Transcript of Heatons Reddish U3A Science Group Mathematical Curiosities.
- Slide 1
- Heatons Reddish U3A Science Group Mathematical Curiosities
- Slide 2
- Mathematics From Greek mthma, "knowledge, study, learning") The
abstract study of topics encompassing Quantity, structure, space
and change and other properties; It has no generally accepted
definition but Aristotles was: The Science of Quantity
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- Mathematics From Greek mthma, "knowledge, study, learning") The
abstract study of topics encompassing : Quantity, structure, space
and change and other properties; It has no generally accepted
definition but Aristotles was: The Science of Quantity
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- Mathematics Quantity, Numbers Arithmetic Structure,
Relationships and Functions Algebra SpaceShape and Form Geometry
ChangeDependency Calculus
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- Numbers Mathematical objects used to count label and measure
Ishango bones 20,000 year old Babylonian Symbols base 60
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- Numbers Egyptians and Romans MMMCCXLIV = 3244 = = 21207 =
MMMCCVII
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- Use of Numbers Counting and measuringCardinal OrderingOrdinal
LabellingIndex (Tag)
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- Real Numbers NegativePositive Irratinalsand zero Fractions
Types of Numbers -5-4-3-2012345
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- Types of Numbers -5-4-3-2012345 5/3 Pi - e Real Numbers
NegativePositive Irrationals Fractions
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- Fun with Numbers: Recurring Decimals Recurring decimals 1/3 =
0.3333333333333......... 1 repeating digit 9/11 =
0.8181818181......... 2 repeating digits 3227/555 = 5.8144144144 3
repeating digits
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- Recurring Decimals Extreme recurring decimals - 7/555 =
5.8144143 repeating digits 1/17 = 0.058823529411764705882352
941176470588235294117647 05882352 94117647................... 16
repeating digits
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- Noreens Numbers
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- By Popular Request
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- Nine curiosities about Nine In base 10 a number is divisible by
nine if and only if its digital root is 9.base 10if and only
ifdigital root i.e add its digits 818+1=9 545+4=9
642516+4+2+5+1=181+8=9 1.
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- Nine curiosities about Nine 12345679 x 9 = 111111111 12345679 x
18 = 222222222 12345679 x 81 = 999999999 Add the missing 8
123456789 x 9 = 1111111101 2.
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- Nine curiosities about Nine a Kaprekar number is a non-negative
integer, the representation of whose square in that base can be
split into two parts that add up to the original number
again.non-negativeinteger e.g.45 45 = 2025 and 20+25 = 45. 9 is a
Kaprekar number because 9 = 81 and 8+1= 9 3.
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- Nine curiosities about Nine Accountants friend Subtracting two
base-10 positive integers that are transpositions of each other
yields a number that is a whole multiple of nine. e.g. 41 - 14 = 27
(2 + 7 = 9) 36957930 - 35967930 = 990000, 4.
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- Nine curiosities about Nine The difference between a base-10
positive integer and the sum of its digits is a whole multiple of
nine. E.g. Sum of digits of 41 = 5, and 41-5 = 36. 3+6 = 9,
divisible by nine Sum of digits of 3596793 is 3+5+9+6+7+9+3 = 42,
3596793-42 = 3596751. 3+5+9+6+7+5+1 = 36. 5.
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- Nine curiosities about Nine Nine is the binary complement of
number sixsix DecimalBinary 60110 91001 6.
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- Nine curiosities about Nine Six recurring nines appear in the
decimal places 762 through 767 of pi. This is known as the Feynman
point.piFeynman point 3.141
5926535897932384626433832795028841971693993751058209749445923078164
06286208998628034825342117067982148086513282306647093844609550582231725359
40812848111745028410270193852110555964462294895493038196442881097566593344
61284756482337867831652712019091456485669234603486104543266482133936072602
49141273724587006606315588174881520920962829254091715364367892590360011330
53054882046652138414695194151160943305727036575959195309218611738193261179
31051185480744623799627495673518857527248912279381830119491298336733624406
56643086021394946395224737190702179860943702770539217176293176752384674818
46766940513200056812714526356082778577134275778960917363717872146844090122
49534301465495853710507922796892589235420199561121290219608640344181598136
2977477130996051870721134 999999
8372978049951059731732816096318595024459
45534690830264252230825334468503526193118817101000313783875288658753320838
14206171776691473035982534904287554687311595628638823537875937519577818577
805321712268066130019278766111959092164201989 7.
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- Nine curiosities about Nine Six recurring nines appear in the
decimal places 762 through 767 of pi. This is known as the Feynman
point.piFeynman point 3.141
5926535897932384626433832795028841971693993751058209749445923078164
06286208998628034825342117067982148086513282306647093844609550582231725359
40812848111745028410270193852110555964462294895493038196442881097566593344
61284756482337867831652712019091456485669234603486104543266482133936072602
49141273724587006606315588174881520920962829254091715364367892590360011330
53054882046652138414695194151160943305727036575959195309218611738193261179
31051185480744623799627495673518857527248912279381830119491298336733624406
56643086021394946395224737190702179860943702770539217176293176752384674818
46766940513200056812714526356082778577134275778960917363717872146844090122
49534301465495853710507922796892589235420199561121290219608640344181598136
2977477130996051870721134 999999
8372978049951059731732816096318595024459
45534690830264252230825334468503526193118817101000313783875288658753320838
14206171776691473035982534904287554687311595628638823537875937519577818577
805321712268066130019278766111959092164201989 7.
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- Nine curiosities about Nine If you divide a number by the
amount of 9s corresponding to its number of digits, the number is
turned into a repeating decimal.repeating decimal e.g. 274/999 =
0.274274274274... 8.
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- Nine curiosities about Nine There are nine circles of Hell in
Dante's Divine Comedy.nine circles of HellDivine Comedy 9.
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- Pi - The Ratio of the circumference of a circle to its Diameter
C= D C= 2 R
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- Pi - Archimedes Polygons of increasing order approximate to a
circle. 3.141873 3.141643
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- Pi - Madhava 14 th century Indian mathematician Leibnitz 17 th
century German mathematician
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- Pi - Dart Board Method Area: Square 4R 2 Circle R 2
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- Relationship between pi and e Worlds most beautiful equation Ee
i Euler's identity Where: e is the base of natural logarithm i is
the square root of -1 is the ratio of the circumference of a
circles to its diameter
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- What is the Mathematical Link?
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- 276 951 438 Magic Squares 15 Magic Number for n x n square
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- Magic Squares
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- 712114 213811 163105 96154 Chautisa Yantra 10 th Century
Panmagic Square Jain Temple Order 4 x 4 Magic Constant 34
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- 712114 213811 163105 96154 Chautisa Yantra 10 th Century
Panmagic Square Jain Temple Order4 x 4 Magic Constant 34
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- 712114 213811 163105 96154 Chautisa Yantra 10 th Century
Panmagic Square Jain Temple Order4 x 4 Magic Constant 34
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- 10 th Century Jain Temple 712114 213811 163105 96154 Chautisa
Yantra
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- 10 th Century Panmagic Square Jain Temple 712114 213811 163105
96154 Chautisa Yantra
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- How to populate a magic square 1 Odd 1 up 1 right
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- How to populate a magic square 1 Odd 1 up 1 right 2
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- How to populate a magic square 1 2 Odd 1 up 1 right 2
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- How to populate a magic square 1 2 Odd 1 up 1 right 3
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- How to populate a magic square 1 3 2 Odd 1 up 1 right 3
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- How to populate a magic square 1 3 42 Odd 1 up 1 right If cell
is full 1 down
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- How to populate a magic square 1 35 42 Odd 1 up 1 right If cell
is full 1 down
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- How to populate a magic square 16 35 42 Odd 1 up 1 right If
cell is full 1 down
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- How to populate a magic square 16 357 42 Odd 1 up 1 right If
cell is full 1 down
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- How to populate a magic square 816 357 42 Odd 1 up 1 right If
cell is full 1 down
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- How to populate a magic square 816 357 492 Odd 1 up 1 right If
cell is full 1 down
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- How to populate a magic square 1234 5678 9101112 13141516 Even
Fill left to right top to bottom But only diagonals
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- How to populate a magic square 115144 12679 810115 133216 Even
(4x4) Fill right to left bottom to top
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- A Geometric Paradox 8 8
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- 3 35 5
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- 5 8 5
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- 5 13 5
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- Geometry in Three Dimensions Strange Things Happen
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- Geometry in Three Dimensions Mobius Ring
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- Geometry in Three Dimensions Mobius Ring 1 Surface 1 Edge
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- Geometry in Three Dimensions Mobius Ring 1 Surface 1 Edge
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- A Stitch Up
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- Numbers in Geometry: The Golden Section ACB AB/AC = BC/AB00
AB/BC = 0.618034
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- Leonardo of Pisa 13 th Century Mathematician 0,1,
2,3,5,8,13,21,34,55,89,144.... Son of Bonnaccio Hence -
Fibbonacci
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- (1)1/1=1.000000(2)1/2=0.500000 (3)2/3=0.666667(4)3/5=0.600000
(5)5/8=0.625000(6)8/13=0.615385 (7)13/21=0.619048(8)21/34=0.617647
(9)34/55=0.618182(10)55/89=0.617978
(11)89/144=0.618056(12)144/233=0.618026 0.618034 Fibonacci Series
0,1, 2,3,5,8,13,21,34,55,89,144....
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- Golden Rectangles
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- Logarithmic Spiral
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- Golden Arcs 360 O x 0.618 = 222.5 O 222.5 O x 0.618 = 137.5
O
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- Fibonacci Flower Classification No. of Petals Flower 3Iris,
Lily 5Buttercup, Columbine, Pink 8Coreopsis,Delphinium 13Cineraria,
Marigold, Ragwort 21Aster, Chicory 34Plantain,Daisy, Pyrethrum
55Daisy, Sunflower 89Daisy, Sunflower 144Sunflower
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- Pineapple Spirals 5 8 13 Clockwise Anti Clockwise
Clockwise
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- Sunflower Spirals 34 55 Clockwise Anti Clockwise
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- Sunflower Spirals From : Mathematics of Life Ian Stewart
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- Infinity A concept describing an unbounded set. The reciprocal
of Zero ?
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- Infinity A concept describing an unbounded set. What is the
reciprocal of Zero ? Only two things are infinite, the universe and
human stupidity, and I'm not sure about the former. Albert Einstein
Albert Einstein
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- Paradoxes of the Infinite Zenos Paradox Achilles Tortoise
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- Achilles Tortoise Paradoxes of the Infinite Zenos Paradox
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- Achilles Tortoise Paradoxes of the Infinite Zenos Paradox
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- Paradoxes of the Infinite Why carpet fitters like stairs ! 6
Units
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- Paradoxes of the Infinite Why carpet fitters like stairs ! 6
Units
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- Paradoxes of the Infinite Why carpet fitters like stairs ! 6
Units
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- Paradoxes of the Infinite Why carpet fitters like stairs ! 6
Units Or is it 4.242641
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- Paradoxes of the Infinite A shape is bounded by a line. e.g.
The length of a line bounding a square of unit area is 1
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- Paradoxes of the Infinite A shape is bounded by a line. e.g.
The length of a line bounding a square of unit area is 1+1+1+1 = 4
But for a finite sized shape: is the boundary always finite?
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- Snowflake Area 1 (unit) + 3/9 + 3/9 x (4/9) + 3/9 x (4/9) 2
Circumference 3 +1 +(4/3) +(4/3) 2
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- Snowflake Area 1 (unit) + 3/9 + 3/9 x (4/9) + 3/9 x (4/9) 2
Circumference 3 +1 +(4/3) +(4/3) 2
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- Snowflake Area 1 (units) + 3/9 + 3/9 x (4/9) + 3/9 x (4/9) 2
Circumference 3 +1 +(4/3) +(4/3) 2
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- Snowflake Area 1 (units) + 3/9 + 3/9 x (4/9) + 3/9 x (4/9) 2
Circumference 3 +1 +(4/3) +(4/3) 2
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- Snowflake Circumference 3 +1 +(4/3) +(4/3) 2 Area 1 (units) +
3/9 + 3/9 x (4/9) + 3/9 x (4/9) 2
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- Snowflake
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- The Mathematics of Hair Combing A Problem of Topology
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- A Bald Surface
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- A Hairy Surface
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- A Combed Surface
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- A Smooth Disk
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- A Hairy Disk
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- A Combed Disk
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- A Smooth Ball
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- The Hairy Ball Problem
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- A Failed Attempt to Comb a Hairy Ball Two Tufts
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- A Torus
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- A Combed Torus
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- Monty Hall Problem 123 How to Win A Car
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- Monty Hall Problem 123
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- 123 ?
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- 123 Dont Switch: P(W) = 1/3
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- Monty Hall Problem 123 Dont Switch: P(W) = 1/3 P(L) = 2/3
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- Monty Hall Problem 1.Initial pick wrong 2.Other wrong door is
opened 3.Switching gets the prize 123 Dont Switch: P(W) = 1/3 P(L)
= 2/3 Always Switch:
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- Monty Hall Problem 123 Dont Switch: P(W) = 1/3 P(L) = 2/3
Always Switch: P(W) = 2/3 P(L) = 1/3 1.Initial pick wrong 2.Other
wrong door is opened 3.Switching gets the prize
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- Monty Hall Problem 123 Dont Switch: P(W) = 1/3 P(L) = 2/3
Always Switch: P(W) = 2/3 P(L) = 1/3 1.Initial pick wrong 2.Other
wrong door is opened 3.Switching gets the prize 1.Initial pick
right 2.Either wrong door is opened 3.Switching gets the other
wrong door and loses
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- Monty Hall Problem 123 1/3 2/3
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- Mathematical Curiosities Further Reading of