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H2 Mathematics Complex Numbers, Series and Statistics

Key results for H2 Mathematics: complex numbers, vectors, series, differential equations and statistics.

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Modulus of a complex number
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1
Modulus of a complex number
|a + bi| = root(a^2 + b^2).
2
Argument of a complex number
The angle from the positive real axis, in (-pi, pi].
3
Complex conjugate
The conjugate of a + bi is a - bi.
4
Product of conjugates
(a + bi)(a - bi) = a^2 + b^2.
5
Roots of real polynomials
Non-real roots occur in conjugate pairs.
6
Scalar product
a . b = a1 b1 + a2 b2 + a3 b3 = |a||b| cos theta.
7
Vector product magnitude
|a x b| = |a||b| sin theta, which is the area of the parallelogram.
8
Area of a triangle from vectors
1/2 |AB x AC|.
9
Vector equation of a line
r = a + lambda d, where d is the direction vector.
10
Sum of an arithmetic progression
S_n = n/2 [2a + (n - 1)d].
11
Sum of a geometric progression
S_n = a(1 - r^n)/(1 - r). Sum to infinity a/(1 - r) when |r| < 1.
12
Method of differences
Write the general term as f(r) - f(r + 1), so that most terms cancel.
13
Maclaurin series of e^x
1 + x + x^2/2! + x^3/3! + ...
14
Maclaurin series of ln(1 + x)
x - x^2/2 + x^3/3 - ..., valid for -1 < x <= 1.
15
Maclaurin series of sin x
x - x^3/3! + x^5/5! - ...
16
Maclaurin series of cos x
1 - x^2/2! + x^4/4! - ...
17
Integration by parts
The integral of u dv/dx dx = uv - the integral of v du/dx dx.
18
Volume of revolution about the x-axis
V = pi times the integral of y^2 dx.
19
Separable differential equation
Write all y terms with dy and all x terms with dx, then integrate both sides.
20
Exponential growth
dP/dt = kP gives P = P0 e^(kt).
21
Binomial distribution
X ~ B(n, p): E(X) = np and Var(X) = npq.
22
Standardising a normal variable
Z = (X - mean)/standard deviation.
23
Distribution of the sample mean
Mean of the sample mean is the population mean, and its standard deviation is sigma/root n.
24
Unbiased estimate of variance
s^2 = sum (x - mean)^2 / (n - 1).
25
Confidence interval for a mean
Sample mean plus or minus z x sigma/root n.
26
Type I error
Rejecting the null hypothesis when it is true.
27
Type II error
Not rejecting the null hypothesis when it is false.
28
p-value rule
Reject the null hypothesis if the p-value is at most the significance level.
29
Conditional probability
P(A given B) = P(A and B)/P(B).
30
Correlation coefficient r
Lies between -1 and 1. Its sign gives the direction and its size the strength of the linear relationship.
31
Regression line of y on x
y = a + bx, with b = Sxy/Sxx. It passes through the point (mean x, mean y).

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