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Further Mathematics Revision
GROUP THEORY
Closure: no new elements created
Identity: element that leaves other elements unchanged: e
Inverse element: element, which if applied to another element creates the identity
Self-inverses have order 2 a2=e
Associativity: a ж (b ж c) = (a ж b) ж c
Commutative: a ж b = b ж a; Latin square property holds
Finite group: number of elements is not infinite.
Order of a group: number of elements
Order of an element: how many times must be combined with itself to make e
Element a, order of group n  an=e
Generator: element a, that powers of a make up all the members of the group
If order of group is prime, all elements are generators.
Cyclic group: has at least one generator
Klein group: group order 4, all self-inverses
Prime order groups: no sub-groups, all elements generators, cyclic group
Lagrange: order of subgroup: factor of group order
Isomorphic: same as…; a=x, b=y,…;
Operator: binary operation
Cancellation laws:
ab=ad  b=d
(a-1 a)b=(a-1 a)d
eb=ed b=d QED
-1
-1
fg=hg  f=h
f(g g)=h(g g)
fe=he f=h QED
RINGS
Two rules of combination:
- Under addition –> forms a group (commutative)
- Under multiplication -> closed and associative (may not have identity)
Commutative ring: if multiplication is commutative  Distributive laws apply
a(b+c) = ab + ac
(b+c)a = ba + ca
Ring with unity: has a multiplication identity -> each element has en inverse
FIELD: Ring whose non-zero element forms a group under multiplication.
Further Mathematics Revision
PROOF
Proof by contradiction: assume the opposite and prove wrong
√3 is irrational
Assume √3 is rational

√3= a/b
a, b have no common factor
3= a²/b²
3b²=a²
 a² = 3m  a=3m …
3b²=(3m)²
3b²=9m²
b²=3m²
 b² = 3m  b=3m
Contradiction  a and b have a common factor  √3 cannot be rational.
√2 is irrational
Assume √2 is rational. It can then be expressed as a fraction in it’s lowest form.

√2=p/q
p, q have no common factor
2=p2/q2
2q2=p2
 p2 is even  p is also even  p=2k …
2q2=4k2
q2=2k2
 q2 is even  q is also even  q=2m
√2=2k/2m  p and q have common factors.
Contradiction √2 cannot be rational  √2 is irrational
Direct Proof/Algebraic Proof
Sum of 3 consecutive integers is a multiple of 3
n, n+1, n+2  n + n + 1 + n + 2 = 3n + 3 = 3(n +1) = 3m,
Sigma: sum of…
5r=1∑r = 1 + 2 +3 +4 +5
4r=1∑r(r-1)= 0 +2 +6 +12
mεZ
5r=1∑r2=
nr=1∑r=
1 +4 +9 +16 +25
1 +2 +3 +4 +…+n
…Prove that if n2=even then n=even too.
 even= 2m
show even number squared is multiple of 2
 2m=(n+1)2
n is odd
2
2
2m=n +2n+1=odd + even + odd= odd + even + odd=even
 QED
Proof by induction:
- Show true for n=1
- Assume true for n=k
- Show is true for n=k+1 (do not assume!)
- True for K implies truth for k+1  by induction true for all k
Further Mathematics Revision
SEQUENCES
Convergent: sequence that has a limit
Divergent: sequence with no limit
Periodic sequence: sequence with pattern
Arithmetic sequences: Un=a +(n-1)d  if Un – Un-1 is a constant
Geometric sequences: Un=arn-1  if Un/Un-1 is a constant eg Un+1=4Un ->Un=a4n-1
Difference equations:
General solution:
Particular solution:
n-1
n-1
Geometric: Un=ar (=a4 )
Un=5.4n-1
Arithmetic: Un=a+d (=3n+(n-3)) Un=3n +3
Categorisation of difference equations
Difference equation: recurrence equation
Order: number of previous terms needed to find given term
An+1=(2n +3)an  order 1
Linearity: if each term mentioned is to the power of 1
An+1= n2an – 3an-1  linear
Constant coefficients: if all coefficients are constant
An+1= 3an – 5an-1 +n  constant coefficients
An+1= nan + 3  not constant coefficients
Homogeneity: if equation contains at least 1 previous term
An+1 = 5an – n2√an-1  homogenous; a to the power is not valid.
An+1 = 2an2  not homogenous
Monotonic sequences: an increasing or decreasing sequence;
If Un+1 – Un > 0  increasing sequence
If Un+1 – Un < 0  decreasing sequence
n +1
n +2
n
(n + 1 )2
n(n + 2 ) n 2 + 2n + 1 n 2
=
=
n +1
(n+1)(n+2)
(n+1)(n+2)
2n
1
=
(n+1)(n+2)
n+1, n+2, 1 are positive. Positive/positive is positive.  increasing monotonic
LIMITS: divide by dominant terms + factorise
lim(Un)= L1 lim(Vn)=L2 L1, L2 need to exist
lim(Un + Vn)= L1+ L2
lim(Un x Vn)= L1x L2
p
lim(1/n )= 0
lim(rn)=0 if –1<r<1
n2
+ 1
n 2 + 2n 2n
lim
=
3
3.2 n
=
n2
2
n
+1
3
=
1
3
lim(Un/Vn)= L1/L2
Further Mathematics Revision
COMPLEX NUMBERS
√-8 = (√8)i
De Moivre’s Theorem: (cosθ +isinθ)n= cosnθ +isinnθ
cos(A+B)= cosAcosB – sinAsinB
sin(A+B)= sinAcosB + cosAsinB
cosθ= cos(-θ)
sinθ= -sin(-θ)
Complex polynomials  use formula (–b+-√(b²-4ac))/2a
Real coefficients  answers in conjugate pairs
Examples:
(a) Find (√3 + i)5 using De Moivre  θ= п/6
r=2
z=√3 + i= 2(cosп/6 +isinп/6)
z5=25(cosп/6+isinп/6)5 =32(cos5п/6 +isin5п/6)=32(-√3/2+i/2)=-16√3+16i
(b) Find z5= 3+3i
 θ=п/4
r=√18
z 5 = 18 cos
5
z 5 =(r 1 ) (cos
5
r=
p
p
+ isin
4
4
+ isin ) =
;
z=r 1 (cosq + isin q )
18 cos
4
+ isin
4
2pp p
2pp
p
p
18 =10 18 ; 5 q = ; q =
+
=
+
4
5ґ4
5 20
5
5solutionswhen q =
p 9 p 17 p 25 p 33 p
,
,
,
,
20 20 20 20 20
Exponential form:
eiθ=cosθ +isinθ
z= r(cosθ + isinθ) = reiθ
e2-3i = e²e-3i = e²(cos(-3) + isin(-3))
2=eln2
2i=(eln2)i=eiln2= cosln2 + isinln2
ax+iy=axaiy=ax(cos(ylna) + isin(ylna))
Further Mathematics Revision
VECTOR SPACES
Closure:
- Addition: u+v is in V
- Scalar multiplication: cu is in V, for all u, c
Identity:
- Addition: there exists an element 0 in V such that u+0=u
- Scalar multiplication: 1u=u
Associativity:
- Addition: (u+v)+w=u+(v+w)
- Scalar multiplication: c(du)=(cd)u
Distributivity:
- Distributive 1: c(u+v)=cu+cv
- Distributive 2: (c+d)u=cu+du
Commutativity under addition: u+v=v+u
Inverses under addition: u+ -u=0
Subspaces: subset of a vector space (must contain zero vector)
- Closed under addition and scalar multiplication
- Subspace inherits properties from vector space
Basis: set of vector, when combined can generate all other vectors in V;
Very efficient, e.g. M22 n(S)=4
1
0
0
0 , 1 , 0
Standard basis: e.g. V3= i, j, k:
0
0
1
Spanning Sets: basis, but less efficient
Span: set of all linear combinations
Dimension: number of elements in a basis of that vector space
Summative results: let V be a vector space with dimV=n, then:
- Any linear independent set in V contains at most n vectors
- Any spanning set for V contains at least n vectors
- Any linearly independent set of exactly n vectors in V is a basis for V
- Any spanning set for V of exactly n vectors is a basis for V
- Any linearly independent set in V can be extended to a basis for V
- Any spanning set for V can be reduced to a basis for V
Linear dependent: if there are scalars (c1c2c3…) such that c1v1+c2v2+…=0
-If at least one of its elements can be written as a combination of the others
Linear independent: not linear dependent
Isomorphism of vector spaces: all vectors are isomorphic
Further Mathematics Revision
MATRICES
Matrix: rectangular array of numbers, not commutative under multiplication
A matrix is defined to be the number of rows x number of columns
1 0
0 9
 3 by 2  order of a matrix
8 4
1xn: row matrix
nx1: column matrix nxn: square matrix
Element: each number in the matrix
Adding and subtracting: can only be done with matrices of the same order.
 corresponding elements
Multiplying matrices: associative but not commutative
1 0
1ґ0 + 0ґ2 1ґ1 + 0ґ5 1ґ1 + 0ґ6
0 1 1
0 1 1
0 9 ґ
= 0 ґ 0 + 9 ґ 2 0 ґ 1 + 9 ґ 5 0 ґ 1 + 9 ґ 6 = 18 45 54
2 5 6
8 4
8ґ0 + 4ґ2 8ґ1 + 4ґ5 8ґ1 + 4ґ6
8 28 32
nxm by mxp  nxp, 2x4 by 4x3  2x3
1 0 0
Identity matrix: I, only exist for square matrices.
1 0
, 0 1 0
Multiplication by identity is commutative: IA=AI=A
0 1
0 0 1
Inverse of a matrix:
inverse
p q
r s ®
s
r
q
s
determinant ґ
®
p
r
1
q
s
®
p
ps qr r
q
p
Singular matrix: one that has no inverse the determinant is 0 (ps-qr=0)
M22 forms a vector space under normal matrix addition and scalar multiplication