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
Slide 3
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
Slide 4
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
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.
Slide 24
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.
Slide 25
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.
Slide 27
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
Slide 36
712114 213811 163105 96154 Chautisa Yantra 10 th Century
Panmagic Square Jain Temple Order4 x 4 Magic Constant 34
Slide 37
712114 213811 163105 96154 Chautisa Yantra 10 th Century
Panmagic Square Jain Temple Order4 x 4 Magic Constant 34
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
Slide 86
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
Slide 87
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
Slide 89
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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Slide 103
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
Slide 110
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