CIE Centre A-level Pure Maths

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CIE Centre A-level Pure Maths P3 Chapter 16 CIE Centre A-level Pure Maths © Adam Gibson

The “depressed cubic”; other cubics can be expressed in this form. COMPLEX NUMBERS Girolamo Cardano: 1501-1576 A colourful life! For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. The “depressed cubic”; other cubics can be expressed in this form.

Cardano “stole” the method from Tartaglia. Finding x you must write: COMPLEX NUMBERS Cardano “stole” the method from Tartaglia. Finding x you must write: but the solution is a real number! can be less than zero For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Square roots of negative numbers can be useful, just as negative numbers themselves.

By introducing one number COMPLEX NUMBERS By introducing one number we can solve lots of new problems, and make other problems easier. It can be multiplied, divided, added etc. just as any other number; but the equation above is the only extra rule that allows you to convert between i and real numbers. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Start by noticing that So the square root of any negative number can be expressed in terms of i.

Therefore we can solve equations like: COMPLEX NUMBERS Therefore we can solve equations like: The answer is But what is The two types of number cannot be “mixed”. Numbers of the form For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. are called imaginary numbers (or “pure imaginary”) Numbers like 1, 2, -3.8 that we used before are called real numbers. When we combine them together in a sum we have complex numbers.

COMPLEX NUMBERS For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

a is the “real part” of z; Re(z) b is the “imaginary part” of z; Im(z) COMPLEX NUMBERS To summarize, a and b are real numbers a is the “real part” of z; Re(z) b is the “imaginary part” of z; Im(z) The sum of the two parts is called a “complex number” For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Adding and subtracting complex numbers: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. For addition and subtraction the real and imaginary parts are kept separate.

Multiplying and dividing complex numbers: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Notice how, for multiplication, the real and imaginary parts “mix” through the formula i2 = -1.

Multiplying and dividing complex numbers: Remember this trick!! For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Read through Sections 16.1 and 16.2 to make sure you understand the basics.

Now there are always two solutions, albeit they can be COMPLEX CONJUGATES Now that we have introduced complex numbers, we can view the quadratic solution differently. Now there are always two solutions, albeit they can be repeated real solutions. If the equation has no real roots, it must have two complex roots. If one complex root is For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. what is the other? These two numbers are called “complex conjugates”.

What are the solutions to ? COMPLEX CONJUGATES What are the solutions to ? * means conjugate If we write Then the complex conjugate is written as Calculate the following: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. This will be discussed later.

square a complex number z? And then square its conjugate, z*: COMPLEX POWERS What happens if we square a complex number z? And then square its conjugate, z*: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Compare the two results; they are complex conjugates! Later we will understand this result geometrically

Continuing these investigations further (you may study COMPLEX POWERS Continuing these investigations further (you may study in your own time if you wish): This will be easy to justify later. Examine the argument on page 228. This is a key idea, although you don’t have to understand the proof. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Non-real roots of polynomials with real coefficients always occur in conjugate pairs.

Find all the roots of the following two polynomials: COMPLEX POWERS Tasks Find all the roots of the following two polynomials: The first example can be attacked using the factor theorem. Examining +/-1,+/-5,+/-13 gives one root as -5. Equating coefficients therefore gives: For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

The roots are therefore -5, 2-3i, 2+3i. COMPLEX POWERS The roots are therefore -5, 2-3i, 2+3i. The second example looks simpler but is, in a way, more difficult. First set w = z2. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. It seems we have to find the square root of a complex number!

Algebra is not the best way to do it, but let’s try anyway. COMPLEX POWERS Algebra is not the best way to do it, but let’s try anyway. The next step is important to understand. It is called “equating real and imaginary parts”. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Simultaneous equations. Let’s apply it to our problem.

COMPLEX POWERS For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Special properties of complex conjugates: z How do we know it must be a real number? z* For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. What is ?

This is a very important result. COMPLEX CONJUGATES This is a very important result. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

How many complex roots do the following polynomials have? COMPLEX CONJUGATES How many complex roots do the following polynomials have? 10 3 5 See page 229. We always have n roots for a polynomial of degree n. If the coefficients are real numbers, then we also know that any non-real roots occur in complex conjugate pairs. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. If 1-8i is a root of polynomial B, what are the other roots?

POLAR COORDINATE FORM z r For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. The modulus is the length of the line from 0+0i to the number z, i.e. r. The argument is the angle between the positive real axis and that line, by convention we use

Find, to 3 s.f. the modulus and argument of the following POLAR COORDINATE FORM Find, to 3 s.f. the modulus and argument of the following complex numbers: modulus For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. To find θ we have two equations:

POLAR COORDINATE FORM For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.

Which function does this? EXPONENTIAL FORM or For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Which function does this?

So (not proof but good enough!) EXPONENTIAL FORM So (not proof but good enough!) If you find this incredible or bizarre, it means you are paying attention. For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is. Substituting gives “Euler’s jewel”: which connects, simply, the 5 most important numbers in mathematics.

We can write any complex number in this form EXPONENTIAL FORM We can write any complex number in this form As before, r is the modulus and θ is the argument. Examples: Do you see how easy it is to calculate powers? Find For the 4th order example given, ask students to give values of the ak coefficients The graph is a good subject for discussion of the nature of higher order polynomials Don’t forget to ask the students what a 1st order polynomial is.