Adding Spice to A level Maths Lessons

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Adding Spice to A level Maths Lessons
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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

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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.

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Average Point Scores

• Mathematics 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
• Althon College 2340 points from 60 students 39
  average
• Basing College 1200 points from 40 students
  30 average
• Total Maths point average
• Althon College 4900 points from 70 students 70
  average
• Basing College 4800 points from 60 students
  80 average

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Obtaining a formula for p
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Obtaining a formula for p
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Obtaining a formula for p
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Obtaining a formula for p
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Obtaining a formula for p
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Rearranging
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Rearranging
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.
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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.

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? 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.

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• 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

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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

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Solve 2 cos x sin x cos x, 0 ?
x lt 360

• 2 cos x sin x cos x
• ? 2 sin x 1
• ? sin x ½
• ? x 30 o or 150 o

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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

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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

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• is the Golden ratio.
• This was widely used in architecture and art.

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A formula for any sequence

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

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A formula for any sequence

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

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A formula for any sequence

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

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A formula for any sequence

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

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Student cancelling
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Student cancelling

• works here

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Algebraic symbols

• Before the 17th century, algebraic manipulation
  was very cumbersome.
• The following slide is a copy of part of
  Cardans 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.

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Cardans solution of a cubic equation, 1545
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• 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.

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• 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.

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• 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.

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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

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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 ).

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• 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
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• 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
  earths 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.

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• 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.)

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• You will be given a surprise test in one of
  your lessons next week.
• When the students enter Fridays lesson, if the
  test has not been given, it will not be a
  surprise when they get it.
• So the surprise test cant be on Friday.
• So when they enter Thursdays lesson, if the
  test has not been given, it will not be a
  surprise when they get it.

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• This sentence is false
• This sentence is true

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(No Transcript)
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• and its graph.

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The graph of y sin 47x on Autograph,
41
The graph of y sin 47x on Autograph,

• and on the Texas TI-82.

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• 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.

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• 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.

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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
  wasnt clear until rewritten by Camille Jordan in
  1870.

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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.

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• 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.

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• The number e is the number such that

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• The number e is the number such that

This can be obtained on a calculator thus
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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.

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• 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.

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• 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.

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• 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
  earths surface.

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• 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
  earths surface.
• The first transcendental number discovered was

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From a textbook from 1830.
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The discovery of large prime numbers is often
reported in the press,
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• though the prime itself is not always explicitly
  revealed.

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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.

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• 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.

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• 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.

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• 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.

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• 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
  Neumanns middle square method.
• Today, the linear congruential random number
  generator is commonly used.

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• A widely used choice of random number generator
  is
• un1 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 un1 (2 31 1)

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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

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• 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.

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• 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.

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• 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.

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• 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.

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• The smallest infinity is ? 0.
• This is the cardinality of the integers.
• ? 0 ? 0 ? 0
• ? 0 ? 0 ? 0
• but ? 0 ? 0 gt ? 0
• The continuum hypothesis states that
• ? 0 ? 0 ? 1 but this has not been proved.

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

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Fin

• Graham Winter 2007