adding spice to a level maths lessons

Adding Spice to A level Maths Lessons

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 ScoresMathematics A2 point average:

Althon College 2560 points from 10 students: 256 average

Basing College 3600 points from 20 students: 180 average

Advanced FSM point average:



Total Maths point average:



Obtaining a formula for π


212108642

1

11

xxxxxxx


xx

xxxxxxx d1

1d1

2

1

0

121086421

0

212108642

1

11

xxxxxxx


xx

xxxxxxx d1

1d1

2

1

0

121086421

0

212108642

1

11

xxxxxxx

1

011

013

13111

1119

917

715

513

31 tan xxxxxxxx


xx

xxxxxxx d1

1d1

2

1

0

121086421

0

212108642

1

11

xxxxxxx

1

011

013

13111

1119

917

715

513

31 tan xxxxxxxx

π1tan1 411

131

111

91

71

51

31

Rearranging:

134

114

94

74

54

344π

Rearranging:

134

114

94

74

54

344π

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 , . . . . . . .

u 1 = 1 , u 2 = 1

u n + 2 = u n + 1 + u n

A formula for the Fibonacci sequence

1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , . . . . . . .

u 1 = 1 , u 2 = 1

u n + 2 = u n + 1 + u n

nn

nu

2

51

5

1

2

51

5

1

is the Golden ratio.

This was widely used in architecture and art.2

51

A formula for any sequence

e.g. 2 , 4 , 8 , 30 , , . . . . . .


e.g. 2 , 4 , 8 , 30 , , . . . . . .

2)51)(41)(31)(21(

)5)(4)(3)(2(

nnnn

u n


e.g. 2 , 4 , 8 , 30 , , . . . . . .

2)51)(41)(31)(21(

)5)(4)(3)(2(

nnnn

u n 4)52)(42)(32)(12(

)5)(4)(3)(1(

nnnn


e.g. 2 , 4 , 8 , 30 , , . . . . . .

30)54)(34)(24)(14(

)5)(3)(2)(1(8

)53)(43)(23)(13(

)5)(4)(2)(1(

nnnnnnnn

2)51)(41)(31)(21(

)5)(4)(3)(2(

nnnn

u n 4)52)(42)(32)(12(

)5)(4)(3)(1(

nnnn

π)45)(35)(25)(15(

)4)(3)(2)(1(

nnnn

“Student” cancelling

xx

x 72

2

742

“Student” cancelling

works here

xx

x 72

2

742

98

49,

95

19,

65

26,

64

16

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 thenf (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

223361 )7(for results of Table xxy

x –3 –2 –1 0 1 2 3

y –3 –2 –1 0 1 2 3

and its graph.

223361 )7(for results of Table xxy

x –3 –2 –1 0 1 2 3

y –3 –2 –1 0 1 2 3

The graph of y = sin 47x on Autograph,

The graph of y = sin 47x on Autograph,

and on the Texas TI-82.

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.

...!11!9!7!5!3

sin119753

xxxxx

xx

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

xx

xe)e(

d

d

The number e is the number such that

xx

xe)e(

d

d

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

!7!6!5!4!3!2 10

1

10

1

10

1

10

1

10

1

10

1

10

1

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 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

20 996 010 × 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.A

A’

B

B’

C

C’

D

D’

O

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 0 א .

This is the cardinality of the integers.

0 א = 0 א + 0 א

0 א = 0 א × 0 א

but 0 א > 0 א ^ 0 א

The continuum hypothesis states that

1 א = 0 א ^ 0 א but this has not been proved.

Is it possible to draw a line that misses every point with integer coordinates?

Fin

Graham Winter 2007

adding spice to a level maths lessons

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