Further Pure Mathematics

347 Pages·2013·14.85 MB·English

Further
Pure

WETUUELL CES

BRIAN AND MARK GAULTER

OXFORD
Further
Pure

Mathematics

BRIAN AND MARK GAULTER
OXFORD

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Contents

Preface

1

Complex Numbers
Whut isa oinples ausnber?
Calculaving with complex numbers
Arguod diagram

Levi in the enuplex plane

Cube roots of unity

Further Trigonometry with

Calculus

‘Genera solutions of tsisonometris
equations

Hanimanie for

Inverse tigonometr: fenetions

Polar Coordinates
Position of a point

2

3

4g

48

Connection botvcen polar and cartesian

coordinates

Sketching cures given in polar
soordimates

Arai of a sector of a curve

Equations of the tangents toa curve

Differential Equations
Firstorder equations requiring an

Second-order differential equations
Solution of cifferemial equations by
substation

Determinants

Definition of 2x 2nd 3 «3
‘determinants

Rules for the wanipalation of
determinants

Factorisation of dete=minans

Soliton of three eeuations it these

Vector Geometry
Vetar equation of Fine

0

Cuntesan equation of a line 98
Resolved pur: of a vestor 88
Direction ratios 98
Diteetion cosines 99
Vector procuct 102
Area afm triangle 105,
Equation of w plane 106
Distance of a plane from the origin 112
Distence of a plane froma point 113
‘Scalar triple product and its

applications 121

‘Curve Sketching and Inequalities 130

Cune skotehing bo

Sketching tational functions with a
‘quadratic cerominat

Inccualities 10

Roots of Polynomial Equations 147

Roots of a quadratic eqantion 147
Rovts of a cubic sjastion Ko
Roots of polynomis equation of
egree m 190
Equations with related roots 132
Complex roots of « polynomial
‘uation It

Proof, Sequences and Series 159

Proof by 159
Proof by contradiction tot
Summation of series 168
Convergence
Maclaurin's series 07
Using power seres 182
185
Hyperbolic Functions 199
Definicions 189
Graphs of cosh x, sink xand tanh x 190
Slandird byperbolie identities 191

Differentiation of hyperbole furesions192
Integration of hyperbolic functions 193
Inverse hyperbolic functions 194
"

12

4

Logarithnnic form

Dilferectistion of inverse hyperbole
functions

‘Double-engk formulae

Power series

Oshore’s rule

Conies
Generating conics
Parabols

Ellipse

Hyperbols

Polar oqustion of conic

Further Integration
everse function of a fonction rule
Integration by parts

Integration of Fractions

Reduction forma

Ate engl

‘Area of a surface of revolution,
Improper integrals

Summation of series

‘Numerical Methods

Solution of polynomial equations

Evaluation of arces under curves

Stepeay-step soluion of differential
equations

Taylor series

Matrices
Notion

‘The onder of a matrix

Addition and subtrection of matrices
Mi tion oF matrices
Determinant of a matrix

Identity matrices and 2ero matrices
Inverss matrices

‘Transformations

Figenvestors and eigenvalues

1S

16

"7

Ansivers,

Dingonalisation
‘The charactersie equation: 23

Further Complex Numbers

De Moivte's theorem

atk roots oF unity

Exponential form os complex
‘number

Trigonomeirie entities

Transforeaations in & complex plane

Intrinsie Coordinates
‘Trigonore:tic functions of y
Radius of curvature

Finding intrinsic cquations

Groups
Binsny aad weary op
Modular arithmetic
Definition of a group 370
Group tab
Symmets
polygon
Non-fnite groups
a notation
Permutation groups
Generator of group
Cyclic aroups
Abelian groups
Onder ofa group
Order of an clement
Subgroups
Tgomoephic groups
Laarange’s theorem
Groups of orces 3
Groups of ont
Groups of order
Groups of onde: 6
Real vector spaces

fa regular nse!
1 Complex numbers

Tie wonder of arabs, tha portent o the del world, tat npn Beso tng ad
ret being which we cll he imagnar 700 f wits

In all oxt previous mathematics w.
to have a square root of a negative number. For example, on page 26 0
Incvolacing Pare Mashematis were we considered the sslution of quacratic
equitions, ae? + Ar+e-= 0, we noted that when # ~ dae is less than reo, the
equation is said vo have no real roots

I fact, such an equation has two complex roots,

‘Take, for example, the solution of x? + 29-48 =, Using the quadratic
formule, we cbt

wre have assumed that {1 not possible

=-tiv3

There is no real number which is VT. as the squsre of way’ reall number is
always pos

Theretote, we s

¥7 1s am imaginary number. We denote JT by i
So, using i, we cam expres the roots of the ecuation above in the form
lt vii

7

or ~ Vi and ~

[Note jis also used to represent VT.

What is a complex number?

A complex number is am ec oF the form
ai

where a and bare real numbers and i? =~}

For example, 3+ Siis-a complen number

a0, the musher ie said to be wholly imaginary. IF = 0, the number is
real. If complex number is 0, both a and b ar 0
We usually use x+y to represent an unknown compicx number, and = to

‘The complex conjugate

‘The complex number ¥~iy i called she complex canjugate (oF often just the
conjugate) of x-+ is, and is denoted by £0

For example, ? ~ 3) isthe complex conjugate of 2 +3), and the complex
conjugate of 8 - 91 is -8 +91.

Calculating with complex numbers

‘When we work with complex numbers, we use ordinary algebraic methods
That means that we eammot combine a real number with an Herm. For
‘example, 2+ 21 cannot be simpilticd.

For tro complex numbers to be equal, thelr real parts must be equal and their
fimaginary parts rust be equal.

‘This is a necessary condition for the equality of 1wo complex numbers.
+i, then a

Hence, if a+ ib

For exampls, if2 = 31.4 in, then x = 2 andy

Addition and subtraction

When adding two complex numbers, we add the real terms and separately adel
the i-terms. For example,

BEN +4-61=3+4)=(i- 8)
=T+i
‘Generally, for addition we have
[ (tis (uti sire mstigey

aad for subieas

[ Obi) = a bid =e) =i 9)

Example 1 Subtract 8 ~

7421-64)
Example 2 Find x and y Hx + 21+

real terms, we get

ting imaginary terms, we get

y= =13
Ss 10
= y=

‘Multiplication

‘We upply the generat algebraic method for multiplication, For example,
ays S43
= 8-10-2135

Since # — ~1. this simplifies to
SG) LA Isx Ls

Bow

Generally, we have

(oF ibKe +d = ae = M+ iad +e) since =

Note [is sap
this formula

os multiply out the

numbers every time tham so remorse

Division

To be able to divide by a complex number, we have to changeit to a real
faumiber. Take, for example, the fraction

4

In the simplification of surds on page 408 of fatraduing Pure Mathematies, we
rote that 1 coakt he splifie hy wenn the mmonstor and the
denominator of this traction by 1 — V3,

Similarly, to simply 2 we rultipy sts numerator and its denominator by

oH

sich i the complex conjayate of the denominator, Thus, we have

24H _ C= 3Nd— 5)
417 @ SKE 5)
UNote: (317 = -(-25

16425

Example 3 Simplify 3+

Maliplying the numerator and the denominator by the complex
‘conjugate of 7 —3i, which is 7+ 3i, we obtain
Bei G4I7+3)
TOR Oa
Berar sy
Por

D41G-3 Age “y= 1-9) =
12 Nowe 65) = -(-9) = 49)

1
or Lossy
pore)

First, we simplify the eumeracor:

(= 3749) _ 35+ 51-211—3!

Qe Fe