- Why is ?
- By looking at and at what can you say about ?
- Can (the zeroth root of x) be defined? If so, how; if not, why?
- Criticise the following ‘proof’: If then
The proof should show you how this generalises. If you have studied group theory you can extend this even further by showing that , where is defined by , is an abelian group.
Why can’t 1 be an element of this group?
Inspired by a test posting on S.O.S. Mathematics CyberBoard
Factorials are fascinating. They are obtained by multiplying the numbers 1, 2, 3, 4, 5 … together and are written using the ! sign. Thus
Factorials get large very quickly so 25! = 15 511 210 043 330 985 984 000 000 and as it gets larger there are more and more zeros.
Question: How can you find the number of zeros in 102! without calculating it?
Answer: All you do is count the number of times powers of 5 divide into 102 and add them up. So
Throw away the remainders and we get that there are zeros in 102!
Now find how many zeros there are in 2004!
Can you show why this method works? Can you generalise it? Have a go, then look for answers on the internet.
Mathematicians will recognise this as a special case of
which is not difficult to prove by induction.
More on factorials at mathworld
Press ‘read more’ below for the answers
1. Give an example of an easy calculation that most simple calculators get wrong
2. If find [Hint: think carefully about before you differentiate]
Mathematicians’ prime motive for research is not money which is just as well as there’s little money in it. However, there are 7 problems, the millenium problems for which there is a prize of $1 million for the solution to any of these problems. You can find out all about them at the Clay Mathematics Institute and in Keith Devlin’s book The Millennium Problems
Why do mathematicians agree that is their favourite formula? Read why at Euler’s Formula which is an excerpt from Keith Devlin’s excellent book Mathematics: the Science of Patterns then read the book!
The Great International Mersenne Prime Search (GIMPS) wants you to help in their search for Mersenne primes. These are prime numbers of the form (where p is itself a prime) and they can be found by anyone! The largest was found last November by a student. It has 6,320,430 decimal digits.
You could win fame and fortune so go along to http://www.mersenne.org/prime.htm where it says:
“You could discover one of the most coveted finds in all of Mathematics – a new Mersenne prime number. We’ve found six already. Join in on this fun, yet serious research project. All you need is a personal computer, patience, and a lot of luck.
In addition to the joy of making a mathematical discovery, you might win some cash. The Electronic Frontier Foundation is offering a $100,000 award to the first person or group to discover a ten million digit prime number! See how GIMPS will distribute this award if we are lucky enough to find a ten million digit prime.“