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Heatons Reddish U3A Science Group Mathematical Curiosities

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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 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