4 Unit Maths – Complex Numbers Complex Laws.
Let z1 = a + ib, and z2 = c + id, where a, b, c and d are real numbers.
√√
Square root of a complex number.
Solve for x and y by inspection. If unable to do through inspection use the identity; And then perform simultaneous equations.
Conjugate.
Adding vectors.
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If
then
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̅ ̅ ̅ ̅ ̅ ̅ ̅̅̅ ̅ ̅ ̅ () ̅
Complete Parallelogram
Head to Tail
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Subtracting vectors.
Complete Parallelogram
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4 Unit Maths – Complex Numbers Modulus-Argument form.
If equation is not in correct two conditions are met.
Rules of Modulus.
– , (eg,
) use the unit circle – just think of when the
The modulus of a complex number is its length.
|| | | || | | || || || || ̅̅|| |||| Rules of Argument.
Further vector properties.
The argument of a complex number is the angle made with respect to the positive x-axis.
() () ̅ ⃗ ⃗
If tail is at the origin, only one letter is u sed. .
However, if tail is not at origin, two letters are used. Note: goes from A to B. T he arrow ‘starts’ at A and ‘ends’ at B.
Angel between two vectors.
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Rules
Separate the argument like above. Draw vectors z1 and z2. Look at the heads of the vectors. Determine the direction of angle. Note: angle is between the two heads, and the head of the angle is on the ‘head’ line.
Rotations.
To rotate a complex number, z, by
Reducing/Enlarging.
To reduce/enlarge a complex number, by x, multiply z by x to get xz.
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anticlockwise, multiply z by cis to get zcis .
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4 Unit Maths – Complex Numbers Purely real/imaginary.
Consider the complex number; z = x + iy. Purely imaginary When x = 0. When
Purely real When y = 0. When
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De Moivre’s Theorem.
Trigonometry
Questions such as: Find cosnp in terms of powers of cos. Start off by writing: = ___ On the left hand side of the equation, use D e Moivre’s theorem, whilst on the right hand side, expand normally using Pascal’s triangles. Compare the real and imaginary parts, d epending on question.
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Questions such as: Find cos x in terms of multiples of x.
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Start off by writing: Group powers of x with their i nverse, and then use the two above equations.
Non-root of unity.
In the form; . Here, . Turn the complex number into cis form, and then equate the equations.
Root of unity.
In the form;
. . The aim is to find all the possible values of – and consequently find all the values of z. Must always represent roots on a unity circle. The roots form a regular polygon, and are conjugate pairs.
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Factorise over complex field.
Locus
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Done geometrically. Circle, centre (a, b), radius c.
Done geometrically. Perpendicular bisector of interval joining (a, b) and (c, d).
Done algebraically. Change z into x + iy.
Done geometrically. A ray coming from (a, b) Let equation of locus be y = mx + b. (angle with respect to the positive x-axis). Open circle at (a, b). Restrictions do apply.
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4 Unit Maths – Complex Numbers
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Done geometrically. This is a part of a straight line which passes through (a, b) and (c, d). Restrictions: can’t be (a, b) or (c, d).
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Draw diagram remember direction of angle. Redraw diagram, where z is the perpendicular bisector of the interval made from (a, b) and (c, d) so that it is easier to find the equation of the circle. Draw triangles to work out radius and centre. Restrictions: above or below the line passing through (a, b) and (c, d), not including (a, b) and (c, d).
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