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Further Pure 1
1. Simplify where possible. Remember that when you are working with complex numbers, you should always simplify
i² to -1.
2. Make sure that you know what a complex conjugate is. Remember that for a complex number z = x + iy, the
complex conjugate z* = x iy.
3. Remember that the product of a complex number and its conjugate is real. In particular, when dividing complex
numbers you need to use the fact that zz* is always real.
4. Make sure that you can plot complex numbers correctly on the Argand diagram. Remember in particular that
the points z and z* are reflections of each other in the x axis, and that the points z and z are rotations of each other
through 180° about the origin.
5. Make sure that you know how addition and subtraction of complex numbers is shown in the Argand
diagram, using vectors. You need to understand that a complex number can be represented not only by a point in
the Argand diagram, but alternatively by a vector.
6. Make sure that you know how to find the modulus of a complex number. In particular, don't forget to take the
7. Make sure that you understand what the modulus represents in the Argand diagram. Remember that |z| is the
distance of the point representing z from the origin, and |z w| is the distance between the points representing z
8. Be very careful when you find the argument of a complex number. Always decide first which quadrant the
complex number is in. It's a good idea to make a rough sketch of the number on an Argand diagram, so you can `see'
9. Make sure that you get the sign of the argument right. The argument is measured in an anticlockwise direction
from the positive real axis. Complex numbers below the real axis have negative arguments and those above or on the
real axis have argument greater than or equal to zero.
10. Remember that complex roots of polynomial equations with real coefficients always occur in conjugate
pairs. This is important when solving polynomial equations. If you know one complex root, then you know a second
complex root (its conjugate) and you can then find a quadratic factor with real coefficients for the equation.
11. Make sure that you can divide a polynomial by a linear or quadratic factor.
12. Check your work carefully. It is easy to make mistakes in the algebra when solving polynomial equations.
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Make sure that you can do matrix multiplication confidently. This is fundamental to the whole chapter.
2. Remember that matrix multiplication is not commutative. In general, AB BA. This is an easy mistake to make as
we are all used to ordinary multiplication being commutative.
3. Remember the useful result about the columns of a matrix.…read more
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You can then be reasonably confident that the iteration is correct to that
number of decimal places. To be certain, check for a change of sign either side of the root you have found.
3. Use enough decimal places in your working. When using the Newton-Raphson method, you need to work with
more decimal places than you need in your final answer. The best approach is to store each approximation in your
calculator, so that you have maximum accuracy at each stage.