the Further Mathematics network
www.fmnetwork.org.uk
1
the Further Mathematics network
www.fmnetwork.org.uk
Big Ideas in Core
Mathematics
Let Maths take you Further…
2
The big ideas
„
„
„
„
„
„
Proof
Algebra: linear algebra, abstract algebra
Numerical Methods
Vectors
Limits and infinite series
Differential Equations
3
Aims of session
„
„
„
„
„
To take the time to reflect on why each of
these topics is important
To find ways to motivate students to study
each of these topics
To consider where these topics are found in
the real world
To consider where these topics lead in higher
level mathematics
To look at lots of nice mathematics
4
the Further Mathematics network
www.fmnetwork.org.uk
Idea 1: Proof
Let Maths take you Further…
5
Thoughts about proof
Pictures, diagrams and examples may make a
statement plausible but only a proof makes it
a theorem, students need some memorable
lessons of this.
Sometimes when we teach proof we like to
teach interesting mathematical facts rather
than good examples illustrating proof itself.
At very high levels, the author of a theorem has
typically convinced one or two other experts
that it is true!
6
Why is proof important
„
„
„
„
We need to know that things will always
work.
So that we can build new mathematics from
things that we are certain are true.
Google needs to know this in a very real
sense, it uses high level theorems about
matrices which ensure that its programs
won’t crash.
Encryption via the RSA algorithm requires
theorems from number theory. We need to
know they will work every time.
7
Useful Examples:
Goldbach’s Conjecture
„
„
„
„
Every even number can be written as the
sum of two prime numbers
One of the Clay Mathematics Institute
Millennium Prizes $1,000,000.
It’s ok for as many even numbers as
humankind has ever checked but will it be ok
forever?
Thinking about proof strategies.
8
Langton’s Ant
Langton's ant travels around in a grid of black or
white squares. If she exits a square, its colour
inverts. If she enters a black square, she turns
right, and if she enters a white square, she turns
left.
9
Langton’s Ant
If she starts out moving right on a white grid, for
example, here is how things go:
10
Choosing examples to do with
students
„
Existence of infinitely many primes
‰
‰
„
this can confuse students because it needs
concept of infinity
most proofs also need uniqueness of factorisation
of integers into products of primes
Square root of two is irrational
‰
‰
This can confuse students because it is proof by
contradiction
Notion of a number being irrational is quite
sophisticated too.
11
First examples
Conjecture: Suppose n is an integer larger than 1 and
n is prime. Then 2n – 1 is prime
n
Is n prime? 2n – 1
Is 2n – 1
prime?
2
Yes
3
Yes
3
Yes
7
Yes
4
No
15
No
5
Yes
31
Yes
6
No
63
No
7
Yes
127
Yes
8
No
255
No
9
No
511
No
10
No
1023
No
12
First examples
Conjecture: Suppose n is an integer larger than 1 and
n is prime. Then 2n – 1 is prime
n
Is n prime? 2n – 1
Is 2n – 1
prime?
2
Yes
3
Yes
3
Yes
7
Yes
4
No
15
No
5
Yes
31
Yes
6
No
63
No
7
Yes
127
Yes
8
No
255
No
9
No
511
No
10
No
1023
No
11
Yes
2047
No (=23x89)
13
Useful examples
Theorem: Suppose n is an integer larger than
1 and n is not prime. Then 2n – 1 is not prime.
14
Useful examples
Theorem: Suppose n is an integer larger than
1 and n is not prime. Then 2n – 1 is not prime.
Proof. Since n is not prime n = ab where a, b
are positive integers with a < n and b < n. Let
x = 2b and let y = 1 + 2b + 22b + …. 2(a-b)b.
Then xy = 2ab – 1 = 2n – 1 . QED
Still some loose ends - why are x and y both
less than 2n – 1.
15
Useful examples
Theorem For every positive integer n, there is
a sequence of n consecutive positive integers
containing no primes.
16
Useful examples
Theorem For every positive integer n, there is
a sequence of n consecutive positive integers
containing no primes.
Proof
Consider the list
(n + 1)! + 2, (n + 1)! + 3, (n + 1)! + (n + 1).
This is a list of n consecutive positive integers,
none of which are prime.
17
Useful examples
Theorem For n ≥ 3, if n distinct points on a
circle are connected in consecutive order with
straight lines, then the interior angles of the
resulting polygon add up to (n – 2)180°
18
Proof
„
n = 3, need to show angle sum is 180.
Obvious.
19
Proof
„
n = 4, need to show angle sum is 360.
20
Proof
„
n = 5, need to show the angle sum is 540
21
Strange tilings!
Imagine I am given a square checkerboard whose
number of squares in each row or column is a
power of two.
So, for example:
2 by 2
4 by 4
22
Strange tilings!
And
8 by 8
23
Strange tilings!
And the list goes on forever:
16 by 16, 32 by 32, 64 by 64, 128 by 128, …, 4096 by
4096, …
For any of these infinitely many boards I am going to
show how if precisely one square is removed at
random from such a board, you can always tile what
remains using the below tiles!
24
Strange tilings!
With a 2 by 2 board this is easy: whichever square is
removed you are left with 3 squares which form an Lshape!
25
Strange tilings!
With a 2 by 2 board this is easy: whichever square is
removed you are left with 3 squares which form an Lshape!
26
Strange tilings!
With a 2 by 2 board this is easy: whichever square is
removed you are left with 3 squares which form an Lshape!
I’m now going to show you how it’s done with an 4 by 4
board.
27
Strange tilings!
28
Strange tilings!
Imagine a square has
been removed at
random.
I have marked this in
blue.
29
Strange tilings!
The trick is to break the
board up into four 2
by 2 boards…
30
Strange tilings!
The trick is to break the
board up into four 2
by 2 boards…
as indicated by the red
lines (notice that 2 is
one power of two
down from 4).
Three of the 4 by 4
boards do not have a
square removed.
31
Strange tilings!
Place a tile which
covers the middle
corner square of
each of the 4 by 4
boards which have
not yet had a square
removed. Now let’s
look at 8 by 8
boards….
32
Strange tilings!
Imagine a square has
been removed at
random.
I have marked this in
grey.
33
Strange tilings!
The trick is to break the
board up into four 4
by 4 boards…
34
Strange tilings!
Place a tile which
covers the middle
corner square of
each of the 4 by 4
boards which have
not yet had a square
removed.
We now just have the
problem of dealing
with 4 by 4 boards
each with one square
removed!
35
Strange tilings!
But we know how to do
4 x 4 boards…etc
etc.
36
the Further Mathematics network
www.fmnetwork.org.uk
Idea 2 : Algebra
Let Maths take you Further…
37
The need for algebra
„
„
Sometimes we don’t know what a quantity is
but we do know things about that quantity or
properties that it should have. In these
situations we need algebra.
This takes many people back to problems
such as
“Henry is five times as old as his son but in
four years time he will only be three times as
old, how old is Henry?”
38
What Google does..
Google has to do three things
a) Crawl the web to find all the publicly
accessible web pages
b) Arrange the data it finds so that it can be
searched quickly
c) Rank the pages in order of importance,
so that the most important can be
presented to the user first.
39
Linear Algebra
Web
page
2
Web
page
1
Web
page
Webpage
Linked to
from
1
3
2
1,3
3
2
2
3
3
1
40
Linear Algebra
y
2
z
x
1
3
41
Linear Algebra
y
2
y
x
z
2
z
x
1
3
z
2
42
Linear Algebra
y
2
y
x
z
2
z
x
1
3
z
2
43
Linear Algebra
Site
2
3
1
Votes Received
1
0.5z
2
x + 0.5z
3
y
The clever bit is that each sites voting power is
determined by the votes it receives.
This gives us some equations…….
44
Linear Algebra
2
3
1
x=
0.5z
y=
x + 0.5z
z=
y
45
Linear Algebra
2
x = 0.5z
3
1
y = x + 0.5z
z= y
Algebra!
46
the Further Mathematics network
www.fmnetwork.org.uk
Idea 3: Numerical Methods
Let Maths take you Further…
47
Numerical Methods
„
„
„
Equations are sometimes not very nice in the
real world.
Approximation is the best that we can do.
Numerical Methods are very important in our
computer-driven world where the trade off
between speed and accuracy demands
careful consideration.
48
Linear Algebra
2
The ranks x, y, z are a solution of
3
1
⎛ 0 0 0.5 ⎞⎛ x ⎞ ⎛ x ⎞
⎜
⎟⎜ ⎟ ⎜ ⎟
1
0
0.5
⎜
⎟⎜ y ⎟ = ⎜ y ⎟
⎜ 0 1 0 ⎟⎜ z ⎟ ⎜ z ⎟
⎝
⎠⎝ ⎠ ⎝ ⎠
How does Google calculate x, y and z so
quickly? We need to find a quick way to
calculate this point, and hence the ranks.
49
Linear Algebra
⎛ 0 0 0.5 ⎞
⎜
⎟
Let T = ⎜ 1 0 0.5 ⎟
⎜0 1 0 ⎟
⎝
⎠
2
3
1
⎛1/ 3 ⎞
⎜
⎟
Starting with the point x = ⎜1/ 3 ⎟
⎜1/ 3 ⎟
⎝
⎠
We look at the sequence of
points Tx , T2x, T3x, T4x,…..
The limit of this sequence
gives the ranks….
50
The Google Equation v.2
Google needs to make sure that this process always converges and
make the convergence as rapid as possible.
They came up with this equation x = (1 – δ)s + δTx
⎛ 1/ 3 ⎞
Where s = ⎜ 1/ 3 ⎟ is the rank source and δ is the damping factor.
⎜
⎟
⎜ 1/ 3 ⎟ Notice that when δ = 1 this gives x = Tx
⎝
⎠
δ = 0.95
δ = 0.85
51
Linear Algebra
4
2
3
1
52
Linear Algebra
4
2
3
Site
1
2
3
4
1
0
0
1
0
2
1
0
1
1
3
0
1
0
0
4
0
1
0
0
1
⎛ 0 0 0.5 0 ⎞
⎜
⎟
1
0
0.5
1
⎟ Gives ranks 1/9, 4/9, 2/9, 2/9
Let T = ⎜
⎜ 0 0.5 0 0 ⎟
⎜
⎟
⎝ 0 0.5 0 0 ⎠
53
Linear Algebra
4
2
5
3
1
Site
1
2
3
4
5
1
0
0
1
0
0
2
1
0
1
0
0
3
0
1
0
0
0
4
0
1
0
0
0
5
0
1
0
0
0
0
0.5 0 0 ⎞
⎛0
⎜
⎟
0
0.5 0 0 ⎟
⎜1
Let T = ⎜ 0 0.333 0 0 0 ⎟ Gives ranks 0, 0, 0, 0, 0. The problem
⎜
⎟ of dangling nodes!
0
0.333
0
0
0
⎜
⎟
⎜ 0 0.333 0 0 0 ⎟
⎝
⎠
54
Linear Algebra
4
2
5
3
Site
1
2
3
4
5
1
0
0
1
0
0
2
1
0
1
1
1
3
0
1
0
0
0
4
0
1
0
0
0
5
0 1
0
0
0
0
0
0.5
0
0
⎛
⎞
⎜
⎟
1
0
0.5
1
1
⎜
⎟
Let T = ⎜ 0 0.333 0 0 0 ⎟ Gives ranks 0.08 0.46, 0.15, 0.15, 0.15
⎜
⎟
⎜ 0 0.333 0 0 0 ⎟
⎜ 0 0.333 0 0 0 ⎟
⎝
⎠
1
55
Role of proof
„
It is a theorem that any matrix like the ones
that Google meets with non-zero determinant
will have a fixed point. This is a non-trivial.
The fact that mathematicians have proved it
means that Google knows its programme
won’t crash.
56
57
Social Networking
„
„
„
A lot of social interaction today captured
through e-mail and instant messaging
People are flocking to social networking sites,
where they volunteer information about
themselves and their social networks.
These new services promise to help
individuals utilize their social networks to find
the social and business contacts they seek.
58
59
Social Networking
Mathematics is used to
„ Identify groups with common interests
„ Identify the most influential nodes (this is
similar to page ranking with google)
„ Identify possibilities to extend groups with
common interests.
How this data is can be used is still a
subject of some debate.
60
Company Organisation
61
Abstract Algebra
„
„
Just as elementary algebra uses letters to
symbolise numbers and study their
properties, the theory of groups, rings, fields,
vector spaces carry the abstraction
considerably further.
The point of isolating abstract structures like
groups is simply that once a theorem is
proved about an abstract group it applies to
any and all of the disparate examples of
groups.
62
Abstract Algebra
„
The 100 Prisoner Problem provides some
insight into the value of looking at more
abstract structures and developing language
and notation for dealing with them.
63
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
2)
3)
4)
3)
4)
4)
64
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
8. (1
9. (1
10. (1
2)
3)
4)
3)
4)
4)
2 )( 3 4 )
3)( 2 4 )
4 )( 2 3)
65
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
8. (1
9. (1
10. (1
11. (1
12. (1
13. (1 4 2 )
14. (1 2 4 )
2)
3)
4)
3)
4)
4)
2 )( 3 4 )
3)( 2 4 )
4 )( 2 3)
3 2)
2 3)
66
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
8. (1
9. (1
10. (1
11. (1
12. (1
13. (1
14. (1
15. (1
16. (1
17. ( 2
18. ( 2
2)
3)
4)
3)
4)
4)
2 )( 3 4 )
3)( 2 4 )
4 )( 2 3)
3 2)
2 3)
4
2
4
3
4
3
2)
4)
3)
4)
3)
4)
67
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
8. (1
9. (1
10. (1
11. (1
12. (1
13. (1
14. (1
2)
15. (1
3)
4)
16. (1
3)
17. ( 2
4)
18. ( 2
19. (1
4)
2 )( 3 4 ) 20. (1
3)( 2 4 ) 21. (1
4 )( 2 3)
3 2)
2 3)
4
2
4
3
4
3
4
3
4
2)
4)
3)
4)
3)
4)
3 2)
4 2)
2 3)
68
Using this strategy for the
four prisoner problem
Let’s count how many ways to shuffle four numbers!
1 2 3 4
1.id
2. (1
3. (1
4. (1
5. ( 2
6. ( 2
7. ( 3
8. (1
9. (1
10. (1
11. (1
12. (1
2)
3)
4)
3)
4)
4)
2 )( 3
3)( 2
4 )( 2
3 2)
2 3)
13. (1
14. (1
15. (1
16. (1
17. ( 2
18. ( 2
19. (1
4 ) 20. (1
4 ) 21. (1
3)22. (1
23. (1
24. (1
4
2
4
3
4
3
4
3
4
2
3
2
2)
4)
3)
4)
3)
4)
3
4
2
4
2
3
2)
2)
3)
3)
4)
4)
69
Different numbers of prisoners…
Numbers of
prisoners?
2
4
6
8
10
20
500
1000
1500
2000
Proportion of time the prisoners
survive (2d.p.)
Using our
Playing
strategy
randomly
50%
41.67%
38.33%
36.55%
35.44%
33.12%
30.79%
30.74%
30.72%
30.71%
25%
6.25%
1.56%
0.39%
0.10%
0.00%
0.00%
0.00%
0.00%
0.00%
70
the Further Mathematics network
www.fmnetwork.org.uk
Idea 3: Vectors
Let Maths take you Further…
71
Thoughts on vectors
„
„
Vectors are useful for describing directions
using numbers. This is really important in the
modern computer age.
Doing multi-dimensional problems.
Sometimes mathematicians like to talk about
doing problems in 27 dimensions as if they
can picture this. Really all they mean is that
they have to handle 27 variables (which
doesn’t sound anywhere near as exciting).
72
Lighting the scene - world without lighting
73
World with lighting
74
Use of vectors in lighting
What is a vector?
⎛5⎞
a=⎜ ⎟
⎝1⎠
⎛2⎞
⎜ ⎟
b = ⎜ −3 ⎟
⎜ −1 ⎟
⎝ ⎠
75
…we can calculate the angle
between two vectors….
⎛
θ = cos −1 ⎜
⎜⎜
⎝
⎛2⎞
⎜ ⎟
b=⎜3⎟
⎜ -1⎟
⎝ ⎠
⎞
⎟
(12 + 22 + 42 ) × ( 22 + 32 + (−1)2 ) ⎟⎟⎠
(1× 2) + (2 × 3) + (4 × −1)
θ
⎛ 4⎞
⎜ ⎟
a = ⎜1⎟
⎜ 2⎟
⎝ ⎠
This uses a technique
called the scalar
product
76
Use of vectors in lighting
Angle between
normal vector
and light
source
determines
how light the
surface should
appear
77
Use of vectors in lighting
78
Use of vectors in lighting
79
Use of vectors in lighting
80
Use of vectors in lighting
81
Lighting a Plane
82
Problems in higher dimensions
Arts
A
B
6
3
Sport
8
5
Learning
5
8
Socialising
6
3
⎛ 6 ⎞ ⎛ 3⎞
⎜ ⎟⎜ ⎟
⎜ 8 ⎟ . ⎜ 5 ⎟ = 18 + 40 + 40 + 18 = 116
⎜ 5⎟ ⎜8⎟
⎜ ⎟⎜ ⎟
⎝ 6 ⎠ ⎝ 3⎠
⎛6⎞
⎜ ⎟
⎜ 8 ⎟ = 36 + 64 + 25 + 36 = 161,
⎜5⎟
⎜ ⎟
⎝6⎠
Angle
between
them is
27.99°
⎛ 3⎞
⎜ ⎟
⎜ 5 ⎟ = 9 + 25 + 64 + 9 = 107
⎜8⎟
⎜ ⎟
⎝ 3⎠
83
the Further Mathematics network
www.fmnetwork.org.uk
Idea 4: Limits and Infinite
Series
Let Maths take you Further…
84
Limits and Infinite Series
„
„
„
„
„
Limits enable us to deal properly with the maths of
very small and very large quantities
This means that we can make sense of things like
instantaneous rates of change and the exact sum of
a continuously changing quantiy.
Calculus can be used to turn difficult functions into
things that are easier to deal with (e.g. linear,
quadratic,… approximations)
This leads to infinite series
Infinite series are really how difficult (i.e. nonpolynomial) functions are dealt with computationally
85
Limits
„
„
I sometimes have doubts about teaching
differentiation by first principles in the A-level
course, primary consequence of this
sometimes seems to be a drastic shrinkage
in the pool of maths, engineering and science
students.
Ignorance of definitions did not hamper the
work of Newton and Leibniz in this area.
86
For example
„
„
„
Consider the equation x3+x2 = 10.
Suppose x1 is an approximate solution
Then x1 + ε is the exact solution, so…
87
Formal Definition
Let’s look at the kind of formal definition
established by Cauchy, Dedekind and
Weierstrass.
f is differentiable at c if there is a real number M such that
for all ε > 0 there exists δ > 0 such that
if 0< x − c < δ then
f ( x ) − f (c )
−M <ε
x−c
88
Limits and infinite series
„
Rolle’s Theorem. The theory of limits and real
numbers is sophisticated enough for us to
prove the existence of a point like c below
89
Limits and Series
„
Mean Value Theorem. This can be
manipulated to give the first of a series of
Taylor approximations.
90
Linear Approximations
Underneath the
image is a
triangle/polygon
mesh
91
Series
„
„
Finding derivatives and integrals, solving
differential equations and working with
complex numbers are all significantly
simplified if we’re dealing with functions
which are represented by power series.
It’s difficult to overestimate the importance of
series to mathematical analysis.
92
Questions to ask students
„
Do you think your calculator stores all the
value of sin(x), cos(x), tan(x), ln(x), ex for all
the different values of x you might input?
93
Power Series
x3 x5 x 7 x9
sin x = x − + − + − ...
3! 5! 7! 9!
A complicated mathematical function like
sine has been reduced to the two most
fundamental mathematical operations of
addition and multiplication
94
Some amazing series
⎛ 1 1 1 1
⎞
π = 4 ⎜1 − + − + − .... ⎟
⎝ 3 5 7 9
⎠
This is a particular case of Gregory’s
formula discovered in 1671.
95
Some amazing series
426880 10005
π
∞
= ∑ (−1) n
n=0
(6n)!(13591409 + 545140134n)
(n !)3 (3n)!(8 ×100100025 × 327843840)
The above is due to Ramanujan.
Just taking the first term gives π correct to 12
decimal places.
Taking the first two terms gives π correct to 40
decimal places!
96
the Further Mathematics network
www.fmnetwork.org.uk
Idea 5: Differential Equations
Let Maths take you Further…
97
Why do we need differential
equations?
„
„
„
„
Differential Equations are at the core of
calculus.
Traditionally the key to understanding the
physical sciences
Also one of the most essential practical tools
engineers, economists and others have for
dealing with rates of change.
E.g. is C is the cost of producing X widgets
then dC/dX is the marginal cost of producing
the Xth widget
98
Differential Equations
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The idea in solving an algebraic equations is
to determine from conditions on a number
what that number might be.
The idea in solving a differential equation is
to determine from conditions on the derivative
(and higher derivatives) of a changing
quantity what that quantity might be at any
given time.
99
Real life applications
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Inflation is rising but not as rapidly as it was
last month (first derivative is positive, second
derivative is negative).
In Economics you find differential equations
relating the values, rate of change, rate of
rate of change of various economic
indicators.
100
Differential Equations
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Sometimes we don’t know how a quantity
changes with respect to time but with respect
to some other quantity
Many situations can only be described by
collections of interrelated differential
equations
The techniques to deal with these problems
over the last 300 years are among the top
achievements in mathematics
101
Differential Equations
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Newton’s laws of motion
Laplace’s heat and wave equations
Maxwell’s electromagnetic theory
Navier Stokes Equations for fluid dynamics
Volterra’s predator prey systems
The emphasis now is more on numerical
approximation and computing and less on
traditional methods involving limits and
infinite processes.
102
Useful classroom demonstration
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Students don’t seem to necessarily
understand what they are doing when they
solve a differential equation.
103
Ways to introduce ‘e’
104
Ways to introduce ‘e’
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Invest £1 at a rate of 100% interest per
annum.
If the interest is given once at the end of the
year then you get £2 back at the end of the
year.
If you get 50% interest after sixth months and
then another 50% at the end of the year you
get back 1x1.5x1.5 = £2.25
105
Ways to introduce ‘e’
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If you get 25% every three months you get
back 1 x 1.25 x 1.25 x 1.25 x 1.25 = £2.44
With 12.5% every 1.5 months you get
1.1258 = £2.57
Does this sequence keep getting bigger and
bigger? Is it bounded? If not, prove it. If it is,
does it tend to a limit?
106
‘e’
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This shows us e’s key role in banking and
compound interest calculations.
‘e’ makes lots of implausible appearances in
problems.
e is ubiquitous in mathematical formula,
theorems and their proofs. It is intimately
involved with trigonometric functions,
geometrical figures, differential equations,
infinite series.
107