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Maths in music

Tim Harford said that any finite sequence of numbers will inevitably occur in the value of pi because it is an infinite sequence of "random" numbers. Is that proven? Surely the sequence is not random in that it needs to express the value of pi, so might it be possible that some specific finite sequences are not permitted?
Stuart Ash

You said "Since the expansion of pi is infinite, Kate could sing any sequence of numbers she fancied and they would match some part of pi." Really? That may be true for an infinite series of random digits, but is it true for pi? As an extreme example, the decimal expansion of pi can, by pairing the digits and adding 32, be converted to a sequence of ASCII characters (letters, numbers and punctuation marks). Does it follow that pi contains the complete works of Shakespeare somewhere within the decimal expansion? Intriguingly, what you assert as "true" is only one of four possibilities! I would refer you to 'The Emperor's New Mind' by Roger Penrose and the discussion about whether the decimal expansion of pi contains 20 consecutive 7's i.e. pi = 3.141592653589793 ... 77777777777777777777 .... or not.
Trevor Smith

When talking about Kate Bush and pi in tonight's programme, I think it was implied that the decimal expansion of pi would contain every possible sequence of numbers because it is infinite. This was a conclusion some of my students assumed recently but I thought it might not be true. It seems clear that being infinite is not enough (1/3 = 0.33333...) or being irrational (0.101001000100001....) Is there some reason why pi is different?
Alyssa Rowlandson

Your statement that because the number pi is infinite then any sequence, (which I take to mean any finite sequence of numbers) will appear in somewhere in pi is incorrect. Currently the only way to determine if a particular sequence exists is pi is to actually find it. The question is equivalent to asking if every sequence of k digits appear equally often for every positive integer k. A sequence which has this property is normal - The question as to whether pi is "normal" is still a open question - unless you have a proof. An counter example to your statement is the irrational number 10010000100000010000000010000000000100..... where every digit is a zero except where the position is a perfect square i.e. at positions 1,4,9,16,25 etc which contain a 1. The number is irrational and therefore non periodic and it does not contain any of the digits 2,3...9. In fact it does not even contain the sequence "11". Perhaps you can have the "Kate Bush" Conjecture - "the sequence in the Kate Bush song 'Pi' exists in the decimal expansion of Pi".
Mark Anderson

Quoting from last week's programme "Kate [Bush] could sing any sequence of numbers she fancied and they would match some part of pi." So, if we think up a substitution code of digits for letters of the alphabet, preferably using all digits equally, then, ignoring punctuation and spaces, any piece of text is matched somewhere in pi. Trying enough codes, we should be able to find a stretch matching the phrase "life, the universe and everything" for instance. But why stop there... There must be whole books encoded in pi. Find the right code, and all the Harry Potter books should be in there, and always have been. Also complete histories of mankind, from beginning to extinction, at least one version of which will be correct. Some books will require improbable sections of pi, such as Welsh Telephone Directories, requiring lots of repeated strings of digits coding for Jones etc. So we need something like Enigma machine coding where the substitution of digits for letters is always changing. We must start building Bletchley Park 2 immediately; a massive supercomputer dedicated to deciphering pi... the truth is in there!
Chris Webster

I was delighted to hear your article featuring Kate Bush's ? (Pi) song. During the article you point out that Ms. Bush makes a mistake at the 54th decimal place. However, it seems obvious to me that you have fallen for a deliberate mathematical practical joke:- because in scrutinising the lyric to this degree, you have in fact become the "sweet and gentle and sensitive man with an obsessive nature and deep fascination for numbers and a complete infatuation with the calculation of pi" she is singing about. I recall in the week the Aerial album came out a noted Expert/Professor appeared on either the Today programme (or was it PM?) to expose the error: "1415926535 8979323846 2643383279 5028841971 6939937510 58231974944". Made me laugh. She's not just a pretty face.
Jon Cousins

At the end of today's programme Tim said that eventually the 50 or so numbers sung by Kate Bush would occur in that order given that Pi is infinite.. But surely unless you know that these numbers occur in the quoted order within the known portion of pi then you can only state that it is almost certainly probable that the numbers will match a given fragment? Whilst a combination of any two digits is undoubtedly certain the likelihood of any match must decrease the longer the string is?
Richard Mansfield

Stat nav

I got the impression that the driving organisation which came up with the 16% relevance of Traffic Reports, based this on the number of drivers listening. Shouldn't the number of total listeners be used in the calculation, many of whom may be at home, or listening again?
Stewart Hey

While some traffic will certainly be on the road travelling in the opposite direction as the "incident", that does not mean traffic flow is not affected... In my driving experience in the Greater Toronto Area, an accident will slow traffic in the opposite lanes as well... Unfortunately, the "rubber-necking" usually will be proportional to the severity of the crash. A few years ago when an airplane slid off a runway at Pearson International Airport, the highways around the airport were quite hazardous to drive on as drivers were often looking away from the road to look for the crash site. So maybe the "usefulness" of a traffic report is subjectively tied to the information you glean from it.

There is another variable in the "Road Traffic Report" equation. I live a few miles off the M1. If there is a road traffic report about a hold-up on the motorway, I know that traffic on my route to work will increase, even though I do not travel on the motorway, as drivers seek to avoid the hold-up. Personally I'd rather there were no traffic reports, and I would be interested to know the time taken by a driver who seeks to avoid an obstruction by taking to the congested alternative route as against a driver who remains on the motorway, and crawls past the obstruction.
John Rouse

There are a number of mistakes when calculating the probability of a traffic news report being useful to a motorist. I agree that a report that doesn't mention a particular road is useful, but only if all traffic problems are included in a report. However you did not take into account the knock on effect of a problem on other roads. Secondly, you restrict your data sets. Radio 2 is not the only source of traffic bulletins, local radio also gives this service, and most car radios can automatically switch to traffic bulletins. Also you only use roads managed by the Highways agency, I think that most roads are local and not covered by the highways agency. I would be interested to know whether most journeys are on local roads. The usefulness of a traffic report also depends on how timely the information is. If you hear a traffic report that the road ahead is blocked, it is only useful if you haven't already passed the last place that you could divert to avoid the problem. The highways agency person you interviewed implied that they counted vehicles on the section of road affected, the relevant number is those vehicles that avoided the incident. Additionally sometimes travel news bulletins include information about other forms of transport, these of course are completely irrelevant to a motorist.
David Pearce

Given the propensity for traffic in the other direction to also be affected, as drivers slow down to catch a glimpse of the accident, I'd say that you could possibly increase the figure of 16% of people affected by traffic reports.
Neil Tarrant

I think there's an incorrect assumption in your analysis of traffic reports. It seems to me that hearing a traffic report is useful for 100% of those driving at the time. It's best expressed in binary - a report can either mention roads that I am driving on or not mention them. If it mentions them then I know what to avoid or expect. If my roads are not mentioned, then I can assume that they are clear and my journey is unaffected by delays.
Philip Parr

Some 15 years ago I was delivering a rental car to a customer in Horsham. As I approached the town, there was a radio report that the very stretch of road I was about to approach was closed because of an accident. Not knowing the area and not having a map-reader with me, I decided to carry on and not risk getting lost on a detour. I drove right through the supposedly affected stretch of named road without obstruction - without meeting a single other vehicle, rare in the south east - and without seeing any sign of an accident. No crashed cars, no debris. Having delivered my car and collected the one it was replacing, some 20 or 30 minutes later I drove back along the same route. There was another identical radio report saying the road was closed to all traffic. I drove seamlessly back to Brighton.
Anne Powell

Travelling in the rush hour in London the folk in my office often said they would (depending on the reported cause) immediately head for the route said to be congested as all the sheep avoided it and caused alternate routes to be congested while the supposedly congested route was now clear. This could have been because of the news either being out news or the route being able to cope better than the alternatives with the now redirected traffic post news bulletin.
Laurie Smith

Giving it 110%

I am currently acting in The Killing of Sister George by Frank Marcus in which the exotic clairvoyant Madame Xenia tells BBC executive and agony Aunt, Mrs Mercy Croft, that "...your advice is a hundred per cent. A hundred and twenty per cent." The play was written in 1965; somewhat earlier than the first instance of the phenomenon known to your contributor, which I believe she said was in the 1983 Torvill and Dean biography.
Diana Lane

I'm reading John Buchan's The Thirty Nine Steps - which pre-dates Torvill and Dean, I think - and on page 46 of the Wordsworth edition the hero Hannay thinks to himself: I mightn't be much of an orator but I was 1000 per cent better than Sir Harry.
Steve Moore, Istanbul, Turkey

I enjoyed hearing about 110% - surely this is just hyperbole, which you all use now. You quoted Susie Dent saying something like "to use 100% would be fantastic" - she actually meant normal or sensible, not fantastic; I suppose that in my fantasies you all (and I blame the media though I cannot identify who started it) would start to use fantastic to mean 'extremely fanciful' instead of "that's right" or "thank you", and similarly with other hyperbolic adjectives.
Mick Leverton

I worked for a very happy company where the motto was "we aim for perfection, but we will accept excellence". This we interpreted as "we give it 100% and are satisfied with a 90% result", and everybody was content. Then we got new management who wanted "110% from everybody" so what he really got was 85% -- we spent the odd 5% making it look like 110%.
Harry Weston

On "110%" etc.: in Truman Capote's "Breakfast at Tiffany's", published in 1958, Holly Golightly's former friend and flatmate, when asked over the phone about Holly's arrest, replies in very hostile terms and adds "And my husband agrees one thousand percent." I expect the next comment will cite either Dr Johnson or Pepys.
Catherine Atherton, USA

More or less can of course apply to words as well as numbers. Taking your item in last programme on 110% or even 1000% to make your point seems similar for the very widespread use of the adverb "really" i.e. "really, really"
Nigel Tibbs

During the discussion on "110%" at no point did anyone question what the percentage was supposed to represent. This seems an inexcusable error for a programme promoting numerical analysis. There seems to be an assumption that the percentage represents "effort", but "effort" can be interpreted in different ways. The term "110%" is a perfectly valid numerical term to use in reference to work. Suppose I am contracted to do 7.5 hours work per day (and do not get paid overtime), now suppose that in order to get a report completed I work late and end up working 8.25 hours. I think that could be accurately and numerically regarded as "giving 110%". I have given 10% more work than was expected of me, without remuneration. What's wrong with expressing such sentiments using such simple terms ?

Talk of giving 110% etc. is not necessarily evidence of innumeracy. A factory may produce, for a limited period, at levels exceeding its long-term capacity, e.g. by deferring maintenance and exhausting stocks. Similarly, a hospital may try to treat too many patients, over and above its long-term capacity, so that no spare capacity exists to manage - say - an outbreak of MRSA. NB linguistic inflation is a widespread phenomenon as notably media folk try to draw attention to themselves. Ordinary folk do it too - how often do we hear the genuinely upset refer to themselves as "devastated", as if they were Coventry after the blitz?Jonathan Phillips

On the subject of linguistic inflation: I have just heard the Hull City manager, Phil Brown, say that his players are "a million per cent" committed - twice.
Thomas Mansell

I think that you needn't mention 105%; it's just poetic licence. It's not incorrect in the same way as "very unique". What annoys me is professional commentators thinking that increasing by 200% means doubling, when in fact it means trebling.
Bryn Jones

Just caught the tail end of your piece on "giving 110%" and I think I can trace the somewhat surprising use of this way to describe effort back to the maths department at Edinburgh University. In my first year there in 1970 I was astonished to see that one fellow in my class regularly achieved over 100% marks in his exams. The explanation I received was that he had demonstrated different ways of solving the same problem. This was reflected in the mark awarded. Perhaps Torvill and Dean had the same lecturer :o)
Leslie Martin

Surely the use of statistics above 100% pre-dates Torvill and Dean in the 1980s. Montgomery once said to Churchill " I have never smoked in my life and I am 100% fit" and Churchill replied "I have smoked all my life and I am 200% fit."
Tony Walker

In industrial engineering, 100% represents the normalised work rate for an operation. This takes into account motivation, fatigue, physical size and strength. 100% represents an average person. Usually people are only expected to work at 85% across a whole shift. A highly motivated, strong and experienced worker could very easily achieve 110%. A highly skilled and motivated super human can't achieve more than 200% though. If 100% is what could be expected of most people, 110% isn't actually putting in much extra effort!
Philip Hughes

Credit crunch metaphors

I think that taking financial risk is like driving a car: once you leave the city limits and start to increase speed, passengers start to feel uncomfortable when you reach, let's say, 130km/h. At 150km/h they are really nervous. However, if you spend 30 minutes at 200km/h, passengers will get used to that speed and may even fall asleep. Taking financial risk is similar: if there are gains investors will forget about the risk they are taking (high speed). Even if there are warnings (slow down), they won't pay attention because investors have lost sensibility towards risk. Of course, that is until... you crash!
Armando Galindo, Mexico

A metaphor for quantitative easing: As a teenager you go to the pub and spend all your money including the fare for the cab home. Rather than walk back through the dark streets you get a cab, safe in the knowledge that your dad will pay for it when you get home. There may be a few cross words but it's worth it and chances are you will get away without paying it back as he gave you the money to go out in the first place.
Ali Palmer, UK

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