Posted by: John Colby | Monday December 29 2008

Factorials and Big Numbers

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

Factorials

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.

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Responses

  1. very good material… well done !


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