Proof and logic

Sunday 24 October 2004 at 3:15 pm | In Articles | 3 Comments

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 \LaTeX facilities (provided by LatexRender of course 😎 ) to help a friend show how to prove

    \sin{(x+\frac{\pi}{4})}+\cos{(x+\frac{\pi}{4})}=\sqrt{2}\cos{x}

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?

    \sin{(x+\frac{\pi}{4})}+\cos{(x+\frac{\pi}{4})}=\sqrt{2}\cos{x}

    For this problem you need to know the addition formulas:
    \ \sin{(x+t)}=\sin{x}\cos{t}+\cos{x}\sin{t}\ \cos{(x+t)}=\cos{x}\cos{t}-\sin{x}\sin{t}

    Using these formulas in the problem we can turn it into:
    \ (\sin{x}\cos{\frac{\pi}{4}}+\cos{x}\sin{\frac{\pi}{4}})+(\cos{x}\cos{\frac{\pi}{4}}-\sin{x}\sin{\frac{\pi}{4}})=\sqrt{2}\cos{x}

    Then we use the fact that \sin{\frac{\pi}{4}}=\frac{\sqrt{2}}{2} and \cos{\frac{\pi}{4}}=\frac{\sqrt{2}}{2}

    This changes it to:
    \ \cancel{\frac{\sqrt{2}}{2}\sin{x}}}+\frac{\sqrt{2}}{2}\cos{x}+\frac{\sqrt{2}}{2}\cos{x}-\cancel{\frac{\sqrt{2}}{2}\sin{x}}=\sqrt{2}\cos{x}

    Here we cancelled out the terms that equals zero and then adding together what we have we end up with:

    \sqrt{2}\cos{x}=\sqrt{2}\cos{x}

Some textbooks misuse infinity

Monday 18 October 2004 at 3:43 pm | In Articles | Post Comment

It’s happened again! \infty 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 \theta between two lines

    \theta=\tan^{-1}\left |\dfrac{m_1-m_2}{1+m_1m_2}\right |

the following appears:

    Putting m_1m_2=-1 gives an angle \tan^{-1}(\infty)=90^{\circ}, confirming the condition for the lines to be perpendicular

This is of course complete nonsense. As I’ve said before \tan(90^{\circ}) doesn’t exist and \tan^{-1}(x) is only defined on \mathbb{R} ie for -\infty<x <\infty
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 t(t-3)=t^2-4 by putting \frac{1}{m}=t then ‘showing’ t=\infty or t=\frac{4}{3}. 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!

Amazing Formula

Monday 11 October 2004 at 9:47 pm | In Articles | 3 Comments

There are many interesting formulae in mathematics;

    \displaystyle \sum _{i=1}^\infty \frac{1}{n^2} = \frac {\pi^2}{6}

must be one of the most amazing of all.

The first reaction is where did that \pi 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 \dfrac{6}{\pi^2}

\pi gets everywhere! See Wikipedia for more such as

    \displaystyle \left(\frac{1}{2}\right)!=\frac{\sqrt{\pi}}{2} or \displaystyle \frac{2}{\pi}=\frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \ldots or \displaystyle \int_{-\infty}^{\infty}e^{-x^2}\;dx=\sqrt{\pi}

Primes

Sunday 10 October 2004 at 6:42 pm | In Articles | Post Comment

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 \pi(x) less than x is given by \pi(x)\sim \dfrac{x}{\log x} which means that the larger the value of x the closer (in a well-defined mathematical sense) \dfrac{x}{\log x} is to \pi(x). 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 n and 2n.

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 N. Then we look at the numbers

    N!+2,N!+3,N!+4,\ldots,N!+(N-1),N!+N

Then each of these numbers is not prime. Why? Look at N!+a where 2 \leq a \leq N. Then a divides both N! and a and so divides N!+a. Clearly a<N!+a so N!+a=a \times \frac{N!+a}{a} shows N!+a is not prime.
So we have a series of N-1 numbers all of which are not prime; thus the gap between a prime less than N!+2 and a prime more than N!+N is at least N.

Error

Saturday 2 October 2004 at 10:04 pm | In Articles | 2 Comments

Six months ago in an article on the LambertW function I wrote:

    Thus \ln is defined by \displaystyle \ln x=\int ^x _0 \frac{dt}{t}\text{ for } x>0 and it is then clear that, for example, \ln 1=0

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 😕

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