### To Prove or Not to Prove

## INTRODUCTION

*called*a leg, doesn’t change the fact that it is

*not*a leg, so the answer is

*four*‘. ‘Excuse me,’ said a passing zoologist, ‘if a trunk is classified as a leg, clearly this will also apply to the tail, so it has

*six*legs, and it’s an insect’. A logician joined the conversation: ‘A

*normal*elephant has four legs, but you did not actually say that

*this*elephant is normal, so there is insufficient evidence…’

^{o}is to carry out an experiment. A mathematician wants more — simply predicting and testing is not enough — for there may be hidden assumptions (that the water boiling is always carried out at normal atmospheric pressure and not, say, on the top of Mount Everest).

## PROBLEM SOLVING AND CONVINCING ARGUMENTS

*seven*squares.

*if*I could cut a square into

*six*smaller squares, then I could do 6+3=9, 9+3=12, and so on, to get the sequence 6, 9, 12, 15, …

*If*I could cut a square into five smaller squares, then I could get the sequence 5, 8, 11, 14, …, and so on. But

*can I*?

*five*squares, it would be possible to get the sequence of possibilities 5, 8, 11, 14, … Then it would be possible to do all possible numbers 4 and above, using the combination of the three sequences

5, 8, 11, 14, …

6, 9, 12, 15, …

*can*you cut a square into five smaller squares? One student, Paul, suggested to me that if you can do n squares you can do n-3 by joining a block of four squares together as in Figure 3. Is Paul’s suggestion correct? It is certainly true for n=9, as Figure 3 shows, but is it true for

*all*whole numbers n?

^{o}C because we never have the experience of trying to boil water on the top of Mount Everest. Scientific proof depends on the

*predictability*of experiments: that we conjecture that when we carry out an experiment it will have a predicted outcome. Such proof is not appropriate in mathematics where we must provide a logical argument that the conclusion follows from explicitly stated assumptions.

- Convince yourself.
- Convince a friend.
- Convince an enemy.

*five*sub-squares.

*but perhaps some other method will*…

(a) Find all numbers n such that a square can be cut into n smaller sub-squares and

provethat this is actually possible for every such number n.(b) For all the numbers n not included in part (a), prove that it isnotpossible to cut a square into such a number of smaller squares.

## MAKING PRECISE STATEMENTS

*not*possible?

*at least four*. There cannot be two or three. Perhaps you might like to try to extend this argument to cover other cases which you suspect cannot be done (if there are any…).

*eventually*agreed on which values of n cannot be done. It has become quite a party piece which I have also tried out with many sixthformers.

*explicitly*stated that the paper could not be cut and then glued together again in a different way. Her solution for n=2 is given in figure 5.

Square problem version 2: A square is cut into n smaller squares by making single straight line cuts, without joining together cut parts into larger wholes. What are the possible values of n?

Square problem version 3: Into how many subsquares n is it possible to cut a square, if it is allowed to join cut parts into large wholes?

*can*re-form the square in Figure 4 into

*five*squares of different sizes: one large, one medium, and three little ones. With a little ingenuity perhaps you can solve the case n=3 for version 3 of the problem. Perhaps now you can specify the solutions of

*both*problems. They will be different. This shows that precision in making mathematical statements is all important.

### Making appropriate deductions

*not*made. Here a proof in the form IF P THEN Q simply requires that if P is true, then Q must be true also. If P is false, then no implication as to the truth or falsehood of Q is necessary.

If x > 6 then x > 3.

## COMMON ERRORS IN PROOF

A particle mass M rests on a rough plane with coefficient of friction μ, inclined to the horizontal at an angle α. Show that if the particle slides down the plane then tanα>μ.

What students often do is to assume tanα>μ and deduce that the particle slides. They have been asked to prove IF P THEN Q where P is “the particle slides” and Q is “tanα>μ”. They often prove IF Q THEN P. In this case it happens that the two things are equivalent. P happens if and only if Q happens. But the question only asks for the implication from P to Q and the students only prove the implication from Q to P.

*converse*of the statement IF P THEN Q. It is important to distinguish between the proof of a statement and the proof of its converse. One may be true and the other may be false. Another example occurred with the case of “into how many squares can I cut a square” (version 2). It is true to say that if a square can be cut into n pieces then it can be cut into n+3 pieces. The converse, that if it can be cut into n+3 pieces it can be cut into n pieces is false, as can be seen from the case n=3,5.

F(x)+c

where c is a constant. This is usually deduced from the fact that the derivative of a constant c is zero. Hence the derivative of

F(x)+c

is the same as the derivative of

F(x).

However, the deduction is false. Let P be the statement that

G(x)=F(x)+c

and Q be the statement

G′(x)=F′(x).

Then, because the derivative of a constant is zero we can deduce that IF P is true THEN Q is true. What we cannot do is to deduce the converse: IF Q is true THEN P is true.

It is actually possible to have Q true and P false. As an example, let

G(x)=1/x

and let

^{o}C.

Reblogged this on Helping Students in Maths and Creating Better Tomorrow.

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