Computations

Question 1

Negating statements

Answer 1

The negation of a statement is the opposite statement.

When forming negations you change Ǝ to Ɐ and Ɐ to Ǝ.

Question 2

Proof by contradiction

Answer 2

Assume the statement is false. (The negation of the statement is true).
Deduce from this a statement which we know to be false.
This shows that “not p” implies q hence “not q” implies p.

Question 3

Disproof by counterexample

Answer 3

Proving a single counterexample disproves the statement by proving the negation.

Question 4

Converting between decimal and rational form of rational numbers

Answer 4

x = a0.a1a2a3…

x = a0 + a1/10 + a2/10^2 + a/10^3 + …

Question 5

Geometric series

Answer 5

Let x be a real number.

If x does not =1, then
x + x^2 + x^3 + · · · + x^n = x(1−x^n )/(1-x).

If, -1

Question 6

Sum of a geometric series

Answer 6

a

Question 7

Inequality rules

Answer 7

1) If x∈R, then either x>0 or x<0 or x=0 (and just one of these is true).
(2) If x>y then −xy and c∈R, then x+c>y+c.
(4) If x>0 and y>0, then xy>0.
(5) If x>y and y>z then x>z.

Question 8

Polar form of complex numbers

Answer 8

For z ∈ C, suppose we know the modulus |z| and the argument of z. Then, letting z = a + bi we can find a and b using geometry. That is, a = |z|cos(θ) and b = |z|sin(θ)

Question 9

convenient polar form

Answer 9

the polar form is z = r (cos(θ) + i sin(θ)), where r = |z| and θ = arg(z). There is a much more convenient way to write this: z=re^(iθ) where r,θ ∈ R with r>0.

Question 10

Quadratic formula

Answer 10

x = -b±√(b^2 - 4ac)/2a