Having spent the last several weeks, months and years trying to get students to at least be interested in accurate mathematics then this is a real kick in the teeth. Back in 2005 Ruth Kelly, then Minister for Education, “… announced that pupils taking GCSE maths and English would have to pass a test in functional skills, such as writing a letter or working out their family budget.” (Source – BBC)

Today, at the same time that the world was waiting for the outcome of the G20 final commnique, a little bit of bad news got buried. This was “The government has dropped the key part of its pledge to improve teenagers’ functional English and maths skills” because “qualifications advisers have said that making GCSE results dependent on a separate skills test could bring the qualifications into disrepute.” (Source – BBC)

Isn’t that fairly obvious conclusion that the GCSE IS in disrepute anyway because it was originally thought necessary to have to include these skills assessments? What has happened to examinations that you can pass them without having at least a basic understanding of the subject?

Someone, somewhere, has screwed up and is failing a generation that for whom they’e supposed to be the guardians of the standards. Adn we, in universities, are having to pick up the pieces in order to generate the graduates with the talents needed to help the country on the road to recovery.

A Lesson in Vertical Stratigraphy

In conventional archaeology, if evidence is found then the age increases the deeper in the stratigraphy you excavate. Exactly the same principle applies in geology, and, naturally, in the papers on my desk. The deeper you go the older the deposit.

However in kitchen cupboards the stratigraphic succession is marked by vertical stratification, the older stuff is usually towards the back. Just how old is only revealed when you clean out the cupboard because you’re sure you have some of that vital ingredient somewhere and you need it, and THEN you find it’s out of date. (And then you have to make a quick trip to the overpriced local shop, but that’s another matter). As kitchen cupboards are usually full to overflowing (well, ours are, anyway) you then realise that it’s time you need to make some space.

And then the memories start flowing. Tins of black treacle and now solid packets of dark sugar for puddings of Christmases past; exotic spices from the near and far Orient, but sadly now having lost their pungency; the dregs of custard powder; the dead garlic that you were sure you had but couldn’t find some months ago; honey, the product of labouring bees from summers past, now solid in the bottom of the jar; packets of something or other that might have been a good idea at the time… 

And now they inhabit the depths of the bin bag. It’s a bit sad really and you start making plans about making sure that you utilise the stuff you buy, dont waste it, a repeat of all the good resolutions  you made last time you cleaned out the cupboard.

But in archaeology and geology you can only estimate dates. With the kitchen cupboard the science is a little more accurate as you have the sell by date to go from. And some of them are embarassingly far in the past.

I’ll stop there.

Factorials

This article is written primarily for use with Business Students who deal with chance and randomness.

Ever wondered what that key on your calculator with x! on it did? If you chanced to look it up in the manual you’d have found it was the factorial key, and if you’d pressed it something less than comprehensible may have turned up.

A factorial is the product of all integers from the number you want the factorial for down to one, subtracting one each time.

x! = x ×  (x – 1) × (x – 2) × … × 3 ×  2 × 1

The ellipsis (…) in the middle means that you just continue with the same maths until you meet up with the other end.

So:

4! = 4 × 3 × 2 × 1

4! = 24

And:

7! = 7 × (7 – 1) × (7 – 2) × … × 3 × 2 × 1

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 5040

Now that’s a bit of a difference – possibly more than you’d expect. By the time you get to 10! you’re in the millions. It seems such a simple, innocent sum, really.

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

10! = 3,628,800

We’ll look at why all this is necessary after you’ve had a look at the table below.

Factorial and other values

Factorial	Number	Setting the scene with examples
0!	1	The factorial of zero is 1 Source
1!	1	The factorial of 1 is also 1
2!	2	2! = 2 × 1 = 2
3!	6	3! = 3 × 2 × 1 = 6
4!	34	4! = 4 × 3 × 2 × 1 = 24
5!	120	5! = 5 × 4 × 3 × 2 × 1 = 120
6!	720	6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
7!	5,040	and so on
8!	40,320
9!	362,880
10!	3,628,800
11!	39,916,800	Chance getting six numbers in the UK National Lottery: 1 in 13,983,816 There were 49,138,831 people in England at the 2001 census. Source There were 60,975,000 people in Britain in mid-2007 Source Chance getting five numbers and two stars in the Euro Millions Lottery: 1 in 76,275,360
12!	479,001,600	Population of the European Union on 1st January 2009 (Estimate) 499.7 million Source
13!	6,227,020,800	The 2008 estimate for the world population is about 6.7 billion. Source
14!	87,178,291,200	About 100 billion (1011) people ever lived SourceThere are also about 100 billion stars in the galaxy Source
15!	1.31×1012	And about 500 billion galaxies in the Universe Source
16!	2.09×1013	Wealth of world in dollars – $44 trillion dollars (that’s 4.4 x 1013) Source
17!	3.56×1014	And we’re still only at Factorial 17
18!	6.40×1015	How far is a light year? 9,460,536,207,068,016 metres (9.4 x 1015) Source
19!	1.22×1017	The age of the Universe, i.e since the Big Bang, is between 13.6 and 13.8 billion years (Source) A year, currently is about 365.25 days and there are 24 × 60 × 60 seconds in a day. So if we multiply all these together we get the age of the Universe in seconds, a mere 4.3 x 1017 seconds. And we’re not up to factorial twenty yet!
20!	2.43×1018
21!	5.11×1019
22!	1.12×1021
23!	2.59×1022	So how many stars are there in the Universe? You can only estimate, but if we take ours as a sort of normal, run of the mill galaxy, with 100 billion stars, and there are 500 billion galaxies, what we have to do is multiply them together. Then we get about 5 × 1022 stars. If we write this out we get 50,000,000,000,000,000,000,000 stars.  Quite big, but we’re still only at factorial 23.
24!	6.20×1023
25!	1.55×1025	In chemistry a mole is the number of grams of a substance equivalent to its molecular mass. For water, H2O, this is 18g, so there are 1000/18 = 55.56 moles in a litre of water. Each mole of any substance contains 6.02 x 1023 molecules (Avogadro’s Number Source), so to find the number of molecules of water in a litre you multiply them together and get 3.34 x 1025.
26!	4.03×1026	If you want to find the size of universe then there are many different theories. Figures vary, but about 1027 metres across may sound reasonable. What is certain is that no-one will ever know for sure.
27!	1.09×1028
28!	3.05×1029
29!	8.84×1030
30!	2.65×1032
31!	8.22×1033
32!	2.63×1035
33!	8.68×1036
34!	2.95×1038
35!	1.03×1040
36!	3.72×1041
37!	1.38×1043
38!	5.23×1044
39!	2.04×1046
40!	8.16×1047
41!	3.35×1049	If you ask how many atoms there are in the earth you get to about 1.3 × 1050 Source
42!	1.41×1051	We’re only at factorial 42.
43!	6.04×1052
44!	2.66×1054
45!	1.20×1056
46!	5.50×1057	Atoms in the sun? about 1.19 × 1057 Source
47!	2.59×1059
48!	1.24×1061
49!	6.08×1062
50!	3.04×1064
51!	1.55×1066
52!	8.07×1067
53!	4.27×1069	Atoms in our galaxy? About 1.2 x 1068 Source
54!	2.31×1061
55!	1.27×1073
56!	7.11×1074
57!	4.05×1076
58!	2.35×1078
59!	1.39×1080	If we’re talking about atoms in universe then a simple answer might be ‘a lot’. How you define the universe is central to the argument, but about 1080 seems to be a reasonable guess. No one’s ever going to be able to count them, anyway.
60!	8.32×1081
61!	5.08×1083
62!	3.15×1085
63!	1.98×1087
64!	1.27×1089
65!	8.25×1090
66!	5.44×1092
67!	3.65×1094
68!	2.48×1096
69!	1.71×1098	At 9.99 x 1099 most calculators give up.
70!	1.20×10100	A Googol is defined at 10100, 1 with a hundred zeros after it. Source
80!	7.16×10118	From now on numbers go up in tens
90!	1.49×10138
100!	9.33×10157
110!	1.59×10178
120!	6.69×10198
130!	6.47×10219
140!	1.35×10241
150!	5.71×10262
160!	4.71×10284
170!	7.26×10306	At this point Excel 2003 gave up.
	About 1012,978,189  This is a number with close on 13 million digits	A prime number can only be exactly divided by itself and 1 – it has no other factors. Large prime numbers have significance in computer security and encryption. Mersenne primes have specific properties (Source) and the latest and largest was discovered in August 2008. The largest Mersenne Prime is 243,112,609 -1 precisely
	10googol  1010100	10 raised to the power of a googol is a Googloplex.To see it written out you’ll have to go to the source. If you were to attempt to write out a googolplex, you have to use more space than known universe occupies. But in number terms we’re not finished – not by a long chalk. An infinitely long chalk.
	∞  The symbol for infinity	Infinity is a very long way off! In fact, no one will ever get there.  The universe is finite. Numbers go on for ever.

Why?

In searching out probabilities for events to happen you must be able to know the likelihood of events happening, even though those events coud happen in a random manner. Suppose that you are in charge of maintenance for a call centre and need to predict the likelihood of failure of PCs. A failure of a PC means lack of productivity, so you keep a number of spare PCs in stock. How many spares you need can be calculate, and this calculation involves factorials.

Chance

If you’ve ever played the UK National Lottery main game, you’ll know that you choose six numbers from 49. The number selection is made using a machine with identically sized, differently numbered balls. The first ball our can be selected from a set of 49, the second from a set of 48 (because one has already gone) the third from 47 and so on. In order to calculate the chance of getting all your six numbers in the first six balls to drop out is an application of combinatorics, the branch of pure mathematics that studies discrete objects. The full arguments can be found from this source.

A future post on this blog will deal with chance and choice.

The story starts on Monday last, 15^th December. A posting was made on one of the forums run by the Dogs Trust, UK rehoming charity, that a dog was threatened with destruction. Now this happens every day in this and a load more countries. But something was different. We couldn’t have more dogs – the campsite we go to has a limit of two dogs per pitch. No way could we help.

That situation lasted for less than 24 hours when we learned that the reason that the dog was being put for destruction was that he wagged his tail too much and was in danger of harming a three year old. One phone call to my wife, her phone call to the campsite to see if they’d bend the rules in the circumstances and we said we’d have him. This was on Tuesday 16^th December.

Now here’s the insane part. Not only had we not seen him, the person rescuing him (in Lancashire, a hundred and twenty miles away) had not seen him either. The only description was that it was a collie cross. Some frantic e-messages and one phone call and we’d arranged to collect him on the Thursday evening. So why were we doing it? We don’t know.

So Max (the dog) duly arrived at the house of sanctuary and was temporarily assimilated into that household of family and two border collies. Thanks to the web we had some photos to look at by Wednesday evening. Thursday came, brought a meeting forward and set off.

In the rush hour.

And rain.

And discovered that there was an accident blocking the motorway.

And followed the route that everyone else was following to try to avoid the blocked motorway.

And it took more than four hours!

So then in a Lancashire village a knock on the door was answered by multiple dogs barking. I’d arrived. And an hour later following refreshments and loads of talk we were off again. And at midnight arrived home with a dog that we didn’t know. He’d not have been here had he not been taken in.

We’ve had the expected dominance issues with the other two which are being resolved. He’s an older boy – needs his rest and peace.

Max, the furry footwarmer (he’s taken up residence in the dog basket that lives under my desk) is OK. He’s obviously been traumatised by the events of the past week for him – having been moved twice. He’s not eating very well – certainly not dog food, so we’re tempting him with sausages and roast chicken. We think he’s holding out for this, sometimes. The person who rescued him found that he wouldn’t eat much, either, so we have progress.

When we went to the vet to have him checked over and microchipped she said that if owners have made the decision to have a dog put down then they withdraw from it, just giving food and shelter, rather than interaction and attention as well. The dog then reacts to this by putting themselves on a care and maintenance basis, just eating what is necessary to survive. In Max’s case this may have been going on for some time. Sue, who rescued him, was told that he only ate a small amount and only in the evenings. We’ve introducing him to our routine which involves breakfast and a certain number of treats during the day. We’ve got to build him up a bit as he’s quite thin.

He’s got a heart murmur, leaky valves probably, but he’s ten and some sort of medical condition was expected. However many dogs have such a murmur – and if he contracts a cough we’re to get him back there sooner rather than later. Medication for things like that can help immensely.

Sandy, the top dog, is being a little possessive of tennis balls, and is showing it. However we’re working on it. Kerry, on the other hand, wants to play. Max is still very unsure, as apart from two days in Lancashire he hasn’t lived with other dogs before.

Our Christmas preparations, which were ahead of plan last weekend, have now descended into an almighty rush, much as usual. The dust can wait. The dog is more important.

Why have we done it? Still don’t know.

But we could not see a dog put down because its tail wagged too much. We’re very glad to have him.

Max the happy dog. Also known as Sir Maxwell Whiffalot as we've discovered he likes self-fragrancing in deposits left by foxes!

Link to Max’s Kennel on Doggysnaps

Link to Max just after Rescue

Link to Max’s personal blog on Doggysnaps

After the investiture - or - The Yeomen of the Guard appear to have caught a troublemaker.

In June I wrote about my wife’s aunt, Dorothy Charles, being awarded the MBE (Member in The Most Excellent Order of the British Empire) in the Queen’s Birthday Honours list. Today she was invested with the honour by Her Majesty the Queen at Buckingham Palace.

They went down yesterday and stayed at The Rubens Hotel, five minutes walk away fron the Palace, ‘they’ being Dorothy, Sandy my wife and Val Holyman. Four poster beds were the order of the day. I was in receipt of a phone call last night where the giggles were decidedly girly, belying the chronological age of the perpetrators. Imbibation of certain varieties of alcohol were to blame, I assumed, and detail provided later on their return home has not made me change my mind.

The ceremony is held in the Ballroom, and it was an honour for them to share this occasion with the awarding of the George Cross to Lance Corporal Matthew Croucher (BBC report and Video). In the queue  prior to the investiture Dorothy was chatting up (or being chatted up by) Henry Sandon (of Antiques Roadshow) who was also receiving an MBE.

There was a walk across the forecourt of Buckingham Palace in from of the Guard, with the gathered tourists outside the railings photographing and videoing. There was the ascent into the ballroom via a lift as Dorothy’s eyesight is failing as falling down the Queen’s staircase would have added incident to the day which was quite overwhelming enough. The investiture is a personal moment between The Queen and the recipient.

Dorothy after the ceremony flanked by Val (right of picture) and Sandy (left of picture)

In the courtyard, after the formality there’s time to make the day complete with photographs. Apart from being captured by the Yeomen (above) it is one of the times in a life when the place is as important as the people. And not just for the recipient of the honour – those accompanying shared the immense sense of occasion and the history.

The medal is cruciform, surmounted by a crown. The inscription reads FOR GOD AND THE EMPIRE. The ribbon is rose pink with pearl grey edges.

Needless to say this day will stay with Dorothy for ever.

Bank rescues are in the news, and of course we’re all affected by them. But the numbers involved are somewhat larger than we’re used to. We can glibly talk of the £500 billion rescue package for UK banks but the number may not be comprehensible. Even though it’s broken down into £50 billion capital injection, £200 billion short term loans and £250 billion loan guarantees, can you really imagine those sums of money? Can you even imagine the net worth amassed by Bill Gates (who’s been toppled from the top of the richest people list)? Can you imagine a million pound bonus?

When I was regularly leading field trips in geology the question of millions came up, because “millions of years” is the usual measure of time in that discipline. We used units of time to build up millions. Here it is – with modifications for the current climate.

Think of a second.

Think of a minute. Could you close your eyes for a minute plus or minus a few seconds? Very few people can, unless they count seconds while they have their eyes closed.

So what about multiples of a thousand?

• A thousand seconds is 16 minutes 40 seconds
• A thousand minutes is a long waking day, 16 hours 40 minutes.
• A thousand hours is just short of six weeks.
• A thousand days – 2.75 years – just about the time for someone to join an English university in a September and graduate three years later in June.
• A thousand weeks is the time from being born and about the average age for leaving home – 19 ¼ years. Parents may well start counting weeks if they realise this.
• A thousand months is a good lifespan for the people in the UK, just over 83 years.

But that’s a thousand – what about multiples of a million?

• A million seconds is just over 11 ½ days
• A million minutes is just short of 23 months.
• Very few people indeed live to be a million hours old, as it’s over 114 years.
• A million days takes us back 2749 years, back to 740BCE, around the time of the foundation of Rome.
• A million weeks ago we’d all have been very cold. It was the middle of the coldest Devensian Glaciation , over 19,000 years ago.
• A million months takes us back more than 83,000 years, At that time Britain was in the warmer Ipswichian period, when forests covered much of Britain and animals like elephant, hippopotamus, beer, bear, lion, boar and hyena would have been present. But no humans or horses.

Translating our time based approach to money takes a little thought. It should be apparent that we’re dealing with big numbers. It’s imagining them that takes the effort.

While we’re here – how about billions?

• A billion seconds -32 years
• A billion minutes - 1,900 years – if we go back we’re in early Roman Britain.
• A billion hours – over 114,000 years
• A billion days – 2,7 million years ago – humans of any type hadn’t evolved.
• A billion weeks – 19,000,000 (that’s nineteen million) years ago.
• A billion months takes us back more than 83million years.

So what about the total financial rescue package, £500 billion? Converting to units of time we get:

• 500 billion seconds we ‘re back in the Devensian Ice Age.
• 500 billion minutes – get towards a million years
• 500 billion days and that 1,347 million years. 540 million years is when the first shelly life appears on earth.
• 500 billion weeks is almost double the age of the solar system.
• 500 billion months is three times the age of the Universe.

So we’re dealing with some pretty big numbers.

There’s about 60 million people in the UK, and of them about 27.3 million pay income tax. If we divide the 500 billion by the 27.3 million we get the liability on each taxpayer for this bailout of over £18,300 each.

And that’s still a lot of money.

But before you think there’s something strange about them have a look at their picture.

The Celtic Terrors on the beach at Porth Kidney, near St Ives, Cornwall, August 2008, They have four paws each, quite normal.

In fact they’re quite normal, but that doesn’t stop them having greater than the average number of paws, taking the whole population of dogs into account. In making sense of this you have to consider how many paws dogs could have and still be able to function. Dogs have a maximum of four paws – that’s determined by the species. But a leg may have had to be amputated because of disease or accident, but there are plenty of dogs going round on three legs – they’ve just learned to adapt. Then again, some dogs can survive with just two functional legs, as long as they’re the front pair as they can be fitted with wheeled supports so that they can still lead a decent life. Hence some dogs have two paws.

So not all dogs have four paws.

So how come that my dogs have more paws than the average?

Just consider what an average is. Formally the expression we’re considering is a mean, expressed by:

The mean is calculated from the sum of all values in the set divided by the number of those values.

So what if we take a sample of 1000 dogs and find that 999 of them have four paws, but the other one has had an accident and has had a leg amputated, and now only has three paws. The average number of paws for this sample of dogs is therefore:

That’s 3.999 paws per dog. That’s the mean, or the average. But 3.999, although close to 4, isn’t 4. If you do the calculations with actual figures instead of the ones that I’ve just made up then the average number of paws per dog is always going to be less than 4 – simply because you don’t get dogs with more than four paws.

In fact, the vast majority of dogs have an above average number of paws. Let’s go further – all dogs with 4 paws have an above average number of paws.

However if you use the data given above (999 dogs with 4 paws and 1 with 3) then two other expressions of an average, the mode and the median, say that my dogs are not unusual. The mode means the most common value, so as dogs with 4 paws are the most common, the modal average is 4. The median is the middle value of the data. You rank the data in order and take the middle value – that’s also going to be 4, and is the median average.

This hasn’t been about unusual dogs, just about interpretations of averages. It’s here because it forms part of my teaching, and has been inspired by Michael Blastland and Andrew Dilnot’s book The Tiger That Isn’t. Andrew Blastland has also written a BBC Magazine series, one of which deals with the question of averages and how they can be misrepresented.

The Tiger That Isn’t was a birthday present to me from my wife. It is on the same plane as the 1954 classic by Darrel Huff, How to Lie with Statistics. I’m going to point our librarians to The Tiger That Isn’t.

The moral of this tale? That when dealing with averages you have to be careful.

In a post a couple of weeks ago on his blog, my friend and colleague Andy Hollyhead wrote about PC World Sutton Coldfield and the quality of service along with the attention that he was given by the staff.

Yesterday I wanted another external disk drive and surfed round to find that the one (I wanted Maxtor 1TB) was out of stock on Tesco Direct but in stock for the same price at PC World, when I could get to collect in-store. Normally I’d hesitate, but I need the drive to back up my increasingly large range of machines so that I can upgrade to XP-SP3. Most people are content with a single PC, and maybe a laptop. I’m having to run various configurations and purposes, so different machines for different tasks is the norm in my office. Anyway, back to the story.

In the afternoon I’d reserved the drive, obtained the email receipt with confirmation number, printed it and called on on my way home. The store lighting is blue. Maybe they think it looks hi-tech but to me it just looks blue. I couldn’t woirk in that environment. It’s not pleasant at all. Having located the correct area I was accosted by a young man in a PC World shirt who took my email printout without many words and started looking for the disk on the shelves behind him. A colleague of his, a young lady (although I use this term advisedly) then managed to persuade herself to wander back, almost snatched the printout with the number and wandered (not walked) across the store to fetch my reserved item from the retail shelf. So much for the reservation system.

During the remainder of the transaction the staff kept up banter, if that it the correct term, between themselves. One phrase that I remember was that the young lady (remember my use of the word, advisedly) offered to “punch the face in” of her colleague. I, as the mere customer, felt like I was interrupting their ongoing conversation by daring to want to actually buy something from the store. Looking round I didn’t see many other customers, the only person not in PC world garb was a rather bored looking security guard on a pedestal beside the door.

When I was eventually deigned to be spoken to a rather strange sound emanated from between the lips of the young lady. I had to ask for a repeat, and failing to understand that then had to stress that I found it impossible to understand what had been said. Shifting the gum to the other side of her mouth I eventually ascertained that she was telling me the price – no preamble, just the price. I paid and left.

DSG, owners of PC World, are finding that profits aren’t what they used to be. The Guardian on June 26th reported that “DSG made a loss of £193m – compared with a profit of £114m last year and more than £300m a year earlier”.

Judging by this performance is it any wonder things are going the way they are? It was just as well that I wasn’t offered extended warranty, the Techguys services for transferring data or anything else I can do better myself because if I had I would have refused and given my reasons which would have been connected with their level of customer service  – I just wanted to get out.

Maybe it’s the light that’s attracting them.

Associated with the Old Money post there’s another page on measurements that may have gone out of use.

Before 1971, when Britain adopted a decimal currency system, financial transactions were carried out using pounds, shillings and pence. There were twelve pennies to the shilling and twenty shillings to the pound.

When  researching old account books and leases (as is our wont) we sometimes have to convert old money to the new equivalents.

To cut a long story short I’ve produced a page of pre-decimal coinage equivalents and explanations.