Some Notes for MS221 12J TMA 04 Cut-off date 22nd May 2012

Please remember that there is no extension given on this final assignment except in exceptional circumstances.

This assignment covers a very large amount of material, all very interesting, however don’t worry if you can’t cover all of it - it would be better to concentrate on one or two areas more thoroughly when time runs short. It is much more important to leave adequate time for examination revision. Particularly try to make sure you have covered most of the Section A questions.

Q1(a) It is important that you have mastered examples of the four rules with complex numbers in Cartesian form (a very likely topic in the examination too). Exercise 2.1 p 21 should help.

(b) What do you notice about the two complex numbers here? Pages 34,35 D1 are relevant. Exercise 4.2 p40. D1 is useful.

TWO IMPORTANT FACTS: For two complex numbers, both of the expressions are positive.

Note: the equation with roots is where and may be real or complex numbers.

(c) Some general comments. You should practise drawing the complex numbers in the Argand diagram until you can convert to the polar form and back again (see p 25 D1). Always draw the complex number to be sure of which quadrant it lies in – getting the wrong quadrant is a frequent cause of mistakes!

For example Activity 3.4 pictures the complex number -2 – 2i. The real part is negative, imaginary part also so the complex number is drawn in the third quadrant.

Notice that when you multiply two complex numbers together, you multiply the moduli

and add the arguments so that This is a fantastic rule.

You may prefer to think in degrees (I often do);

Extending this to the complex number , then

Note that the polar form of a complex number is not unique as the angle given can be increased by any multiple of . Thus can be expressed as or in general For example is the same as and (I have worked in degrees to make it simpler, but note that radians are required in your answer).

Here is an example of finding sixth roots. This is not asked in the TMA but could be useful for the examination.

To find one sixth root of (say) , take the sixth root of the modulus and divide the argument by 6.To find all 6 roots, write down 6 forms of and take the sixth root of each of these. Spelling this out (and using degrees for clarity)

are equivalent.

are sixth roots.

They will be equally spaced around the unit circle (draw them!). You would now express these in conjugate pairs. Look at Activity 3.4 on p27 D1. If you have drawn them as suggested above the conjugate pairs should be obvious.Values of angles are normally given in radians.

Note carefully that when you find the roots of an equation with real coefficients, the roots occur in conjugate pairs, of the form . If the order of the equation is even for an equation of odd order there is one real root and the rest are in conjugate pairs.

If you were solving there would be 7 roots (or solutions). Every polynomial equation has at least one real root so there would be 3 conjugate pairs and one more root. All the roots would have the same modulus and they will be equally spaced around the circle with this radius in the Argand diagram.

Activity 4.7 p.39 D1 is helpful. Looking at this activity we see that the basis of the method is that = -16. In this example you are finding 8th roots.First express -16 in 8 different ways and take the 8th root in each case.It is useful to call the roots

When you have identified the conjugate pairs, then find the polynomial. Exercise 4.1, page 40 may help here.

Remember that(i) (ii) and These are conjugate.(iii). Evaluating for example

Q2 (a) The rules for divisibility are given in the Handbook p.82.

(b) Fermat’s Little Theorem. See p27 D2. Activity 3.7 p 28 may be useful.

(c) Have a look at p.21 and following in D1. Also Activity 3.5.

Q3 (a) The Euclidean algorithm is important and worth mastering (see example p 24 and Activity 3.6 p 25). This is very likely to be on the exam paper too!This produces multiplicative inverses.As you found with functions and matrices, the idea of an inverse is very important.

(b)There is an interesting application to CODES. The RSA cipher is asked for here.. The multiplicative inverse is used in the deciphering rule. See Activity 4.3 p 38.

It would be very useful to start using your Handbook from now in preparation for the examination. Remember that you may write in any examples or other useful material.

Q4 Groups is a fascinating topic and those of you who intend to do M208 will meet it there.

Thinking ahead to the examination, the key to understanding here is familiarity with the small groups and their properties.Page 40 lists up to order 8. You should try to become familiar with group tables up to order 6 as there are only a few of these.There is one group of order 1 consisting of the identity element alone.There is only one group of order 2 (up to isomorphism of course). This is There is one group of order 3. This is

There are two groups of order 4. These are most useful groups for you to know. It would be helpful for you to recognise both of these groups and know one set of elements to represent each with a suitable operation such as + or multiply.

(i) - the cyclic group of order 4(ii) S (rectangle) see p.37 - usually called the Klein group in other texts.

* the way to tell these two apart is to look down the main (top left to bottom right) diagonal where all the entries in S are the identity.

* looking down the main diagonal picks up inverses as it shows element which combine with themselves to produce the identity.

* looking down the main diagonal shows self inverse elements,. e.g. 2 in Self-inverse elements are very useful in identifying groups.

* cyclic groups have a generating element, that is an element that combines with itself a certain number of times to produce the identity. In there are two generating elements.These are 1 and 3 in

There is one group of order 5 called , equivalent to . Remember that for each order there is a cyclic group represented by modular addition.

There are two groups of order 6. (i) . A cyclic group equivalent to (ii) S (triangle) the symmetry group of the triangle. The elements are transformations

and you need to understand and use the r, q notation. This is the first non-commutative group. You can see this from the table by checking the result of the operation on two elements in different orders. There are other representations of course.

There are 5 groups of order 8 – you won’t remember all these but there is of course. Another useful one is the symmetry group of the square which has four rotations and four reflections in the axes of symmetry. The tables are nice to look at if you have plenty of time, and to examine for subgroups.

Before tackling the question, make sure you understand activity 1.6 on p 14 and the work on Cayley tables on p 15.

(a) Here you have a symmetry group.

(b) Here you have a group of order 6 It is good to know that there are only two groups of order 6 – so which one is it??(i) Activity 2.5 p 27 is useful to show how you test the group axioms.

(b) (ii) one very helpful way to decide whether groups are isomorphic is to look at the number of self- inverse elements by making a table of inverses as on p.20. Activity 3.4 is very useful here.

‘Keyphrase’ : look down the main diagonal, i.e. top left to bottom right Self inverse elements give the identity on the main diagonal when multiplied by themselves). This is a good way to distinguish between groups.

(iii)Now you can be creative here.

Q5 This is a topic where we like you to get the argument correct as this is about proof.

Mathematical Induction is a most important method of proof. The only others are the direct proof, sometimes by considering all cases, and proof by contradiction. (for example the famous proof that the square root of 2 is irrational is done by assuming it is rational and showing the contradiction,)

There is no need to read what follows if you are happy with the argument in the unit. It is just to clarify.

You start with a proposition – often given as a formula. You are able to check that it holds for any specific n but your task is to prove that it is true for ‘all n’.

Step 1 is to check that the result holds for some n, usually n = 1.

Step 2 is to assume that the result holds for a value n=k. Building on this supposition, you show that if it was true for n=k, then it is also true for n = k+1.For a sequence you should look at the result for n = k, add the next term and simplify and show that you have the result appropriate to n = k + 1.But because it was true for n=1, we accept that the result is true for all n by the Principle of Mathematical Induction.

You need to be precise in what you write! Follow the setting out in the proofs in the unit.Your example is not a series so you have to apply the principle of mathematical induction. However a similar example is Exercise 3.2 in Exercise Booklet D.

Step 1. This is to check that the result is true when n=1.

Step 2. Assume the result is true for ‘k’ Now see what happens when you produce the k+1 th term. Is it the same as the formula with k replaced by k+1.

If so you can continue the induction argument.

There is no point in giving lists of examples where the result is true and you will not receive any marks for this.

Here is a lighthearted example of the argument behind mathematical induction. A castle has n rooms (n belongs to the infinite set of positive integers) and the rooms are ordered. Each room contains the key to the next room. What do you need to visit each room.However in mathematics we accept the mathematical induction argument as a principle.

Q6 (a) Remember that no amount of examples will ever prove a result (unless you exhaust all possibilities). However a single counterexample will disprove a result.

(b) Truth tables and compound propositions are found in D4, p 20 – 22.

Good luck with the assignment and with the examination. You can get past papers from OUSA (telephone or on-line) There are concise solutions to past papers in the student MS221 site (alternatively The Black Badge Press publishes nice solutions – see OUSA site).

Exam revision is what gives you the grade – try typing some exam marks into the assessment calculator to see how it works.

Remember you must submit TMA 04 on time!