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Heatons Reddish U3A
Science Group
Mathematical
Curiosities
Mathematics
From Greek
μάθημα máthēma, "knowledge, study, learning")
The abstract study of topics encompassing
Quantity, structure, space and change and other
properties;
It has no generally accepted definition but Aristotle’s
was:
“The Science of Quantity”
Mathematics
From Greek
μάθημα máthēma, "knowledge, study, learning")
The abstract study of topics encompassing :
Quantity, structure, space and change and other
properties;
It has no generally accepted definition but Aristotle’s
was:
“The Science of Quantity”
Mathematics
Quantity,
Structure,
Space
Change
Numbers
Arithmetic
Relationships and Functions
Algebra
Shape and Form
Geometry
Dependency
Calculus
Numbers
Mathematical objects used to count
label and measure
Ishango bones
20,000 year old
Babylonian Symbols –
base 60
Numbers
Egyptians and Romans
= 3244 =
MMMCCXLIV
= 21207 =
MMMCCVII
Use of Numbers
• Counting and measuring
• Ordering
• Labelling
Cardinal
Ordinal
Index (Tag)
Types of Numbers
Real Numbers
Negative
Positive
Irratinalsand zero
-5 -4 -3 -2 -1 0
1
Fractions
2
3
4
5
Types of Numbers
Real Numbers
Negative
Irrationals
-5 -4 -3 -2 -1 0
Positive
Pi - p
e
1
2
3
5/3
Fractions
4
5
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
Recurring Decimals
• Extreme recurring decimals 7/555 = 5.8144143 repeating digits
1/17 = 0.058823529411764705882352
941176470588235294117647
05882352 94117647...................
16 repeating digits
Noreen’s Numbers
Noreen’s Numbers
Noreen’s Numbers
Noreen’s Numbers
By Popular Request
玖
Nine curiosities about Nine
1.
In base 10 a number is divisible by nine if and
only if its digital root is 9.
i.e add its digits
81
54
64251
8+1=9
5+4=9
6+4+2+5+1=18
1+8=9
Nine curiosities about Nine
2.
12345679 x 9 = 111111111
12345679 x 18 = 222222222
12345679 x 81 = 999999999
Add the missing 8
123456789 x 9 = 1111111101
Nine curiosities about Nine
3.
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.
e.g. 45
45² = 2025 and 20+25 = 45.
9 is a Kaprekar number because
9² = 81 and 8+1= 9
Nine curiosities about Nine
4.
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,
Nine curiosities about Nine
5.
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 .
Nine curiosities about Nine
6.
Nine is the binary complement of number six
Decimal
6
9
Binary
0110
1001
Nine curiosities about Nine
7.
Six recurring nines appear in the decimal places 762
through 767 of pi. This is known as the Feynman
point.
3.1415926535897932384626433832795028841971693993751058209749445923078164
06286208998628034825342117067982148086513282306647093844609550582231725359
40812848111745028410270193852110555964462294895493038196442881097566593344
61284756482337867831652712019091456485669234603486104543266482133936072602
49141273724587006606315588174881520920962829254091715364367892590360011330
53054882046652138414695194151160943305727036575959195309218611738193261179
31051185480744623799627495673518857527248912279381830119491298336733624406
56643086021394946395224737190702179860943702770539217176293176752384674818
46766940513200056812714526356082778577134275778960917363717872146844090122
49534301465495853710507922796892589235420199561121290219608640344181598136
29774771309960518707211349999998372978049951059731732816096318595024459
45534690830264252230825334468503526193118817101000313783875288658753320838
14206171776691473035982534904287554687311595628638823537875937519577818577
805321712268066130019278766111959092164201989
Nine curiosities about Nine
7.
Six recurring nines appear in the decimal places 762
through 767 of pi. This is known as the Feynman
point.
3.1415926535897932384626433832795028841971693993751058209749445923078164
06286208998628034825342117067982148086513282306647093844609550582231725359
40812848111745028410270193852110555964462294895493038196442881097566593344
61284756482337867831652712019091456485669234603486104543266482133936072602
49141273724587006606315588174881520920962829254091715364367892590360011330
53054882046652138414695194151160943305727036575959195309218611738193261179
31051185480744623799627495673518857527248912279381830119491298336733624406
56643086021394946395224737190702179860943702770539217176293176752384674818
46766940513200056812714526356082778577134275778960917363717872146844090122
49534301465495853710507922796892589235420199561121290219608640344181598136
29774771309960518707211349999998372978049951059731732816096318595024459
45534690830264252230825334468503526193118817101000313783875288658753320838
14206171776691473035982534904287554687311595628638823537875937519577818577
805321712268066130019278766111959092164201989
Nine curiosities about Nine
8.
If you divide a number by the amount of 9s
corresponding to its number of digits, the
number is turned into a repeating decimal.
e.g. 274/999 = 0.274274274274...
Nine curiosities about Nine
9.
There are nine circles of Hell in Dante's Divine
Comedy.
Pi The Ratio of the
circumference of a
circle to it’s Diameter
C= pD
C= 2pR
p
Pi • Archimedes
• Polygons of increasing order
approximate to a circle.
• p= 3.141873
• p= 3.141643
p
Pi -
p
• Madhava
• Leibnitz
• 14th century Indian
mathematician
• 17th century German
mathematician
Pi -
p
• Dart Board Method
Area:
Square
Circle
4R2
pR 2
Relationship between pi and e
World’s most beautiful equation
•Ee
ip
= -1
Euler's identity
Where:
e
i
π
is the base of natural
logarithm
is the square root of -1
is the ratio of the
circumference of a circles to
its diameter
What is the Mathematical Link?
Magic Squares
Magic Number
for n x n
square
2
7
6
15
9
5
1
15
4
3
8
15
15 15 15 15 15
Magic Squares
Chautisa Yantra
10th Century
Jain Temple
Panmagic Square
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
Order
4x4
Magic
Constant
34
Chautisa Yantra
10th Century
Jain Temple
Order
Magic Constant
Panmagic Square
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
4x4
34
Chautisa Yantra
10th Century
Jain Temple
Order
Magic Constant
Panmagic Square
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
4x4
34
Chautisa Yantra
10th Century
Jain Temple
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
Chautisa Yantra
10th Century
Jain Temple
Panmagic Square
7
12
1
14
2
13
8
11
16
3
10
5
9
6
15
4
How to populate a magic square
Odd
1 up
1 right
1
How to populate a magic square
2
Odd
1 up
1 right
1
How to populate a magic square
2
Odd
1 up
1 right
1
2
How to populate a magic square
Odd
1 up
1 right
1
3
2
How to populate a magic square
Odd
1 up
1 right
1
3
3
2
How to populate a magic square
Odd
1 up
1 right
If cell is full
1 down
1
3
4
2
How to populate a magic square
Odd
1 up
1 right
If cell is full
1 down
1
3
4
5
2
How to populate a magic square
Odd
1 up
1 right
If cell is full
1 down
1
3
4
6
5
2
How to populate a magic square
Odd
1 up
1 right
If cell is full
1 down
3
4
1
6
5
7
2
How to populate a magic square
Odd
1 up
1 right
8
1
6
3
5
7
If cell is full
1 down
4
2
How to populate a magic square
Odd
1 up
1 right
8
1
6
3
5
7
If cell is full
1 down
4
9
2
How to populate a magic square
Even
Fill left to right
top to bottom
But only
diagonals
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16
How to populate a magic square
Even (4x4)
1
Fill right to left
bottom to top
12
8
13
15 14
6
7
10 11
3
2
4
9
5
16
A Geometric Paradox
8
8
A Geometric Paradox
3
5
5
3
A Geometric Paradox
5
5
8
A Geometric Paradox
5
5
13
Geometry in Three Dimensions
Strange Things Happen
Geometry in Three Dimensions
Mobius Ring
Geometry in Three Dimensions
Mobius Ring
1 Surface – 1 Edge
Geometry in Three Dimensions
Mobius Ring
1 Surface – 1 Edge
A Stitch Up
Numbers in Geometry:
The Golden Section
A
B
AB/AC = BC/AB00
AB/BC = 0.618034
C
Leonardo of Pisa
13th Century
Mathematician
0,
1,
1,
2,
3,
Son of Bonnaccio
Hence - Fibbonacci
5,
8,
13, 21, 34, 55, 89,
144
....
Fibonacci Series
0,
(1)
(3)
(5)
(7)
(9)
(11)
1,
1,
2,
3,
5,
1/1= 1.000000
2/3= 0.666667
5/8=
13/21=
34/55=
89/144=
0.625000
0.619048
0.618182
0.618056

0.618034
8,
13, 21, 34, 55, 89,
(2)
(4)
(6)
(8)
(10)
(12)
144
....
1/2= 0.500000
3/5= 0.600000
8/13=
21/34=
55/89=
144/233=
0.615385
0.617647
0.617978
0.618026

0.618034
Golden Rectangles
Logarithmic Spiral
Golden Arcs
360O x 0.618 = 222.5O
222.5O x 0.618 = 137.5O
Fibonacci Flower Classification
No. of
Petals
3
5
8
13
21
34
55
89
144
Flower
Iris, Lily
Buttercup, Columbine, Pink
Coreopsis,Delphinium
Cineraria, Marigold, Ragwort
Aster, Chicory
Plantain,Daisy, Pyrethrum
Daisy, Sunflower
Daisy , Sunflower
Sunflower
Pineapple Spirals
5
8
13
Clockwise Anti Clockwise Clockwise
Sunflower Spirals
34
55
Clockwise Anti Clockwise
Sunflower Spirals
From :
Mathematics of Life
Ian Stewart
Infinity
A concept describing
an unbounded set.
The reciprocal of Zero ?
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
Paradoxes of the Infinite
Zeno’s Paradox
Achilles
Tortoise
Paradoxes of the Infinite
Zeno’s Paradox
Achilles
Tortoise
Paradoxes of the Infinite
Zeno’s Paradox
Achilles
Tortoise
Paradoxes of the Infinite
Why carpet fitters like
stairs !
6 Units
Paradoxes of the Infinite
Why carpet fitters like
stairs !
6 Units
Paradoxes of the Infinite
Why carpet fitters like
stairs !
6 Units
Paradoxes of the Infinite
Why carpet fitters like
stairs !
6 Units
Or is it
4.242641
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
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?
Snowflake
Area
Circumference
1 (unit)
3
+ 3/9
+1
+ 3/9 x (4/9)
+(4/3)
+ 3/9 x (4/9)2
+(4/3)2




Snowflake
Area
Circumference
1 (unit)
3
+ 3/9
+1
+ 3/9 x (4/9)
+(4/3)
+ 3/9 x (4/9)2
+(4/3)2




Snowflake
Area
Circumference
1 (units)
3
+ 3/9
+1
+ 3/9 x (4/9)
+(4/3)
+ 3/9 x (4/9)2
+(4/3)2




Snowflake
Area
Circumference
1 (units)
3
+ 3/9
+1
+ 3/9 x (4/9)
+(4/3)
+ 3/9 x (4/9)2
+(4/3)2




Snowflake
Area
Circumference
1 (units)
3
+ 3/9
+1
+ 3/9 x (4/9)
+(4/3)
+ 3/9 x (4/9)2
+(4/3)2





-(n1)
A = 1  3   4 9
n =0
n

C = 1  3   4 / 3
n =0
n
Snowflake
The Mathematics of Hair Combing
A Problem of Topology
A Bald Surface
A Hairy Surface
A Combed Surface
A Smooth Disk
A Hairy Disk
A Combed Disk
A Smooth Ball
The Hairy Ball Problem
A Failed Attempt to Comb
a Hairy Ball
Two Tufts
A Torus
A Combed Torus
A Combed Torus
Monty Hall Problem
1
2
3
How to Win A Car
Monty Hall Problem
1
2
3
Monty Hall Problem
1
?
2
3
Monty Hall Problem
1
Don’t Switch:
P(W) = 1/3
2
3
Monty Hall Problem
1
Don’t Switch:
P(W) = 1/3
P(L) = 2/3
2
3
Monty Hall Problem
1.Initial pick
wrong
2.Other wrong
door is opened
3.Switching gets
the prize
1
Don’t Switch:
P(W) = 1/3
P(L) = 2/3
2
3
Always Switch:
Monty Hall Problem
1.Initial pick
wrong
2.Other wrong
door is opened
3.Switching gets
the prize
1
Don’t Switch:
P(W) = 1/3
P(L) = 2/3
2
3
Always Switch:
P(W) = 2/3
P(L) = 1/3
Monty Hall Problem
1.Initial pick right
2.Either wrong
door is opened
3.Switching gets
the other
wrong door
and loses
1.Initial pick
wrong
2.Other wrong
door is opened
3.Switching gets
the prize
1
Don’t Switch:
P(W) = 1/3
P(L) = 2/3
2
3
Always Switch:
P(W) = 2/3
P(L) = 1/3
P(L) = 1/3
Monty Hall Problem
1
1/3
2
3
2/3
•of
Mathematical Curiosities
Further Reading