Transcript x - NCETM

Adding Spice to A level Maths
Lessons
Graham Winter 2007
5% interest on ¼ d since 1066
1  960 × 1.05
2007 – 1066
= £90 543 898 922 419 141.99
Total GDP for world in 2003
= £25 000 000 000 000
Fold a piece of paper in half.
Then fold it in half again.
And again, fifty times in all.
It now has a thickness of 78 000 000
miles, which is 4/5 of the distance to
the sun – a 7½ year trip on Concorde.
Average Point Scores
Mathematics A2 point average:
Althon College
Basing College
2560 points from 10 students:
3600 points from 20 students:
256 average
180 average
Advanced FSM point average:
Althon College
Basing College
2340 points from 60 students:
1200 points from 40 students:
39 average
30 average
Total Maths point average:
Althon College
Basing College
4900 points from 70 students:
4800 points from 60 students:
70 average
80 average
Obtaining a formula for π
Obtaining a formula for π
1 x  x  x  x  x
2
4
6
8
10
1
 x 
1 x 2
12
Obtaining a formula for π
1 x  x  x  x  x
2

1
0
4
6
8
1 x  x  x  x  x
2
4
6
8
10
10
1
 x 
1 x 2
12
 x  dx 
12

1
0
1
dx
2
1 x
Obtaining a formula for π
1 x  x  x  x  x
2

1
4
6
8
1 x  x  x  x  x
2
4
6
8
10
10
1
 x 
1 x 2
12
 x  dx 
12
0
x 
1
3

1
0
x  x  x  x  x 
3
1
5
5
1
7
7
1
9
9
1
11
11
1
13
1
dx
2
1 x
 
1
x  0  tan
13
1
x

1
0
Obtaining a formula for π
1 x  x  x  x  x
2

1
4
6
8
1 x  x  x  x  x
2
4
6
8
1
 x 
1 x 2
10
10
12
 x  dx 
12
0
x 
1
3

1
0
x  x  x  x  x 
3
5
1
5
1
7
7
1
9
9
1
11
11
1
13
1
dx
2
1 x
 
1
x  0  tan
13
1
1
1         tan 1  14 π
1
3
1
5
1
7
1
9
1
11
1
13
x

1
0
Rearranging:
π  4  43  54  74  94  114  134 
Rearranging:
π  4  43  54  74  94  114  134 
This formula converges very slowly.
A computer performing 10 12 calculations per
second, which began calculating this formula at
the Big Bang 4.4 billion years ago, would have
just established the 29th decimal place.
A graphics calculator can be simply programmed
to calculate  using this formula.
: Clrhome
:4A
:3B
: Repeat 0
: A – 4/B + 4/(B + 2)  A
: Disp A
:B+4B
: End
The calculator would
have to run the
program for 8½
years to establish the
9th decimal place.
 has been calculated to 206 billion decimal
places.
The diameter of the universe is 40 billion light
years.
Hence just 30 decimal places of  are needed
to find the circumference of the universe
correct to the nearest mm.
Let S = 1 + 2 + 4 + 8 + 16 + 32 + 64 + . . .
 S = 1 + 2( 1 + 2 + 4 + 8 + 16 + 32 + . . . )
 S = 1 + 2S
 S – 2S = 1
 –S = 1
 S = –1
To prove 1 = 2
Let







x=y
x 2 = xy
x 2– y 2 = xy – y 2
(x + y)(x – y) = y(x – y)
x+y=y
y+y=y
2y = y
2=1
Solve:
2 cos x sin x = cos x,
0  x < 360
2 cos x sin x = cos x
 2 sin x = 1
 sin x = ½
 x = 30 o or 150 o
A formula for the Fibonacci sequence
1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , . . . . . . .
u1 = 1
,
u2=1
un+2 = un+1 + un
A formula for the Fibonacci sequence
1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , . . . . . . .
u1 = 1
,
u2=1
un+2 = un+1 + un
n
1 1 5 
1 1 5 

 


un 
5  2 
5  2 
n
1 5
2
is the Golden ratio.
This was widely used in architecture and art.
A formula for any sequence
e.g. 2 , 4 , 8 , 30 ,  , . . . . . .
A formula for any sequence
e.g. 2 , 4 , 8 , 30 ,  , . . . . . .
un
(n  2)( n  3)( n  4)( n  5)

2
(1  2)(1  3)(1  4)(1  5)
A formula for any sequence
e.g. 2 , 4 , 8 , 30 ,  , . . . . . .
un
(n  1)( n  3)( n  4)( n  5)
(n  2)( n  3)( n  4)( n  5)
4

2 
(2  1)( 2  3)( 2  4)( 2  5)
(1  2)(1  3)(1  4)(1  5)
A formula for any sequence
e.g. 2 , 4 , 8 , 30 ,  , . . . . . .
un
(n  1)( n  3)( n  4)( n  5)
(n  2)( n  3)( n  4)( n  5)
4

2 
(2  1)( 2  3)( 2  4)( 2  5)
(1  2)(1  3)(1  4)(1  5)
(n  1)( n  2)( n  4)( n  5)
(n  1)( n  2)( n  3)( n  5)

8 
 30
(3  1)(3  2)(3  4)(3  5)
(4  1)( 4  2)( 4  3)( 4  5)
(n  1)( n  2)( n  3)( n  4)

π
(5  1)(5  2)(5  3)(5  4)
“Student” cancelling
4x  7 2  7

2
x
2x
“Student” cancelling
4x  7 2  7

2
x
2x
works here
16
64
,
26
65
19
,
95
,
49
98
Algebraic symbols
Before the 17th century, algebraic manipulation
was very cumbersome.
The following slide is a copy of part of
Cardan’s work on solving cubic equations,
published in 1545, together with a translation.
Note that the translation uses “modern”
symbols e.g. +, not present in the original.
Cardan’s solution of a cubic equation, 1545
Cardan was professor of science at Milan
university. He divided his time equally between
mechanics, astrology and debauchery.
One of his sons was executed for poisoning his
wife, and he cut off the ears of his other in a fit of
rage after some offence had been committed .
He was imprisoned for heresy, became the
astrologer to the Pope, and felt obliged to commit
suicide after predicting the date of his own death.
In his Ars Magna he found a general solution for
cubic equations, introducing negative and
imaginary numbers in the process.
Roman numerals were still used extensively for
accounting until 1600.
One of the first appearances of decimal notation
was in a work by Pitiscus in 1608.
The unknown in an equation was called rei (Latin
for thing) and its square called zensus, so for
example x 2 + 3x – 2 was written Z p 3R m 2 by
Pacioli in 1500.
In 1553 Stifel used AA for A 2.
The German mathematician Jordanus first used
letters for unknowns c. 1200, but there were no
symbols for + or –.
His work Algorithmus was not printed until 1534.
The + and – symbols were first consistently used
by the French mathematician Vieta in 1591.
The × symbol was invented by the English
mathematician William Oughtred in 1631.
The = symbol was invented by the Welsh
mathematician Robert Record in 1557.
RSA Coding and Decoding as a
Function and its Inverse
For RSA coding , two numbers are chosen:
- a product of 2 primes e.g. 1189 = 29  41
- a number coprime to1189 e.g. 3
- The coding function is then
f (x) = x 3 mod 1189
i.e. take the remainder when x 3 is divided by 1189
The inverse function is:
f – 1 (x) = x 187 mod 1189
The number 187 has been calculated using 29
and 41.
It is the number which, when it is multiplied by
3, gives an answer which is exactly one more
than a multiple of the lowest common multiple
of 28 (= 29 – 1) and 40 (= 41 – 1 ).
A 30 tonne lorry travelling at 30 mph collides
with a 1 tonne car travelling at 30 mph.
Let v be the speed of the wreckage
after the collision.


30 × 30 – 1 × 30 = 30v + 1v
870 = 31v
v = 28.1 mph
The value of g is less on the equator (9.76 ms –2) than it
is at the poles (9.86 ms –2 ), due to the greater distance
to the centre of the earth (3963 miles v. 3949 miles) and
also due to the earth’s rotation.
A person is about ½ inch taller when they get up than
when they go to bed.
So to minimize your body mass index, you should
measure your height and weight first thing in the
morning on the equator.
An anorexic should consider taking the measurements
at the Pole just before retiring.
Taking g = 10 may not produce accuracy to 1
significant place.
e.g. v = u + at with u = 5.5 and t = 7
With g = 10, we obtain v = 75.5
or v = 80 (1 s.f.)
With g = 9.8, we obtain v =74.1
or v = 70 (1 s.f.)
“You will be given a surprise test in one of
your lessons next week.”
When the students enter Friday’s lesson, if the
test has not been given, it will not be a surprise
when they get it.
So the surprise test can’t be on Friday.
So when they enter Thursday’s lesson, if the
test has not been given, it will not be a surprise
when they get it.
This sentence is false
This sentence is true
Table of results for y  361 x 3 ( x 2  7) 2
x
–3
–2
–1
0
1
2
3
y
–3
–2
–1
0
1
2
3
Table of results for y  361 x 3 ( x 2  7) 2
x
–3
–2
–1
0
1
2
3
y
–3
–2
–1
0
1
2
3
and its graph.
The graph of y = sin 47x
on Autograph,
The graph of y = sin 47x
on Autograph,
and on the Texas TI-82.
x3
x5
x7
x9
x 11
sin x  x 




 ...
3!
5!
7!
9!
11!
The word sine is from the Latin word sinus for
breast.
This is due to a mistranslation of the Hindu
word for chord-half into Arabic.
Suppose sin A = 3/5 and sin B = 5/13
- then cos A = 4/5 and cos B = 12/13
- and
sin (A + B) = 3/5 × 12/13 + 4/5 × 5/13 = 56/65
cos (A + B) = 4/5 × 12/13 – 3/5 × 5/13 = 33/65
33, 56, 65 is a Pythagorean triplet.
All Pythagorean triplets are of the form
m 2 – n 2 , 2mn , m 2 + n 2 for integers m ,n.
Quintics and higher powered polynomials cannot
generally be solved.
This was proved for quintics by Niels Abel in
1825.
Evariste Galois proved it true for all polynomials
with higher powers, though this wasn’t clear until
rewritten by Camille Jordan in 1870.
Pierre Wantzel resolved a couple of famous
Greek problems in 1837:
- an angle cannot be trisected using only
compasses and a straight edge;
- a cube cannot be doubled using only ruler and
compasses.
That a circle cannot be squared i.e. it is
impossible to construct a square with the same
area as a given circle using only compasses and
a straight edge, followed the proof that  is
transcendental in 1882.
The question arises as to whether such numbers
as e +  , e ×  , e e , e  ,  e etc are
transcendental, and in most cases the answer is
not known.
An exception is e  which was shown to be
transcendental by Alexandr Gelfond in 1934.
It is also known that at least one of e e and e e² is
transcendental.
The number e is the number such that
d x
(e )  e x
dx
The number e is the number such that
d x
(e )  e x
dx
This can be obtained
on a calculator thus:
The coefficients in the
binomial expansion of
(1 + x) 5.
The coefficient of x 6 in the expansion of (1 + x) 49
is 49 C 6 , the number of ways of winning the
jackpot on the National Lottery.
The number of ways of winning the jackpot on
the National Lottery is 13 983 816.
13 983 816 two pence pieces laid end to end
would stretch 220 miles – from London to
Paris.
13 983 816 seconds is 161 days – from 13th
April until 21st September.
A 500 gram Marmite jar comfortably holds
200 two pence pieces.
Were these to fall to the floor, the chances that
they all land showing a head is 1 in 1.6 × 10 60
Which is slightly less likely than the
probability of winning the jackpot on the
National Lottery eight weeks running.
The factorial function gets very big very fast.
60! = 8.3 × 10 81 , which is of the order of the
number of electrons in the observable universe.
The number of permutations of the alphabet is
26! = 4.03 × 10 26 , which is 792 000
permutations for every square millimeter of the
earth’s surface.
The factorial function gets very big very fast.
60! = 8.3 × 10 81 , which is of the order of the
number of electrons in the observable universe.
The number of permutations of the alphabet is
26! = 4.03 × 10 26 , which is 792 000
permutations for every square millimeter of the
earth’s surface.
The first transcendental number discovered was
1
1
1
1
1
1
1







2!
3!
4!
5!
6!
7!
10 10
10
10
10
10
10
From a textbook from 1830.
The discovery of large prime numbers is
often reported in the press,
though the
prime itself is
not always
explicitly
revealed.
Mersenne primes are of the form 2 p – 1,
where p is prime.
The integer part of the log 10 of a whole
number is one less than the number of its
digits.
log 10 2 p = 6 320 429
 p ≈ 6 320 429  log 10 2 = 20 996 010
20 996 010 × log 10
20 996 011 × log 10
20 996 012 × log 10
20 996 013 × log 10
20 996 014 × log 10
2
2
2
2
2
=
=
=
=
=
6 320 428.8
6 320 429.1
6 320 429.4
6 320 429.7
6 320 430.0
20 996 010 × log 10 2 = 6 320 428.8
20 996 011 × log 10 2 = 6 320 429.1
20 996 012 × log 10 2 = 6 320 429.4
20 996 013 × log 10 2 = 6 320 429.7
20 996 014 × log 10 2 = 6 320 430.0
20 996 012 is even
20 996 013 is a multiple of 3
Hence M 20 996 011 = 2 20 996 011 – 1
Suppose 2 20 996 011 – 1 = a × 10 6 320 429
 2 20 996 011 = b × 10 6 320 429 ,
where b ≈ a
 20 996 011 log 10 2 = log 10 b + 6 320 429
 20 996 011 log 10 2 – 6 320 429 = log 10 b
 0.1002909 = log 10 b
 b = 10 0.1002902
 b = 1.25977
M 20 996 011 = 1.25977 × 10 6 320 429
With 3 people, the chance that they all
have different birthdays is 364/365 × 363/365
That is 0.9918
So the probability that two or more of
them share a birthday is 0.0082
The probability that two or more share a
birthday from 23 people is 0.5073
The probability that a passenger on a tube
train is carrying a bomb is 1/1000 000
The probability that two passengers on a
tube train are carrying bombs is
1/1 000 000× 1/1 000 000 = 1/1 000 000 000 000
So to reduce the chances that you are on a
tube train that has a suicide bomber on it,
carry a bomb with you.
In the 4th dimension, the distance d between the
points (w 1 , x 1 , y 1 , z 1) and (w 2 , x 2 , y 2 , z 2) is
given by:
d 2 = (w1 – w2) 2 + (x1 – x2) 2 + (y1 – y2) 2 + (z1 – z2) 2
A 4D hypercube is called a
tesseract, and is bounded by
16 verticies, 32 edges, 24
faces and 8 cubes.
In the 4th dimension, the distance d between the
points (w 1 , x 1 , y 1 , z 1) and (w 2 , x 2 , y 2 , z 2) is
given by:
d 2 = (w1 – w2) 2 + (x1 – x2) 2 + (y1 – y2) 2 + (z1 – z2) 2
A 4D hypercube is called a
tesseract, and is bounded by
16 verticies, 32 edges, 24
faces and 8 cubes.
A tesseract.
A 4D sphere is the set of all points whose distance
from a fixed point is constant.
The volume of a 4D sphere is ½  2 r 4 .
A 5D unit sphere is numerically the largest.
In 4 dimensions, all knots fall apart.
If a left shoe were taken into the 4th dimension, it
could be “turned over and moved” into a right shoe.
Random numbers are used in aeronautics, nuclear
physics and gambling.
In the past cards or dice have been use to generate
them, as well as the middle digit of the areas of the
parishes of England (L.H.C Tippet 1927).
Early computer algorithms for pseudorandom
numbers were not always sayisfactory e.g. Von
Neumann’s middle square method.
Today, the linear congruential random number
generator is commonly used.
A widely used choice of random number
generator is:
un+1 = 16 807 × un (mod 2 31 – 1 )
u 0 = any integer less than 2 31 – 1
The random number displayed on a
calculator screen is then
x = un+1 ÷ (2 31 – 1)
The 142 857 times table:
142 857 × 2 = 285 714
142 857 × 3 = 428 571
142 857 × 4 = 571 428
142 857 × 5 = 714 285
142 857 × 6 = 857 142
142 857 × 7 = 999 999
The reciprocal of 7 is
0. 142 857 142 857 142 . . .
The reciprocal of 17 is
0.058 823 529 411 764 705 882 352 . . .
So the 588 235 294 117 647 times table
behaves in a similar fashion to that of 142857.
This happens when the reciprocal of a prime
has a recurring length one less than the prime.
The set of integers and the set of even
numbers are the same size, since there is a
1 : 1 mapping between them which is
onto.
The set of integers and the set of even
numbers are the same size, since there is a
1 : 1 mapping between them which is
onto.
A finite line and an infinite line have the
same number
of points.
O
A
D
B
A’
B’
C
C’
D’
The Hotel Infinity has infinitely many
rooms.
If it is full, and another guest turns up,
then a room is found for him by asking
every guest to move on one room.
If it is full and infinitely many guests
arrive, each existing guest is asked to
move to a room whose number is twice
their present number.
The smallest infinity is ‫ א‬0.
This is the cardinality of the integers.
‫א‬0 + ‫א‬0 = ‫א‬0
‫א‬0 × ‫א‬0 = ‫א‬0
but
‫א‬0 ^ ‫א‬0 > ‫א‬0
The continuum hypothesis states that
‫ א‬0 ^ ‫ א‬0 = ‫ א‬1 but this has not been proved.
Is it possible to draw a line that misses
every point with integer coordinates?
Fin
Graham Winter 2007