**Proof by induction**

Set out proofs as Step 1 (prove claim true for n=1) and Step 2 (prove claim infectious, i.e. if it’s true for n=k, then it’s true for n=k+1), plus a final line: “Step 1 + Step 2 ⇒ true for all n, by induction”.

Don’t set them out as four steps (Basis, Assumption, Induction, Conclusion) like the book says. There are only two steps in proofs by induction.

**Interval bisection**

Do as in this example for finding roots of f(x)=x^{2}–2=0

f(1.4)=1.962−2=−0.038<0 and f(1.5)=2.25−2=0.25>0

∴ change of sign ∴ root in [1.4,1.5]

f(1.45)=2.1025−2>0

∴ root in [1.4,1.45]

Don’t do a table as in the book. Much too long and complicated. If you do revision questions from the book, only do two interval bisections for each question. Not more.

**Linear interpolation**

Do a diagram like this:

and calculate Δ from similar triangles: ED/EA=CB/CA. Then your next guess is a+Δ

In the book they also draw similar triangles, but in a way that makes a more complicated calculation.

**For complex numbers, swapping between**

**x,y [or Re(z), Im(z)]** and

**r,θ [or |z|, arg(z)]** formats

On your calculator:

**Use Pol( , ) to convert from x,y** [Re(z), Im(z)] **to r,θ** [|z|, arg(z)]

**Use Rec( , ) to convert from r,θ** [|z|, arg(z)] **to x,y** [Re(z), Im(z)]

Always draw a diagram to check

Don’t use r=|z|=√(x^{2}+y^{2}) and θ=arg(z)=tan^{–1}(y/x) and x=r cosθ and y=r sin θ as in book. Those equations are correct, and you should know them. But they are more complicated. Also, tan^{–1} has two values for any given (y/x). Your calculator shows only one of them, and you have to do extra work to find whether to use the number your calculator gives you or add π or subtract π.

**Solving cubics and quartics given one root**

If necessary, divide through the equation so that the term with the highest power (x^{3} or z^{3} for a cubic, x^{4} or z^{4} for a quartic) has coefficient 1. Then use the facts that:

- the conjugate of a root is also a root (so if a+bi is a root, a−bi is too)
- sum of roots = r
_{1}+r_{2}+r_{3}=minus coefficient of second term (i.e. coefficient of x^{2}in a cubic, coefficient of x^{3}in a quartic). - product of roots = r
_{1}r_{2}r_{3}=minus constant term (in a cubic) or plus constant term (in a quartic) [it’s minus if if you have an odd number of roots, and plus if you have an even number].

Don’t use long division, as in the book. It is never simpler, and usually more complicated and longer.

**Square roots of complex numbers**

Example: find √(3+4i)

Two methods, both better than the book

**One:** via Pol(,)

Pol(3,4) ⇒ 5 cis 0.92729521800161

Square-root modulus, halve argument

⇒ √5 cis 0.46364760900081

Rec(√5,0.46364760900081) = 2 + i

and other square root must be 2 − i

**Two:** sum and difference of squares

Call the square root a+bi

Then |(a+bi)|=√|(3+4i)|

So, squaring, a^{2}+b^{2}=5

Also (a+bi)(a+bi)=3+4i

So, equating real parts, a^{2}−b^{2}=3

Solving those two equations, a^{2}=4 and b^{2}=1

a=±2 and b=±1

If the number you’re finding the square root of has a positive imaginary part, then the signs of the real and imaginary parts of the square roots have to be the same; if it has a negative imaginary part, then the signs of the real and imaginary parts of the square roots have to be opposite.

Therefore the roots are 2+i and −(2+i).

[If we were looking for the square roots of 3−4i, they would be 2−i and −(2−i). The square roots of the conjugate are the conjugates of the square roots].

**Converting between matrices (tables of numbers) and linear transformations**

Nothing the book says is wrong, but this basic rule for converting between linear transformations (scalings, rotations, reflections, shearings) and matrices (tables of numbers) will help. *Draw* a picture of where the matrix moves (1,0) and (0,1). Remember also that (0,0) stays where it is. And those three facts tell you what it does to *all* shapes!

If the linear transformation moves (1,0) to (a,c) and (0,1) to (b,d), then the matrix is

a b

c d

If the matrix is

a b

c d

then it moves (1,0) to (a,c) and (0,1) to (b,d).