### Handbook of Mathematics

### For Students and Explorers

Rajesh R. Parwani Editor

An Imprint of the

Simplicity Research Institute, Singapore.

www.simplicitysg.net

Handbook of Mathematics:

For Students and Explorers.

Published by SRI Books, an imprint of the Simplicity Research Institute, Singapore.

www.sribooks.simplicitysg.net Contact: enquiry@simplicitysg.net.

Copyright c 2016 by Rajesh R. Parwani.

All rights reserved.

A CIP record for this book is available from the National Library Board, Singapore.

Edition SRI-2016-1E.

ISBN: 978 − 981 − 09 − 7989 − 8 (pbook) ISBN: 978 − 981 − 09 − 7990 − 4 (ebook)

### MENU

Page

1 Prefixes and Units 1

2 Numbers and Constants 2

3 Some Greek 3

4 Prime Numbers 4

5 Divisibility and Factorisation 5 6 Sequences and Series 6

7 Binomial Theorem 7

8 Combinatorics 8

9 Ratios and Percentages 9 10 Areas and Volumes 10

11 Means 11

12 Inequalities and Approximations 12 13 Quadratic Equations 13 14 Polynomials and Rational Functions14

15 Exponents 15

16 Logarithms 16

17 Matrices 17

18 Vectors 19

19 Kinematics 20

20 Sets 21

21 Probability 22

22 Statistics 23

23 Functions 24

24 Boolean Algebra 25

25 Methods of Proof 26

26 Euclidean Geometry 27 27 Coordinate Geometry 28

28 Circles 30

29 Geometry of Circles 31

30 Triangles 33 31 Congruent Triangles 34 32 Triangle Bisectors 35 33 Polyhedra and Graphs 36

34 Trigonometry 37

35 Trigonometric Identities 38 36 Sine and Cosine Rules 39 37 Elevation, Depression and Bearing 40 38 Differential Calculus 41

39 Derivatives 42

40 Integral Calculus 43

41 Integrals 44

42 Recent Theorems 45

43 Unresolved Conjectures 46

This book is dedicated to one of my first teachers, my late aunt, Parpati.

Preface

In addition to providing reference for the usual topics covered in high-schools and colleges, this book includes eclectic titbits to

stimulate enquiry and investigation.

Acknowledgements I thank Ching Chee Leong, Meer Ashwinkumar, Ng Wei Khim, and Siu Zhuo Bin for helpful feedback on the

draft.

January 2016, Singapore.

References

For details on the material in this handbook, please refer to the relevant entries in

Wikipedia (www.wikipedia.org) or MathWorld (mathworld.wolfram.com).

For exercises related to the content here, check out our other books at www.simplicitysg.net/books.

Feedback: enquiry@simplicitysg.net

Prefixes and Units

• Any non-zero number* may be written in stan-
dard (scientific) notation as ±A × 10^{p}where
1 ≤ A < 10 and p is an integer.

• Some common prefixes:

– nano: 10^{−9}.
– micro: 10^{−6}.
– milli: 10^{−3}.
– kilo: 10^{3}.
– mega: 10^{6}.
– giga: 10^{9}.

• 1 centimetre (cm) = 10 millimetres (mm) .

• 1 metre (m) = 100 cm.

• 1 light-year ≈ 9.46 × 10^{12}km.

• 1 hectare = 10, 000 m^{2}.

• 1 litre (L) = 1000 cm^{3}.

• 1 kilogram (kg) = 1000 grams (g).

• 1 tonne (t) = 1000 kg.

• 1 dozen = 12 units.

• 1 googol = 10^{100}units.

*Note: Unless otherwise stated, we will deal only with real numbers in this handbook .

Numbers and Constants

• A rational number can be written as the ratio of two integers, For example, 0.23 = 23/100 is rational.

• Most numbers are irrational, that is, not ratio- nal. Examples are√

2, π, and e (Euler’s constant).

• Some numbers are suspected to be irrational but
a proof is lacking at the time of writing. Exam-
ples are π ± e, and π^{e}.

• π ≈ 3.142.

• π^{2}≈ 9.87.

• e ≈ 2.718.

• √

2 ≈ 1.414.

• √

3 ≈ 1.732.

• √

5 ≈ 2.236.

• φ (Golden ratio) ≈ 1.618.

Did You Know?

The first 25 digits of π:

3.141 592 653 589 793 238 462 643 3....

Some Greek

Greek letters are often used in mathematics.

The lower case, and some upper case letters are indicated below.

• α alpha.

• β beta.

• γ gamma.

• Γ Gamma.

• δ delta.

• ∆ Delta .

• epsilon.

• ζ zeta.

• η eta.

• θ theta.

• ι iota

• κ kappa

• λ lambda.

• Λ Lambda.

• µ mu.

• ν nu.

• ξ xi.

• o omicron.

• π pi.

• Π Pi.

• ρ rho.

• σ sigma.

• Σ Sigma .

• τ tau.

• υ upsilon.

• φ phi.

• χ chi.

• ψ psi.

• ω omega.

• Ω Omega.

Did You Know?

n→∞lim

1 + 1

n

n

= e, where e is Euler’s constant.

Prime Numbers

• Prime numbers are positive integers that have exactly two factors: 1 and themselves.

• The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, ....

• There are an infinite number of primes, as first shown by Euclid.

• An elementary (but slow) way to check for pri- mality of a number N is to check if the number is divisible by primes less than√

N .

• Prime Factorisation Theorem:

Every integer has a unique decomposition into a product of its prime factors.

• There is no known explicit formula for the n-th prime.

• Prime Number Theorem:

Let π(n) count the number of primes up to n.

For example, π(7) = 4. Then, as n → ∞, π(n) ∼ n

ln n,

meaning that the ratio π(n)/(n/ ln n) approaches 1 as n → ∞. (Note: ln is the natural logarithm).

Divisibility and Factorisation

• A number is divisible by 2 if its last digit is even.

For example, 26498 is even.

• A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 is divisible by 3, but 431 is not.

• A number is divisible by 4 if the number rep- resented by its last two digits is divisible by 4.

For example, 26418 is not divisible by 4 because 18 is not.

• A number is divisible by 5 if its last digit is 0 or 5.

• A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 126 is divisible by 9.

• Let L be the least common multiple of the natural numbers m and n, and let G be their greatest common divisor. Then GL = mn.

• a^{2}− b^{2}= (a − b)(a + b).

• a^{3}∓ b^{3}= (a ∓ b)(a^{2}± ab + b^{2}).

• x^{n}− 1 = (x − 1)(1 + x + · · · + x^{(n−2)} + x^{(n−1)}).

• For n odd,

x^{n}+ 1 = (x + 1)(1 − x + x^{2}− x^{3}+ · · · + x^{(n−1)}).

Sequences and Series

• Arithmetic Progression (AP): The n-th term, an, of a sequence in arithmetic progression is given by an = a1+ (n − 1)d, where d is the common difference between consecutive terms.

• The sum, Sn, of n consecutive terms of an AP is given by n

2(a1+ an).

• Geometric Progression (GP): The n-th term,
an, of a sequence in geometric progression is
given by an= a1r^{n−1}where r = a2/a1 is the
common ratio between consecutive terms.

• The sum, Sn, of n consecutive terms of a GP is
given bya1(r^{n}− 1)

r − 1 .

• For a GP with |r| < 1, the infinite sequence has a convergent sum S∞= a1

1 − r.

• e^{x}=

∞

X

n=0

x^{n}

n! = 1 + x +x^{2}
2! +x^{3}

3! + ...

• For n real and |x| < 1 we have the convergent series

(1 + x)^{n}=1 + nx +n(n − 1)
2! x^{2}
+ n(n − 1)(n − 2)

3! x^{3}+ . . .

Binomial Theorem

• For a positive integer n,
(a + b)^{n}= a^{n}+

n 1

a^{n−1}b +

n 2

a^{n−2}b^{2}+ ...

+n r

a^{n−r}b^{r}+ ... + n
n − 1

ab^{n−1}+ b^{n}

=

r=n

X

r=0

nCr a^{n−r}b^{r}.

• The binomial coefficient is defined by

nCr≡n r

= n!

r! (n − r)!

with n! ≡ n × (n − 1) × (n − 2) × .... × 2 × 1 and 0! ≡ 1.

• The (r + 1)-th term in the expansion is given by

n r

a^{n−r}b^{r}.

• Special Cases:

(a ± b)^{2}= a^{2}± 2ab + b^{2} .
(a ± b)^{3}= a^{3}± 3a^{2}b + 3ab^{2}± b^{3}.
(a ± b)^{4}= a^{4}± 4a^{3}b + 6a^{2}b^{2}± 4ab^{3}+ b^{4} .

Combinatorics

• The number of ordered arrangements of n dis- tinguishable objects on a line is n! .

• The number of ordered arrangements of n ob- jects on a line is

n!

p! q! r!,

where there are p identical objects of one type, q identical objects of a second type, etc.

• The number of ordered arrangements of r ob- jects (on a line), chosen from a collection of n distinguishable objects is

nPr≡ n!

(n − r)! .

• The number of ways of choosing r objects from a collection of n distinguishable objects is

nCr≡ n!

r! (n − r)! .

• Pascal’s Identity: For 0 ≤ r ≤ n,

n r

+ n

r + 1

=n + 1 r + 1

.

Ratios and Percentages

• If a variable y is proportional to another vari- able x, y ∝ x, then y = kx for some constant k.

That is, the ratio y/x is a constant equal to k.

• If a variable y is inversely proportional to another variable x, y ∝ 1/x, then y = k/x for some constant k. That is, the product y · x is a constant equal to k.

• The fraction a/b corresponds to 100a/b per- cent. For example, the fraction 1/2 is 50%

while 3/2 is 150%.

• Money deposited in a bank savings account typ- ically earns interest. Simple interest at R%

per annum on a principal of P dollars will yield P (1 + r) dollars at the end of the year, where r = R/100.

• Compound Interest: The nominal annual in- terest of R% may be divided into N parts, and an interest of (R/N )% paid on the accumulated amount (principal plus interest) at the end of each 12

N months. So at the end of t years an initial savings of P would have grown to P

1 + r

N

N t

where r = R/100.

Areas and Volumes

• Area of a triangle:

1

2b × h, where b is the base and h the height.

• Area of a parallelogram:

b × h, where b is the base and h the height.

• Area of a circle of radius r: πr^{2}.

• Circumference of a circle of radius r: 2πr.

• Volume of pyramid or cone:

1

3(area of base) × (vertical height).

• Volume of solid figure of constant cross-section:

(area of cross-section) × (vertical height).

• Surface area of solid figure of constant cross- section:

2 × (area of cross-section) +

(perimeter of cross-section) × (vertical height).

• Volume of a sphere of radius r: 4
3πr^{3}.

• Surface area of a sphere of radius r: 4πr^{2}.

• Lateral surface area of cone: πrl, where l is the
lateral height, and r the radius of the circle at
the base. (The base area is πr^{2}).

Means

Let a and b be two real numbers. Then the following means may be defined (restricting to positive numbers for the geometric mean).

• Quadratic Mean Q: Q = s

a^{2}+ b^{2}
2
(also known as root mean square).

• Arithmetic Mean A: A = a + b 2 .

• Geometric Mean G: G =√ ab.

• Harmonic Mean H: It is determined by 2

H = 1 a+1

b.

• The following relations hold between the means:

G =

√ AH.

Q ≥ A ≥ G ≥ H.

Did you know?

You can view sample pages of all our books at

www.simplicitysg.net/books

Inequalities and Approximations Let (a1, a2) and (b1, b2) be pairs of real num- bers in the following.

• Cauchy-Schwartz:

(a1b1+ a2b2)^{2}≤ (a^{2}_{1}+ a^{2}_{2})(b^{2}_{1}+ b^{2}_{2}).

• Triangle Inequality:

q

(a1+ b1)^{2}+ (a2+ b2)^{2}≤
q

a^{2}_{1}+ a^{2}_{2}+
q

b^{2}_{1}+ b^{2}_{2}.

• Rearrangement Inequality: If a2≥ a1 and
b2≥ b_{1}then a2b2+ a1b1≥ a_{2}b1+ a1b2.

• Isoperimetric Inequality: A plane (closed)
figure of area A and perimeter P satisfies
4πA ≤ P^{2}. This implies that the circle has the
largest area for a given perimeter.

• e^{a}≥ 1 + a.

• Bernoulli’s Inequality:

(1 + a)^{b}≥ 1 + ab for a ≥ −1 and b ≥ 1.

• For x > 0, x^{x}≥ 1
e

1/e

.

• Stirling’s approximation:

n! ∼√ 2πnn

e

n

as n → ∞.

Quadratic Equations

• The solution of the quadratic equation
ax^{2}+ bx + c = 0, with a 6= 0 is given by the
roots

x =−b ±√

∆

2a .

• ∆ ≡ b^{2}− 4ac is called the discriminant.

• The two roots are real if and only if ∆ ≥ 0; the case ∆ = 0 corresponds to a repeated root.

• The orientation of the parabola y(x) = ax^{2}+
bx + c is determined by the sign of a: When
a > 0, the curve has a minimum point while for
a < 0 it has a maximum.

• The symmetry axis of the parabola is at x =

−b 2a.

a > 0

a < 0 y

Polynomials and Rational Functions

• Factorisation Theorem:

For a polynomial P (x), P (α) = 0 if and only if P (x) ≡ (x − α)Q(x) for some polynomial Q.

• Remainder Theorem: If the polynomial P (x) is divided by (x − α), the remainder is P (α).

That is, P (x) ≡ (x−α)Q(x)+R with R = P (α).

• For a cubic curve y = ax^{3}+ bx^{2}+ cx + d,
the sign of the leading coefficient determines its
main shape. If a > 0, the curve rises upwards for
large positive x and decreases for large negative
x. In between, it might have a local minimum
and a local maximum.

• Partial fraction decomposition of a rational
function P (x)/Q(x): First, use long division to
reduce the degree of the numerator to below
that of the denominator. Next, each factor of
(x − a) in Q(x) would require a partial fraction
A/(x − a). If the factor is repeated in Q, for
example (x − a)^{2}, then one uses two partial frac-
tions A1/(x−a) and A2/(x−a)^{2}for that factor.

If Q contains a term that cannot be factorised
(using real numbers), for example x^{2}+ x + 1,
then the partial fraction for that term is of the
form (Ax + B)/(x^{2}+ x + 1), that is, the numer-
ator is one degree lower than the denominator.

Exponents For a, b > 0,

• a^{−1}=_{a}^{1} .

• a^{0}= 1 .

• a^{xy}= (a^{x})^{y} .

• a^{x}a^{y}= a^{x+y} .

• (ab)^{x}= a^{x}b^{x} .

• a^{1}2 =√
a .

• √ ab =√

a√ b .

• a^{m}n = √^{n}
am

.

• For a > 1, y = a^{x} is positive and an increasing
function of x along the real line; for 0 < a < 1,
a^{x}is a decreasing function of x.

Did You Know?

Euler’s Identity:

e^{iθ}= cos θ + i sin θ ,
where i =√

−1.

Logarithms

The basic relation between exponents and log- arithms is

y = b^{x}⇔ x = log_{b}y .
For a, b > 0, a 6= 1, b 6= 1 and P, Q > 0,

• log_{b}1 = 0 .

• log_{b}P Q = log_{b}P + log_{b}Q .

• log_{b}P

Q= log_{b}P − log_{b}Q .

• log_{b}P^{c}= c log_{b}P .

• log_{b}P = log_{a}P
log_{a}b .

• log_{b}a = 1
log_{a}b .

• For a > 1 and x > 0, y = log_{a}x is an increasing
function of x; it is positive for x > 1, negative
for x < 1 and vanishes at x = 1.

• The graph of y = log_{a}x may be obtained by
reflecting the exponential curve y = a^{x} about
the line y = x.

• The natural logarithm, log_{e}x, where e is Eu-
ler’s constant, is often denoted by ln x.

Matrices

• A m × n matrix A, multiplying from the left to a n × p matrix B, yields a m × p matrix C; that is C = AB. The cijelement of C is obtained by multiplying the i-th row of A to the j-th column of B, term by term, and adding the pieces.

• The pair of simultaneous equations in the variables (x, y),

ax + by = e cx + dy = f ,

may be written as the matrix equation M X = R with

M =

a b c d

,

X =

x y

and

R =

e f

.

• The determinant of the 2 × 2 matrix M is de- fined by det(M)= ad − bc.

• If det(M)6= 0, one may define an inverse ma- trix

M^{−1}= 1
det(M)

d −b

−c a

. (continued →)

• If det(M)= 0 the matrix is termed singular and there is no inverse matrix.

• The inverse matrix satisfies
M M^{−1}= M^{−1}M =

1 0 0 1

, the last matrix being the identity matrix, usu- ally denoted by the letter I.

• The solution of M X = R is
X = M^{−1}R = 1

det(M)

de − bf

−ce + af

.

Did You Know?

The book Integrated Mathematics for Explorers, by Adeline Ng and Rajesh Parwani, contains questions that allow you to test your understanding of most of

the topics in this handbook.

Check it out at

www.simplicitysg.net/books/imaths

MENU

Vectors

• The magnitude of a vector ~a = ax~i + a_{y}~j +
ax~k is |~a| ≡

q

a^{2}_{x}+ a^{2}_{y}+ a^{2}_{z}where i, j, k are or-
thonormal vectors, that is, i · i = j · j = k · k = 1
and i · j = i · k = j · k = 0.

• The scalar or dot product of two vectors is

~a · ~b = ~b · ~a = axbx+ ayby+ azbz= |~a||~b| cos θ , where θ is the angle between ~a and ~b.

• The vector or cross product of two vectors is

~a × ~b = −~b × ~a = |~a||~b| sin θ ~n , where ~n is the unit vector perpendicular to the plane defined by ~a and ~b; its direction given by the right-hand rule (point your right-hand fin- gers in the direction of ~a and close them in the direction of ~b, the thumb points along ~n). In Cartesian coordinates, ~a × ~b is

(aybz− a_{z}by)~i + (azbx− a_{x}bz)~j + (axby− a_{y}bx)~k .

• Scalar Triple Product:

~a · (~b × ~c) = ~b · (~c × ~a) = ~c · (~a × ~b).

• Lagrange’s formula (Vector Triple Product):

~a × (~b × ~c) = ~b(~a · ~c) − ~c(~a · ~b).

Kinematics

• Kinematics is the study of motion, without in- quiring about the causes of the motion.

• Let the vector ~x(t) represent the position of a particle at time t, and ∆~x its displacement in the time interval ∆t. (For one-dimensional problems, ~x may be simply written as x).

• Its average velocity during that interval is then

∆~x

∆t. The instantaneous velocity is obtained in the limit ∆t → 0, and is denoted in calculus notation by ~v = d~x

dt.

• The acceleration is ~a =d~v
dt =d^{2}~x

dt^{2}.

• If the distance moved in the time interval ∆t is D, then the average speed is D

∆t.

Did You Know?

A particle moving at constant (non-zero) speed around a circle has

zero average velocity on completing one round.

Sets

• A set is a collection of items called elements.

For example, the set C = {2, 4, 6} consists of the three elements 2, 4, and 6. The number of elements of a set A is denoted by n(A).

The empty set is denoted by ∅.

x ∈ A means ‘x is an element of set A’.

• Two sets are equal if they have the same ele- ments. Set A is a subset of set B, denoted by A ⊂ B, if every element of A is also in B.

• The universal set, E, consists of all those ele-
ments under consideration. The complement
of A, denoted by A^{0}, includes all elements of E
not in A. For example, in the set of positive in-
tegers, the complement of the set of odd integers
is the set of even numbers (including zero).

• Union of Sets: A ∪ B = {x|x ∈ A or x ∈ B}.

• Intersection of Sets:

A ∩ B = {x|x ∈ A and x ∈ B}.

• A ∩ A^{0}= ∅.

• A ∪ A^{0}= E.

• n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

• (A ∩ B)^{0}= A^{0}∪ B^{0} and (A ∪ B)^{0}= A^{0}∩ B^{0}.

Probability

• The probability of an event A occurring is de- noted by P (A) with 0 ≤ P (A) ≤ 1.

• The value of P (A) may be estimated by not- ing the relative frequency of the occurrence of A in repeated trials of an experiment, or in accumulated data of observations.

• Given p = P (A), then n repeated trials of the same experiment will yield event A np times on average. That is, expectation E(A) = np.

• Independent Events: (See the ‘Sets’ chapter for the ∩ and other notation). P (A ∩ B) = P (A) × P (B).

• P (A ∪ B) = P (A) + P (B) − P (A ∩ B).

• Conditional Probability: The probability of event B given that event A has occurred is P (B|A) =P (A ∩ B)

P (A) .

• Bayes’ Theorem:

P (B|A) = P (A|B)P (B) P (A) ,

where P (A) may be written as P (A|B)P (B) +
P (A|B^{0})P (B^{0}) with B^{0} being the complement
of B; that is P (B) + P (B^{0}) = 1.

Statistics

For a data set xi, i = 1, 2, . . . , n,

• The mean µ is defined by

µ = 1 n

n

X

i=1

xi.

• The variance is

σ^{2}= 1
n

n

X

i=1

(xi− µ)^{2}

= 1 n

n

X

i=1

x^{2}_{i}− µ^{2} ,

where σ is the standard deviation.

• Bhatia-Davis Inequality:

For a bounded probability distribution P (X) with m ≤ X ≤ M , we have

σ^{2}≤ (M − µ)(µ − m).

• Samuelson’s Inequality:

For each xi, µ − σ√

n − 1 ≤ xi≤ µ + σ√ n − 1.

Functions

• A function f maps an element x of a domain set to a unique element y = f (x) in its codomain.

• The range or image of the function is a subset of the codomian.

• A function f is one-to-one (injective) if f (a) = f (b) implies a = b.

• A function f is onto (surjective) if its range coincides with its codomain.

• A function that is both injective and surjective is called bijective.

• The inverse function f^{−1} reverses the map-
ping due to f . f^{−1}(y) = x where y = f (x).

(f^{−1} exists if f is one-to-one, or if the domain
of f^{−1}is suitably restricted.)

• Two functions f and g can be composed to give a new function g ◦ f that acts as g ◦ f (x) = g(f (x)).

Did You Know?

Jacobi’s identity for vectors:

~

a × (~b × ~c) + ~b × (~c × ~a) + ~c × (~a × ~b) = 0.

Boolean Algebra

Classical computers operate according to Boolean logic. Boolean algebra implements Boolean logic using rules described below.

• In Boolean algebra a variable x takes only one of two values to represent true/false states:

1 (TRUE) or 0 (FALSE).

• AND operation: Denoted by · or ∧.

x ∧ y = 1 if and only if x = y = 1; otherwise it equals 0. This operation is commutative, x∧y = y ∧ x, and associative, x ∧ (y ∧ z) = (x ∧ y) ∧ z.

• OR operation: Denoted by + or ∨.

x ∨ y = 0 if and only if x = y = 0; otherwise it equals 1. This operation is also commutative and associative.

• NEGATION: Denoted by ¯x or x^{0}.

¯

x = 1 − x where “ − ” here denotes the usual subtraction.

• Distributive Laws:

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).

and x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z).

Methods of Proof

• Deduction: Proceeds from the initial state- ment to the conclusion through a sequence of logical steps.

• Contradiction: A method of proving the truth of a statement by first assuming its negation, and showing that that leads to a contradiction.

For example, this method is usually used to show that√

2 is irrational.

• Contrapositive: The statement A ⇒ B is log- ically equivalent to the statement ¯B ⇒ ¯A, where B is the negation of B. For example, “all even¯ numbers are divisible by 2” is equivalent to “if a number is not divisible by 2, it cannot be even”.

• Induction: The truth of a formula for all natu- ral numbers n is determined by first showing the formula to be true for n = 1, and then showing that its presumed truth for n = k implies its truth for n = k + 1.

Did You Know?

π 4 =

∞

X

n=0

(−1)^{n}
2n + 1= 1 −1

3+1 5−1

7+ ...

Euclidean Geometry

• Parallel lines do not meet, even when extended.

Perpendicular lines meet at an angle of 90^{◦}.

• If lines AB and CD intersect at O, then ∠AOC =

∠BOD (‘vertically opposite angles’).

• If P is a point not on the line AB produced, then the shortest distance from P to AB produced is along the perpendicular from P to that line.

• Let AB and CD be two parallel lines intersected by the line EF at G and H respectively (with A and C on the same side of EF ). Then ∠AGH =

∠DHG (‘alternate angles’).

B C A

D O

A B

C D

E

F

G

H

Coordinate Geometry

• The straight line joining points (x, y) and (x1, y1)

is y − y1

x − x1

= m = tan α ,

where 0 ≤ α < π is the angle the line makes with the positive x-axis, and m is the slope (gradient). The equation may be re-written as y = mx + c where c is the intercept on the y-axis.

• If another line is perpendicular to the above line, its slope must be −1/m.

• The mid-point of a line joining two points (x1, y1)
and (x2, y2) is x_{1}+ x2

2 ,y1+ y2

2

.

• Given the vertices A(x_{1}, y1), B(x2, y2) and C(x3, y3),
of a triangle, with their relative order being anti-
clockwise, the area of the triangle is

A =1 2

x1 x2 x3 x1

y1 y2 y3 y1

=1

2(x1y2+ x2y3+ x3y1− x1y3− x3y2− x2y1) The terms are generated as follows: Start at the left-edge of the top row and multiply each term in the top row by a term one step to the right in

the bottom row, adding the pieces. Then start at the right-edge of the top row and multiply each term in the top row by a term one step to the left in the bottom row, adding the pieces.

Finally, subtract those two contributions and in- clude the overall 1/2.

• For a polygon with n sides the general shoelace algorithm for the area is

A =1

2|x_{1}y2+ x2y3+ · · · + xn−1yn+ xny1

− x_{2}y1− x_{3}y2− · · · − x_{n}yn−1− x_{1}yn| .

Did You Know?

The book Real World Mathematics, by W. K. Ng and R. Parwani, contains questions on real-world applications of most of the topics in

this handbook.

Check it out at

www.simplicitysg.net/books/rwm

MENU

Circles

• A line joining two points on the circumference of a circle forms a chord. The perpendicular line from the centre of the circle to the chord bisects the chord (conversely, the perpendicular bisector of a chord passes through the centre).

• Uniqueness: Given any three points not on a line, exactly one circle passes through those points.

• A tangent to a circle at any point P on its circumference is perpendicular to the line OP where O is the centre of the circle. Therefore, the normal to the circle at P lies along OP .

• The equation for a circle is
(x − x0)^{2}+ (y − y0)^{2}= r^{2},
where (x0, y0) is the centre and r the radius.

• Given the form

x^{2}+ y^{2}− 2ax − 2by + c = 0 ,

for some constants a, b, c, one can complete the
squares to get (x − a)^{2}+ (y − b)^{2}= a^{2}+ b^{2}− c; if
c < a^{2}+ b^{2}then the equation represents a circle
with centre (a, b) and radius R =p

a^{2}+ b^{2}− c.

Geometry of Circles

• Inscribed Angle Theorem: Let A and B be two points on the circumference of a circle. The angle subtended by A and B at the centre of the circle is twice that subtended at a point C on the circumference, see Fig.(29.1). This implies that two angles subtended by the same chord, on its same side, must be equal.

• Tangent-Chord Theorem (Alternate Seg- ment Theorem): Let the triangle ABC be inscribed in a circle and a tangent drawn at point A. Let D be another point on the tan- gent line such that D and C are on opposite sides of the line AB. Then ∠DAB = ∠BCA.

See Fig.(29.2).

• Intersecting Chords Theorem: Let A, B, C, and D be points on the circumference of a cir- cle and X the point of intersection of the lines AC and BD. Then the triangle ABX is similar to triangle DCX. See Fig.(29.3). This implies AX · CX = BX · DX.

• Tangent-Secant Theorem: Let A, B and C
be points on the circumference of a circle and
let the tangent at A meet the line CB produced
at D. Then (DA)^{2}= DB × DC. See Fig.(29.4).

Figure 29.1: Figure for Inscribed Angle Theorem.

Figure 29.2: Figure for Tangent-Chord Theorem.

Figure 29.3: Figure for Intersecting Chords Theorem.

Figure 29.4: Figure for Tangent-Secant Theorem

Triangles

• In triangle ABC let BC be produced to D as shown below. Then the exterior angle ACD =

∠BAC + ∠ABC.

• In triangle ABC let a, b and c represent the sides opposite the corresponding angles (vertices) de- noted in upper-case. Then A > B ⇔ a > b.

• In a triangle ABC with non-zero area, a+b > c.

• Pythagoras’ Theorem: In triangle ABC with
C = 90^{◦}, a^{2}+ b^{2}= c^{2}.

• Two triangles ABC and P QR are similar if one is a scaled version of the other; see figure below.

That is, if they have the same angles, then their sides are in the same proportion: If QRkBC, then BC/QR = AB/P Q.

• Heron’s formula for the area of a triangle:

Area =ps(s − a)(s − b)(s − c) where 2s = (a + b + c).

A

B C D

A, P

B C

Q R

Congruent Triangles

Two triangles are congruent (identical), if they satisfy any one of the conditions listed below.

S refers to a side while A to an angle.

• SSS. That is, all three sides are the same for both triangles.

• SAS. Two sides and the included angle are the same for both triangles.

• ASA. One side and the two angles adjacent to that side are the same for both triangles.

• Other cases:

(i) AAS reduces to the ASA case since the sum
of angles in a triangle is 180^{◦}.

(ii) In a right-angled triangle, the sides and an- gles are constrained, so it is easy to check for congruence.

Did You Know?

Given a triangle with perimeter P
and area A, we have the inequality
P^{2}≥ 12√

3A, with equality holding for equilateral triangles.

Triangle Bisectors

• Centroid theorem:

In M ABC let P, Q and R be mid-points of the sides AB, BC and CA respectively. Then the medians CP , AQ and BR pass through a point G (centroid) inside the triangle, and GP = CP/3, with similar relations for the other bisec- tors.

• Circumcentre theorem:

In M ABC let P P^{0}, QQ^{0} and RR^{0} be perpen-
dicular bisectors of the sides AB, BC and CA
respectively. The bisectors pass through a point
O (circumcentre), which may lie outside the
triangle. A circle drawn with O as centre cir-
cumscribes the triangle.

• In-centre theorem:

In M ABC let AP, BQ and CR be bisectors of the angles A, B and C respectively. The bisec- tors pass through a point I (incentre) inside the triangle. A circle drawn with I as centre can be inscribed in the triangle.

• The Euler line, connecting the centroid and circumcentre, passes through the orthocentre, which is the point of intersection of the three altitudes of the triangle. An altitude is a line from a vertex that is perpendicular to the op- posite side.

Polyhedra and Graphs

• There are five regular convex polyhedra in three dimensional space (Platonic Solids). ‘Regu- lar’ means that the faces of a polyhedron are identical. The number of faces F , edges E, and vertices V is indicated below for each case.

• Tetrahedron: F = 4, E = 6, V = 4.

• Hexahedron: F = 6, E = 12, V = 8.

• Octahedron: F = 8, E = 12, V = 6.

• Dodecahedron: F = 12, E = 30, V = 20.

• Icosahedron: F = 20, E = 30, V = 12.

• Each case satisfies Euler’s polyhedron for- mula, V − E + F = 2, which holds more gener- ally for non-regular polyhedra. (The boundary of a convex polyhedron can be deformed into the surface of a sphere. The ‘2’ in the formula above is the Euler characteristic of a sphere).

• There is a duality between the solids, in the ex- change V ←→ F .

• Let G be a planar graph, that is, a collection of vertices V on the plane connected by edges E. If G is connected, that is, there is a path between any two vertices, then Euler’s formula above applies to the graph with F counting the faces (including the outer region as one face).

Trigonometry

• For angle measurements in radian, θ ≡ s/R
where s is the arc length of circle, of radius R,
subtended by that angle. Therefore 2π radians
equals 360^{◦}.

• For a right-angled triangle ABC, with C =
90^{◦}, sin A = a/c , cos A = b/c and tan A =
sin A/ cos A = a/b, where the small case letters
denote lengths opposite the corresponding an-
gles.

• The sin and cos functions range over the in- terval [−1, 1] while the tan function ranges over the real line.

• Periodicity: sin(θ + 360^{◦}) = sin(θ), cos(θ +
360^{◦}) = cos(θ) and tan(θ + 180^{◦}) = tan(θ).

• The principal values for the inverse func-
tions sin^{−1}and tan^{−1}are those that lie in the
range −π/2 ≤ y ≤ π/2, while the cos^{−1} func-
tion has the range 0 ≤ y ≤ π.

• The functions csc, sec and cot are reciprocals of the sin, cos and tan functions (csc x is also written as cosec x).

• Phenomena that are described by an equation of the form y(x) = A + B sin(kx + C) are called sinusoidal.

Trigonometric Identities

• Some identities:

sin^{2}A + cos^{2}A = 1.

sin(−A) = − sin A and cos(−A) = cos A.

tan A = sin A/ cos A and tan(−A) = − tan A.

• Addition Formulae:

sin(A ± B) = sin A cos B ± cos A sin B . cos(C ± D) = cos C cos D ∓ sin C sin D . tan(A ± B) = tan A ± tan B

1 ∓ tan A tan B .

• Factor Formulae:

sin A ± sin B = 2 sinA ± B

2 cosA ∓ B

2 .

cos A + cos B = 2 cosA + B

2 cosA − B

2 .

cos A − cos B = −2 sinA + B

2 sinA − B

2 .

• R-Formulae: With R =√

a^{2}+ b^{2}and tan α =
b/a,

a cos θ ± b sin θ = R cos(θ ∓ α) . a sin θ ± b cos θ = R sin(θ ± α) .

Sine and Cosine Rules

• The sides of a triangle ABC are labelled with small case letters a, b and c denoting lengths op- posite the corresponding angles A, B and C.

• The sine rule:

a sin A = b

sin B = 2R ,

where R is the radius of the circle that circum- scribes the triangle (the centre of the circle lies at the intersection of the perpendicular bisectors of the three sides).

• The cosine rule:

c^{2}= a^{2}+ b^{2}− 2ab cos C .

• The area of the triangle is Area =1

2ab sin C .

Did You Know?

The overdot notation ˙x to denote dx
dt,
and ¨x for d^{2}x

dt^{2}, was invented by Newton.

Elevation, Depression and Bearing

• If a point P is above (below) the horizontal through the observation point O, then the angle that OP makes with the horizontal is the angle of elevation (angle of depression) of P .

• An absolute bearing denotes a direction rel-
ative to North. It is usually expressed by a
clockwise angle measured in degrees, for exam-
ple, 065^{◦}.

• Relative bearings define an angle relative to a chosen axis, for example the axis along which an aircraft is pointing.

Did You Know?

L’Hopital’s Rule: If

x→alimf (x) = lim

x→ag(x) = 0 then

x→alim f (x) g(x)= lim

x→a

f^{0}(x)
g^{0}(x)
if the later limit exists. For example,

x→0lim sin x

x = lim

x→0

cos x 1 = 1.

Differential Calculus

• Let ∆f represent the change in a function f (x) as x changes by ∆x. The derivative of f is defined by df

dx = f^{0}(x) ≡ lim∆x→0∆f

∆x. (We assume here and below that the function is twice differentiable).

• The stationary points of y = f (x) are those for which df

dx = 0. They are turning points (local minima or local maxima), or points of inflexion (where the first derivative has the same sign on both sides of the point).

• If d^{2}f

dx^{2} > (<) 0 at the stationary point, it is a
local minimum (maximum). Stationary points
with d^{2}f

dx^{2} = 0 can be examined by checking the
sign of f^{0}(x) on either side of the point.

• The global extrema occur at the boundaries of the domain or at stationary points.

• If ∆x is small but not strictly zero, we may form the approximation

∆y ≈ dy dx

× ∆x .

Derivatives

Let f and g be functions of x, and A, B, n represent constants in the following.

• d

dxAx^{n}= Anx^{n−1}.

• d

dxe^{x}= e^{x}.

• d

dxln x = 1 x.

• d

dxsin x = cos x and d

dxcos x = − sin x.

• d

dxtan x = sec^{2}x.

• d

dx(f + g) = f^{0}+ g^{0}.

• d

dx(f g) = f g^{0}+ gf^{0}.

• d dx

f

g =gf^{0}− f g^{0}
g^{2} .

• d

dxf (g(x)) = df dg×dg

dx.

• d

dxf = 1/ dx df

.

Integral Calculus

• Fundamental Theorem of Calculus:

Z b a

f (x) dx = F (b) − F (a) , where F (x) is a function that satisfies

dF (x)

dx = f (x) .

• Hence Z b

a

dF

dx dx = F (b) − F (a).

• Indefinite integrals: R f (x) dx = F (x) + C, where F and f are related as above, and C is a constant of integration which can be fixed once we have more information about the problem.

• For calculating areas between the curve y = f (x) and the x-axis through the formulaR y dx, note that if f (x) < 0 within a region x1≤ x ≤ x2, the integral in that region would give a nega- tive value and the area there is then the negative of the integral.

• One may also evaluate the area between a curve and the y-axis. In this case the integral would be

Z y2
y_{1}

x dy.

Integrals

In the formulae below, f and g are functions of x while a, b, n are constants; C is a constant of integration.

• Z

(a + bx)^{n}dx = (a + bx)^{n+1}

b(n + 1) + C, n 6= −1.

•

Z 1

a + bx dx = 1

bln(a + bx) + C.

• Z

sin(a + bx) dx =−1

b cos(a + bx) + C.

• Z

cos(a + bx) dx =1

bsin(a + bx) + C.

• Z

e^{(a+bx)}dx =1

be^{(a+bx)}+ C.

• Z

(f + g) dx = Z

f dx + Z

g dx.

• Z c

a

f (x) dx = Zb

a

f (x) dx + Z c

b

f (x) dx.

Did You Know?

Z+∞

−∞

e^{−x}^{2}dx =√
π

Recent Theorems

Some long standing conjectures were only proven true relatively recently:

• Kepler’s Conjecture (proven)

The way to pack equal sized spheres in three- dimensional space, so as to maximise the aver- age density, is the intuitive regular arrangement!

• The Four Colour Map Theorem

No more than four colours are sufficient to colour a map of contiguous countries on the plane, with- out adjacent countries having the same colour.

• Fermat’s Last Theorem

The equation x^{n}+ y^{n}= z^{n}has no positive in-
tegral solution (x, y, z) for any integer n > 2.

Did You Know?

There are an infinite number of
integral solutions corresponding to
Pythagoras’ Theorem, x^{2}+ y^{2}= z^{2}.

Explicitly,

x = m^{2}− n^{2}, y = 2mn, z = m^{2}+ n^{2},
where m, n are any positive integers.

Unresolved Conjectures

Many mathematicians believe the conjectures below to be true, but rigorous proofs are lack- ing at the time of writing.

• Goldbach Conjecture

Every even integer larger than 2 can be written as the sum of two primes. For example, 42 = 5 + 37.

• Twin Prime Conjecture

There are infinitely many twin primes. (A ‘twin’

prime differs from another prime number by 2.

For example, 11 and 13 are twin primes.)

• Collatz Conjecture

Start with any positive integer n. If it is even, apply the rule n → n/2, while if it is odd apply the rule n → 3n + 1. Continue the iteration on each result. You will eventually reach the number 1. For example, 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

• Beal’s Conjecture

Let x^{m}+ y^{n}= z^{p}, with all letters representing
positive integers, and m, n, p each being greater
than 2. Then, if x, y, z are pairwise relatively
prime, the equation has no solutions. (Note:

This conjecture is a generalisation of Fermat’s Last Theorem).