CorePure1 Chapter 1 :: Complex Numbers [email protected] www.drfrostmaths.com @DrFrostMaths Last modified: 6th September 2019 www.drfrostmaths.com Everything is completely free. Why not register? Register now to interactively practise questions on this topic, including past paper questions and extension questions (including MAT + UKMT). Teachers: you can create student accounts (or students can register themselves), to set work, monitor progress and even create worksheets. With questions by:

Dashboard with points, trophies, notifications and student progress. Questions organised by topic, difficulty and past paper. Teaching videos with topic tests to check understanding. Chapter Overview 1:: Understand and manipulate () complex numbers. 2:: Find complex solutions to quadratic equations. Determine giving your answer in the form .

Solve . 3:: Find complex solutions to cubic and quartic equations. Given that is one of the roots of the equation , determine the other two roots. Why complex numbers? Remember when you used the quadratic formula, and it broke when you had a negative discriminant? (the bit) 2 4 = 2 By adding to our number system, we can then represent roots of quadratics that we couldnt previously do so with real numbers (i.e. numbers in ). ! Complex

numbers were originally introduced by the Italian mathematician Cardano in the 1500s for this very purpose, i.e. to represent the roots of polynomials which werent real. An imaginary number is of the form where , e.g.

A complex number is of the form , where , e.g. We say that is the real part and the imaginary part of the number. Notation Notes: Recall that in an algebraic term, we write numbers first, then mathematical constants (e.g. or ), then variables, e.g. . So where does come in? is similarly a mathematical constant, but we tend to put after . So wed write . Therefore the above is technically wrong (if is a variable): we should write . In FP1 you would see the identity . It is not . Confusingly however, the special case of this, Eulers Identity, is written , not , for consistency with the more general . Other Practical Applications 1 Fractals A Mandelbrot Set is the most popular fractal. Well see how it works in the next slide

2 Analytic Number Theory Number Theory is the study of integers. Analytic Number Theory treats integers as reals/ complex numbers to use other (analytic) methods to study them. For example, the Riemann Zeta Function allows complex numbers as inputs, and is closely related to the distribution of prime numbers. 3 Physics and Engineering Used in Signal Analysis, Quantum Mechanics, Fluid Dynamics, Relativity, Control Theory... Complex Number Basics Write the following in terms of : ? ? ?

Simplify: ? ? ? Solving Quadratic Equations Solve 2 = 25 ? Solve Using the quadratic formula: ?

Notation Note: Just as we tend to use as the default real-numbered variable and for integers, we tend to use as the default letter for complex numbers. Exercises 1A-1B Pearson Core Pure Mathematics Book 1 Page 3, 4-5 Multiplying Complex Numbers Given that , it follows that Determine the value of and Express each of the following in the form , where are integers. 1) 2)

( 2+3 )( 32 ) ? ( 5 3 ) ? 2 ? ? We can therefore see that for increasing powers of , we obtain where if is a multiple of 4. ? ? Test Your Understanding Edexcel FP1 June 2010

Expand and simplify ? ++ + Exercise 1C Pearson Core Pure Mathematics Book 1 Page 6 Just for your interest How do Mandelbrot sets work? You may have learnt about recurrence relations/term-to-term sequences, e.g. A Mandelbrot Set is governed by a recurrence relation: Observe that the first sequence diverges (0,

2, 4, ), i.e. approaches infinity, but the second sequence converges, in this case, gradually approaching 10 (0, 5, 7.5, 8.75, ) where is some complex number. Some examples: Suppose we chose Then: Suppose we chose Then: This clearly diverges. This clearly alternates. While it doesnt converge, the value remains bounded within some finite range. ( ) In Chapter 2 you will

learn about Argand Diagrams. A complex number is plotted at . So for example is plotted at . 1 Yes! No! 1 A Mandelbrot Set consists of all points on an Argand Diagram where for the corresponding complex number , the recurrence relation does not diverge, i.e. converges or repeats within some finite range. Based on our examples, wed therefore plot (with coordinate ), but not 1 (with coordinate ).

( ) By considering all possible values of , we end up with this remarkable fractal pattern We can get a coloured diagram by setting the colour to be the rate at which the recurrence diverges (black meaning it doesnt diverge). Complex Conjugation Suppose that and Determine:

? ? What do you notice about both results? They are both rational/surd-free! The second result in particular is useful, because ? a means to rationalise a denominator. we saw in Pure Year 1/GCSE that it gives us Does a similar thing happen with two complex numbers that are similarly related in this way? ? ? What do you notice about both results? They are both real! This similarly gives us a way to real-ise a denominator, and thus do division of complex numbers ? Complex Conjugation ! If then is the complex conjugate of . Write in the form .

5+4 2+3 10+15 + 812 ? =? 2 23 2+3 ? 4(3 ) ? As with rationalise denominators of surds, we multiple numerator and denominator by the conjugate of the denominator. Fro Speed Tip: It is useful to remember that: Use of Technology Monkey says: Mode Complex. You can divide, multiply and raise to a power, any complex number. Use the ENG key to get the . You could also theoretically get the conjugate from the OPTN menu. In the same menu you can find functions to

get the real part of a number of the imaginary part. Test Your Understanding FP1 (Old) Jan 2009 Q9 125 32 36241510 =? 3+2 32 9+4 Exercise 1D Pearson Core Pure Mathematics Book 1 Pages 7-8 Edexcel FP1(Old) June 2013 Q9 Solving: ?

Roots of Quadratics Suppose we are solving . Then solving: and thus and . But the converse is also true: if we knew the roots of a quadratic equation were and , then by the Factor Theorem, and are factors, and thus the original quadratic equation could be written as If and are the roots of a quadratic, then it can be written as If one of the roots of a quadratic was , do we know the other root? It must be its complex conjugate . Can you prove why? If is the root of a quadratic equation with real coefficients and is a complex number, then the other root must be its complex conjugate, . Proof that Roots Come in Complex Conjugates Prove that complex roots of a quadratic, with real coefficients, come in complex conjugates. Proof 1: When we use the quadratic formula: We can see the real part (before the ) is

the same (provided that are real), so if one root is complex, the other is its conjugate. ? Proof 2: In order to be a quadratic with real coefficients, and must be real. Let and Then If we require this to be real then and therefore . Also: ? In order for this to be real,

But therefore: Thus either (i.e. the roots werent complex) or , which combined with , means the roots are conjugates. Example Question [Textbook] Given that is one of the roots of a quadratic equation with real coefficients, (a) state the value of the other root, . (b) find the quadratic equation. a b ? Slow way: Quick (preferred) way: Using , combined with : (recalling that )

? Test Your Understanding FP1 (Old) Jan 2011 Q4 ? ? Exercise 1E Pearson Core Pure Mathematics Book 1 Pages 9-10 Roots of Cubic and Quartic Equations The same principle applies to polynomials of higher degree, e.g. cubics and quartics. A cubic equation always has three roots All complex roots come in conjugate pairs.

(by the Fundamental Law of Algebra). These roots may be repeated, and not all may be real roots ? 3 real roots. Comment on the 3 roots

1 real root.? 2 complex Comment roots on the 3 (which are roots conjugates) ? but two of 3 real roots, Comment on the 3 them the same

value roots root). (i.e. repeated Roots of Cubic and Quartic Equations And the same with quartics 2 real roots. ? 2 complexonroots Comment the 4 (a conjugate

roots pair) 4 real roots. ? Two of them Comment onwith the 4 the same value. roots

0 real roots. ? 2Comment pairs of complex on the 4 conjugate roots. roots Example Question [Textbook] Given that is a root of the quartic equation , solve the equation completely. Another root is . So is a factor of Use (mostly) common sense to determine the other bracket: It must start with in order to get the term in the expansion. It must end with -5 to get the -50 term.

So we know second bracket is of form . To work out the we need to compare either or terms in the expansion, say for example the term: ? Solving So roots are You could also use algebraic long division, but Ive always favoured determining the second bracket by intuition as per the left. Example Question [Textbook] Show that is a factor of Hence solve the equation

Comparing terms: Solving : Solving : ? Test Your Understanding FP1 (Old) Jan 2010 Q6 ? ? Exercise 1F Pearson Core Pure Mathematics Book 1 Pages 13-14