the other end.
To the right and left of this block there are other days, but they are blurred, both in the past and in the future, but the current week is bright and clear.
I can scroll backwards to view events in the recent past, or forwards to picture events in the near future.
Each day has a morning at the base of the rectangle, and an evening at the top. As I progress through the day I know where I am as I couldn't be anywhere but at 'that' point in time.
The days of the week all have their own characters; Monday is white and plain, Wednesday a bit lacy round the edges and Friday a bit like brown wrapping paper!
I have a form of synaesthesia. My form is primarily about time.
I 'see' the week as a ladder, the orientation of which varies, and I (and therefore the current day) are at a point on it. I look up and down the ladder for the different days.
The year is a circle in front of me, and I move round on it. December is always to the right and the summer to the left when it is autumn.
So the New Year always starts in the same place, where the circle starts to bend back from the extreme right.
I see numbers very clearly as colours. I seem to be able to remember patterns of numbers fairly accurately by the colour sequence that appears in my mind.
Sometime these colours clash and sometimes they blend together well. The colours are:
5 and 9 are non-distinct, almost earthy like colours. The closest colour I guess would be brown, but it is not at all strong.
- 0 White
- 1 Black
- 2 Yellow
- 3 Light green
- 4 Medium blue
- 6 Red
- 7 Purple
- 8 Orange
A telephone number might appear to me like a sunset, with a mixture of reds, oranges and yellows but with a dash of blue.
I have always seen numbers as definite colours, thus:
seven is pale blue, one is tawny gold
and 18 is dark green etc.
I cannot imagine all numbers being black - how boring that would be!
Sue Hook, UK
I have always seen numbers in a set pattern and this has made number manipulation quite slow for me as I have to travel along my number lines to get to the right place or to add or subtract numbers.
My pattern is: one to 12 are in a loop from left to right slightly to the left of me and then 12 - 20 are in a straight line going backwards to a point level with my ear more or less; then each group of tens goes in a joining loop till 100; then 100 on is a distant straight line.
I see numbers from one to twelve climbing from left to right in a fairly shallow gradient, then dropping sharply down to twenty, then starting to rise again up to a hundred.
It continues all over again up to a thousand. This applies when I think about numbers, whether numerals, words, or in any language.
Months go similarly, rising very gradually from January, rather more sharply from April. They level out at August (when my birthday is), then fall steeply to December.
Days of the week go in a straight line from Monday on the left to Sunday on the right.
But if I think about Saturday and Sunday specifically they are on the left with Monday etc. continuing rather hazily to the right.
I also 'see' numbers, years, months and weekdays on a 'scale' and I believe this contributes to my ability to remember dates.
I 'see' them at a certain position on my scale, which, for year dates, begins far to my left with the years BC.
From 1 AD the scale rises gradually at a slight gradient.
Around 1600 AD the scale is right in front of me and continues at the same gradient, to my right. At 1900 AD there is a hairpin bend!
I have had this mental image for as long as I can remember, as I have of the colours of the days of the week. These are completely different to my daughters' weekday colours.
They were also the subject of much amused debate with my mother. I thought everyone experienced this until I met my husband, who has no idea what we are talking about and thinks we are completely bonkers.
I see numbers arranged in lines in front of me and thought that everyone else did too.
I see numbers one to 20 from left to right, but I can move along the line.
Numbers 20 - 100 are in a line that curls round after 20 towards the far left; higher numbers float off into the distance.
I have always seen patterns of numbers in my mind for as long as I can remember.
I tried to explain what they were like to my teacher when I was about six, and I also tried to draw them for her, but I could not manage it because they are 3D!
I have patterns for ordinary numbers, the days of the week and the days of the year in months - not unlike a calendar.
I 'see' the patterns whenever I think of numbers at all, and they have always been the same, but I am not aware of any colours.
Maggie Heywood, UK
For me, the number sequences are curvilinear and slightly three-dimensional with highlighting. I either picture the whole sequence or zoom in when doing mental arithmetic, measurement or planning.
I know that the shapes and sequences are directly connected with early childhood learning experience, which I can reconstruct.
I have almost always related simple multiplication tables to my number map rather than reciting them to myself by rote.
If the phenomenon really was unusual, it would be difficult to explain it convincingly to people who have not experienced it, as it is in a sense very close to the core of one's private operating system.
I have always envisaged numbers in a certain way. I have different "time lines" and I can view these from different angles.
For example the numbers one to 10 can be viewed from either end but number one is always the smallest in size. I also associate most of these numbers with a colour:
I've always imagined everybody makes such associations with numbers and see them clearly in their heads.
- 1 white
- 2 yellow
- 3 red
- 4 green
- 5 blue
- 6 (nothing specific)
- 7 brown or purple
- 8 green
- 9 grey
- 10 gold or black.
Numbers also take on characters in my mind. Numbers four and eight are female, and pleasant enough. Six is male, and unpleasant. Seven is male, older than six and pleasant.
The numbers "pleasantness" can change depending on which other number it is with. "Pleasantness", in this case, being its willingness to make "10".
This must have come from early arithmetic at school.
For example - six is much more pleasant with the nice, female four (as they make 10), but five and six wind each other up - making 11 and meaning I have to "carry one" when adding up.
I have always seen pictures in colour for all things numerical. The numbers up to 10 do not even have the number in the picture, but after that they do.
So if I was asked to add three and four, the pictures for them would come up and I would recognise the picture answer as seven.
Birthdays are all pictures within a picture of the year. Bus numbers and money are also the same.
The way I see numbers changes depending on my current view point.
For example, if I am looking at numbers one to 10 from a low number viewpoint, the spatial orientation changes compared with viewing the same numbers from a higher value.
I find that I can estimate the approximate value of and addition or subtraction without actually doing the arithmetic.
I can visualise a number "string" which is like seeing a long winding path spiralling up to the heavens, much like you would imagine a path winding up around a fairytale castle in the clouds.
It is not vividly coloured, but rather monochrome and grey.
It starts at my bottom left and rises in a series of ramps with turning points at 10, 20 etc. Thus it is in base 10.
The path turns frequently but never crosses over itself again.
It describes a rising spiral up to 100 which is at my top right. It continues quite lucidly up to 1,000.
In much the same way I visualise the year.
I visualise the calendar entirely dependant upon where I stand today, thus November looks very different when the current date is in May, than when I look back at it from, say January.
I can easily remember dates, even of events many years ago, by reference to the pictures in my head.
I see numbers in colour, and letters too. Each single digit has its own colour, and keeps it when combined with other numbers.
For example, two is red and four is green, thus 24 is half red and half green.
For adding small numbers up I find this really useful because for me two colours add up to another colour - white + blue = green (you can work out which numbers are white and blue!).
The colours help me remember long numbers like phone numbers and postcodes, because I know if the pattern of colours is right.
Similarly people's names have colours associated with them!
I also experience numbers arranged in a complex shaped line.
I remember being able to "see" the numbers in this way when I was learning to count up to a hundred as a small child.
I also have other mental arrangements for days of the week and the year, but not for the alphabet.
I also hear different vowel sounds as colours. This in turn leads to the numbers having colours, taken from their pronunciation; "one" is red, "five" and "nine", having the same vowel sound are both light green, "seven" and "ten" are brown, etc.
It is nearly 50 years since my maths master was amused to find that I saw numbers in 3-dimensional patterns; e.g. that number "50" is somewhere to the left just below shoulder height.
I wonder if someone in the psychology field has taken this up. Numeracy is a big educational issue, and understanding how people deal with numbers must be of importance.
I have a very strong colour feeling for all numbers from one to nine:
Dorothea McEwan, UK
- 0 no colour
- 1 grey
- 2 pale blue
- 3 straw blonde
- 4 strong blue
- 5 brown
- 6 bright green
- 7 the colour of the ocean
- 8 reddish
- 9 grey
While, sadly, not seeing numbers in brilliant Technicolor, I have long had discussions in the family about the colour of Monday, where is 53 and which direction do you go from 1918 to 1914?
I do certainly associate shade with numbers and date - the shade depends on which side of the number one places oneself - if that makes sense!! The point being I can move myself around the number (or date) series and the shade depends on the direction of viewing.
I raised the subject of "where is 53" with a group of engineers.
I received the usual blank looks, but there was one other person there (in a group of about 15) who admitted to "seeing" numbers.
His format, however, is a Snakes and Ladders board.
Dr David Weaver,
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The avoidance of a numerical scale of belief, is excused on the grounds that ordinary people do not have the required numerical sophistication, or that the measurement of probabilities is too imprecise and too constraining on the decision makers.
The standard of proof in criminal cases is 'beyond reasonable doubt', the suggested formula for judges to use is 'as certain as you would be for an important decision'.
Civil cases are decided on 'the balance of probability', which is usually accepted as equivalent to just over 50% probability or 'more likely than not'.
These two standards of proof, one vague and one specific do not suffice for all judicial decisions
One comment that I would make about the 'A University degree is worth so many thousands over life' is that the wrong number is being calculated.
What you actually want to know is the value at some time of the quantity of money.
So you could calculate the Net Present Value of the income.
This would take into account that graduates may have earnings more heavily weighted towards the end of their life.
On top of that, you would really want to look at the distribution of these numbers as averages can easily be deceptive for two sets of numbers which have different distributions.
If graduates earn an extra £400,000 in their lifetime, they will pay additional income tax.
If they are at 40% marginal, this will be £160,000. Even if they are only paying at 20% this will be £80,000 i.e. greater than what the government want them to stump up.
Perhaps the government should be investing in students to generate greater tax revenue in the future.
Surely the use of the statistic stating that university graduates will earn £400,000 more that non-graduates will not sustain as the number of graduates increase.
How could 50% of the population earn so much more than the 50% who did not graduate?
The more graduates there are, the lower will be the differential.
Russell Davies, UK
We are constantly being told that graduates earn something like £400,000 more, over their lifetime, due to their degree. This seems to be a numerical fallacy.
What is not being done but should be done is to compare the earnings a person has as a graduate with what that same person could have earned without a degree.
For example, my brother left school at 16, I went to a top university and got a degree and followed a career in engineering to a senior post.
The nearest we have to what I might have earned without that degree is the comparison with what my brother earned.
I have little doubt that we earned a very similar total over our working lives.
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I'm afraid your mathematician omitted some vital aero and hydrodynamic information!
True, the spin does have a very important gyroscopic effect (a perfect stone would have all its mass at its outer edge). But the spin also plays a vital role in stopping the stone from sinking into the water on impact.
In the mentioned equation, v (velocity) was taken to equal the forward velocity of the stone. This is not so. v is the relative velocity between the stone surface and the water.
Assuming the stone rolls along the index finger at point of release when the stone impacts the water relative velocities up to 2 x v are achieved. When this is squared, this is an enormous difference to the forces your mathematician was talking about.
Of course you would have to look at the resultant speeds across the whole of the stone's surface and consider the shape of the stone which would govern the fluid flow and resulting thrust achieved.
I was amused by your piece about stone skimming, known to generations of English children as 'playing ducks and drakes.'
Spinning the stone is very important, also crouching low so that it hits the water at a narrow angle.
The choice of stone is also very important, flat ones being much more suitable than round ones - in my childhood, we particularly favoured smooth pieces of slate.
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I was very interested in your item on breast cancer screening.
I had heard of a survey in which two matched groups of women were monitored. One group had screening, one did not.
They were followed over quite a number of years. The death rate in both groups was the same.
However in the screened group fewer deaths occurred from breast cancer, but more deaths from other things.
The explanation was that as screening makes people anxious, this will cause a slight rise in deaths in itself.
Then as reported in More or Less, you get a lot of false positives which causes more anxiety and further examinations both of which are further risk factors.
The further investigations also give quite a lot of false positives, this will mean more invasive investigations with higher risks especially if an operation under general anaesthetic is involved.
So overall it appeared from this study that the considerable cost of screening might be better spent on something else!
I want to endorse the sentiments in your piece on the abuse of statistics in reporting medical stories.
As a regular contributor to the Usenet newsgroup dedicated to breast cancer support I find myself regularly having to explain "what the reporter really meant".
The most common offence is using a difference of differences, like the 6% of 8% you cited in connection with the breast cancer and alcohol story.
The point being that the first number is meaningless (and could be mathematically indefinite) if you don't know the second.
The statistics relating to alcohol consumption and breast cancer in women had worried me a lot.
But I still continued to have two small glasses of wine with most evening meals and two or three halves of bitter in the local pub three times a week.
So I am relieved to know that alcohol is increasing my risk of breast cancer by such a tiny fraction, especially as several other facts over which I have no control are much more significant.
My mother and my father's mother died of breast cancer and my twin sister was treated for it a year ago.
Another set of figures in your programme reminded me of my sister's case too - her cancer was discovered about eight months after an 'all clear' mammogram.
You were certainly right to stress that statistics cannot be applied to individual cases.
To avoid further confusion between the American and English billion perhaps you could adopt the milliard as the correct English term for the American billion?
So we have million, milliard, billion and trillion... and may I suggest for a thousand billion, a billiard?
Big numbers need not be confusing. We should use the English billion instead of the American.
The 'bi' in billion stands for two - as in bicycle (a two wheeled vehicle).
Similarly, the 'tri' in trillion stands for three. It now becomes easy to understand words such as quadrillion, quintillion, etc.
Once we understand this, it becomes easy to calculate the numbers - there are six noughts in a million.
For billion we multiply six by two, for trillion we multiply six by three, etc.
versions of these words are difficult to understand and we should stop using them.
language already contains a perfectly good word - milliard - for a thousand million.
From 1968 to 1970 I used to work in St Christopher House, Southwark Street in London and our moderately lofty daily view was of Bankside Power station.
Day in day out glancing at it whilst trawling through files made me rather fond of it.
When it became Tate Modern I was thrilled to bits!
We often wondered how many bricks it took to build such a vast edifice, guessed, but had no way of knowing the answer.
Many thanks for the programme giving the informed estimate of 4.2 million.
Maths and music
Mathematics is beautiful. Pythagoras said "The whole world is hidden in numbers".
Fibonacci discovered the "Golden Ratio" which is the key to good design.
The major and minor musical scales fit perfectly on the Golden Ratio that Fibonacci discovered.
J.S.Bach, often did not write the full harmonies to his music referring instead to "5/4 or 6/4 or 6/3" chord structures.
Music is another expression of numbers or maths.
The frequencies involved in music are the beauty of music and maths.
D J Edwards,
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