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Complex numbersPictured in a number plane

In the same way as the real numbers you’re used to canbe pictured as points on a number line

plus rules for adding, subtracting, multiplying, and dividingthem, complex numbers can be pictured as points in theplane plus rules for adding, subtracting, multiplying, anddividing them.

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Complex numbersPictured in a number plane

We add a number i =√

(−1). To do that we have to movefrom a number line to a number plane, because no x onthe number line has x2 = −1. Once having added thatone extra number, we get a whole heap of extra numberswhich we describe like 2 + 4i meaning "2 across, 4 up".

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Rules for adding, multiplying, etc.for complex numbers

To add: (2 + 4i) + (3− 3i) = 5 + iTo subtract: (2 + 4i)− (3− 3i) = −1 + 7iMultiply as you would multiply surds, except i2 = −1(2 + 4

√2)(3− 3

√2) = 6 + 6

√2− 12(

√2)2 = −18 + 6

√2

Similarly(2 + 4i)(3− 3i) = 6 + 6i − 12i2 = 18 + 6i

Divide as you would divide surds, except i2 = −1

2+4√

23−3√

2= (2+4

√2)(3+3

√2)

(3−3√

2)(3+3√

2)= 30+18

√2

(−9)2+4i3−3i =

(2+4i)(3+3i)(3−3i)(3+3i) =

−6+18i18

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Sums with complex numbersPractice

Do the surd examples for practice, then the complexnumber sums

1. (5 +√

2)(3 + 4√

2) 5+√

23+4√

2

2. (6 + 3√

2)(7 + 2√

2) 6+3√

27+2√

2

3. (5− 2√

2)(1 + 5√

2) 5−2√

21+5√

2

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Sums with complex numbersPractice

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Solving quadratic equationswith complex numbers

With complex numbers we can solve all quadraticequations (by completing the square or by the quadraticformula), because every negative number now has asquare root. E.g.

√−4 = 2i

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Answers to practice questionsAdding, multiplying, dividing, solving

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New number systemsWhat new sorts of numbers have you learned about before?

What numbers did you know aboutbefore you started primary school?

Since then you have learned aboutquite a few new sorts of numbers.You’ve also learned new thingsabout numbers you knew already -for example, 4 is a square number,5 is a prime, 6 is a triangularnumber - but what new sorts ofnumbers have you learned about?

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Different sorts of numbersExamples

The natural numbers are thecounting numbers 1,2,3, . . ..

Give examples of:I a whole number which is not a natural (counting)

numberI a rational number (fraction) which is not a whole

numberI a surdI a real number in the green region (not rational and

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A number plane, not a number lineDifferent ways of picturing numbers

At primary school you learned how topicture counting numbers in two ways,both as fingers (or other counting things)and as points on a number line. All theother sorts of numbers you’ve learnedabout fit on the number line. Complexnumbers don’t. We need a numberplane.

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Another way of picturing real numbersEnlargements

We could picture, or "code", the real number x in adifferent way, as an enlargement by x and a rotation ofzero.

What would an enlargement of the butterfly by zero (and arotation by zero) look like? What would an enlargementby −1 and a rotation by zero (which we write as:enlargement by 1 and rotation by 180◦) look like?

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Square roots of negative numbersExample: square root of -1

Here’s the butterfly enlarged by −1(or rotated by 180◦). You couldn’tget there by squaring anyenlargement by x and rotating byzero. But by rotating by . . . what?If you moved off the number lineand into a number plane?

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Enlargements-and-rotationsCan also be pictured as points in a plane

What enlargement-and-rotation getsyou from (1,0) to the following points?

I (1,1)I (0,1)I (-1,0)I (2,0)

What points are got from (1,0) by the followingenlargements-and-rotations?

I Enlarge by 3, rotate by zeroI Enlarge by 1, rotate by 180◦I Enlarge by 1, rotate by 90◦I Enlarge by

√2, rotate by 45◦

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Multiplying complex numbersModulus and argument

The complex number "enlarge by r , rotate by θ" is writtenr cis θ or sometimes r∠θ. r is the modulus of the complexnumber, θ is the argument of the number.

Multiply these complex numbersI 3 cis 0× 2 cis 45◦

I 3 cis 45◦ × 2 cis 45◦

I 1 cis 90◦ × 1 cis 90◦

I 2 cis 90◦ × 2 cis 90◦

What’s the rule? When you multiply complex numbers,you . . . . . . the moduluses. You . . . . . . the arguments.

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Complex numbers as pointsa+ib

r cisφ is equivalent to the pointx = r cosφ, y = r sinφ. The symbol imeans "

√−1" or "enlarge by 1, rotate

by 45◦", or "one unit up in the numberplane".

This diagram shows us thatmultiplying complex numbersusing the method for surds, anddoing it by mod × mod and arg+ arg, are equivalent.

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Checking out multiplicationwith the a+bi form

3 cis 0× 4 cis 45◦ = (3) · (2√

2 + 2√

2i) = (in a+bi terms) =(in mod-arg terms)

3 cis 45◦ × 4 cis 45◦ = (3√

2 + 3√

2i) · (2√

2 + 2√

2i) = (ina+bi terms) = (in mod-arg terms)

1 cis 90◦ × 1 cis 90◦ = i · i = (in a+bi terms) = (in mod-argterms)

2 cis 90◦ × 2 cis 90◦ = 2i · 2i = (in a+bi terms) = (inmod-arg terms)

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Converting between different formsa+bi (Cartesian) and mod-arg

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Calculations with complex numbersin a+bi (Cartesian) form

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RadiansThe measure of angle we’ll usually use

Use your calculator to convert theseto radians: 360 degrees, 30 degrees,180 degrees, 60 degrees, 90degrees, 45 degrees.

Convert these to degrees: π6 , π

4 , π, π2 ,

π3 , 2π

3

One of the reasons why mathematicians usually useradians is that if we measure in radians, a lot of formulasbecome neater, for example

sin x = x − x3

3!+

x5

5!− x7

7!+

x9

9!− . . .

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Using radianswith complex numbers

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Dividing complex numbersIn a+bi (Cartesian) and mod-arg forms

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Big facts about complex numbersWhich you don’t need to know for exam

The complex number system, which you get by addingjust√−1 to the real numbers, gives you solutions to every

polynomial equation.

The complex number system is the "final" number systemin the sense that if you keep extending beyond it, thenadding, subtracting, multiplying, and dividing stop workingwell as they do with ordinary numbers.

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Square roots of complex numbersTwo methods to find them

Example: find 2 so that w2 = −16 + 30i

Method 1: w2 ≈ 34 cis 2.06w ≈

√34 cis 1.03 ≈ 3 + 5i (maybe rounding error)

Check: (3 + 5i)2 = −16 + 30i (rounding error fixed ok?)w = ±(3 + 5i)

Method 2: Let w = a + bi|w2| = |w |2 = a2 + b2 = 34Re(w2) = a2 − b2 = −16Solve those simultaneous equations for a2 and b2

a2 = 9 and b2 = 25a and b must have the same sign because Im(w2) > 0So w = ±(3 + 5i)

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Real and imaginary partsof complex numbers; and equating them

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Conjugatesa+bi and a-bi

Conjugate complex numbersare like conjugate surds. Weuse them for division.(a + bi)∗ = a− bi .(r cisφ)∗ = r cis(−φ).

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ConjugatesHow complex numbers are symmetrical

The real number line is not symmetrical. If you have anumber line with the numbers erased, you can tell whichside of 0 is positive and which is negative. How?

The complex plane is symmetrical. If Jill develops thetheory of complex numbers using one square root of −1which she calls i , and Jack develops it using j instead,where j = −i , their theories are the same.Therefore, if a + bi is a root of a polynomial equation withreal coefficients, a− bi is also a root. Roots come inconjugate pairs.

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Working with conjugatesAnd quadratic equations

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Vieta formulasWays of solving quadratics and cubics

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Solving cubics and quarticsUsing conjugates and Vieta formulas

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Solving cubics and quarticsSolutions

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Solving cubics and quarticsUsing conjugates and Vieta formulas

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Geometry in Argand diagrams1

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Edexcel’s proof that mods multiplyand arguments add

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Geometry in Argand diagrams2

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Geometry in Argand diagrams3

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Test1

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Test2

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Test3

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Test4

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Test5

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Test6

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Test7

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Test8

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