Mathematical proof is one of the topics that students find very difficult. Many of them assume what they are trying to prove, end up with a true statement and then think they have proved the result. Studying truth tables, particularly the implication operator may well help. See Logical Operations and Truth Tables
On a mathematics forum a student wanted to use its facilities (provided by LatexRender of course ) to help a friend show how to prove
This is the original ‘proof’ they gave. Although they have now changed it I get the impression that I failed to convince them of the faulty logic; how would you explain what is wrong?
For this problem you need to know the addition formulas:
Using these formulas in the problem we can turn it into:
Then we use the fact that and
This changes it to:
Here we cancelled out the terms that equals zero and then adding together what we have we end up with:
It’s happened again! used in a textbook (unnamed to protect the guilty) as if it were a real number instead of an idea. In a discussion of the formula for the acute angle between two lines
the following appears:
- Putting gives an angle , confirming the condition for the lines to be perpendicular
This is of course complete nonsense. As I’ve said before doesn’t exist and is only defined on ie for
The textbook was written by the examiners (which is one reason why we use it); this worries me even more.
I suppose this is better than one well-known textbook back in the eighties which solved the equation by putting then ‘showing’ or . This seems to show that all linear equations are quadratics in disguise; or cubics, quartics – who knows where this nonsense leads
See also Division by zero shock!
There are many interesting formulae in mathematics;
must be one of the most amazing of all.
The first reaction is where did that come from? You can find 14 different proofs of this in a paper on Robin Chapman’s Home Page [look for Evaluating zeta(2)]
Given this result can you prove another amazing result?
If you pick two positive integers at random, the probability of them having no common divisor is
gets everywhere! See Wikipedia for more such as
- or or
Euclid’s proof that there are an infinite number of primes is a classic and as such appears as the first proof in Proofs from The Book.
Equally well-known is the formula (known as The Prime Number Theorem) which tells you that the number of primes less than is given by which means that the larger the value of the closer (in a well-defined mathematical sense) is to . This is quite hard to prove.
An easier, but non-trivial result, is Bertrand’s postulate which says that there is always a prime between and .
The fact that there are arbitrarily large gaps between successive primes is not difficult to prove. Suppose we want to find a gap between successive primes which is at least of size . Then we look at the numbers
Then each of these numbers is not prime. Why? Look at where . Then divides both and and so divides . Clearly so shows is not prime.
So we have a series of numbers all of which are not prime; thus the gap between a prime less than and a prime more than is at least .
Six months ago in an article on the LambertW function I wrote:
- Thus is defined by and it is then clear that, for example,
There’s a serious error in there which also completely invalidates it is then clear that …. Going back to the article the mistake leapt out at me – is it obvious to you? It’s strange how you read what you want to read rather than what is actually there