The intent of this section was to display a collection of strange but true
facts. The problem I encountered is that 90% of the ones found on the internet
are either Urban Legends or just completely made up (usually very funny). I have diveded them into five categories:
my comments are in italics
If you have three quarters, four dimes, and four pennies, you have $1.19. You
also have the largest amount of money in coins without being able to make
change for a dollar.
coincidentally a quarter has 119 ridges on its edge.
111,111,111 x 111,111,111 = 12,345,678,987,654,321
1,741,725 = 1^7+7^7+4^7+1^7+7^7+2^7+5^7
In the New Guinea highlands each number is associated with a body part, so that
the word for "three" might be the same word used to denote the middle
finger. The attachments vary across the thousands of different communities in
the area.
Among the Yupno the little finger of the left hand is one, the right big toe is
20 and the left ear is 21. When only one man is present (women are forbidden to
count in public) numbers stop at 33 for the penis. With more men present it is
possible to go to higher numbers.
Cognitive neuropsychologist Brian Butterworth, of University College London,
promoting his book The Mathematical Brain, citing evidence from diverse fields.
Pi is the number of times a circle's diameter will fit around its circumference
Here is Pi to 100 decimal places: 3.14159265358979323846264338
3279502884197169399375105820974944592
3078164062862089986280348253421170679
Pi occurs in hundreds of equations in many sciences including those describing
the DNA double helix, a rainbow, ripples spreading from where a raindrop fell
into water, superstrings, Einstein's Gravitational Field Equation, normal
distribution, distribution of primes, geometry problems, waves, navigation....
It is easy to prove that if you have a circle that fits exactly inside a
square, then Pi = 4 x (Area of circle) / (Area of square)
If you drew a circle within a square by hand and split it up into a rough grid
and counted the squares you'd prbably come up with an answer of Pi = 3.13...
Not bad for a rough hand drawing, only 0.37% out!
Pi does not have to be written in decimal (base 10) notation (3.14159265....).
Here it is in binary (base 2) notation:
11.001001000011111101101010
1000100010000101101000
110000100011010011
Westerners tend to think of a base-10 number system as "natural", but
many other cultures have come up with a wide range of bases, most famously the
Sumerian base 60, from which we get our hours.
You can do lots more stuff with Pi when it is in binary format - like drawing
weird pictures of it, or even listening to it. As Pi has an infinite number of
places, it is quite possible that any message you liked could be heard
somewhere in Pi. It has even been suggested it contains the VOICE OF GOD. In
Carl Sagan's book 'Contact' the places of Pi (in base 11) are found to contain
a message from the beings that built the universe.
Satan does not appear in Pi too quickly: The first time '666' appears is at
position 2440
Half the circumference of a circle with radius 1 is exactly Pi. The area inside
that circle is also exactly Pi!
It is impossible to 'square the circle'. i.e: You can't draw a square with the
same area as a circle using standard / Euclidean straight-edge and compass
construction in a finite number of steps. The Greeks were obsessed with trying
to do this.
Pi is a 'transcendental' number. This means that it is not the solution to any
finite polynomial (eg: lots of numbers added in a series) with whole number
coefficients. This is why it is impossible to square the circle.
In ancient Greece the symbol
p
was used to denote the number 80
The digits of Pi appear 'random' and yet they describe something essential to
the universe - No pattern emerges and yet it is predestined. The sequence of
digits has so far passed all known tests for randomness. 'Approximate Entropy'
can be used to establish just how random a number is - it turns out that Pi is
more random than the square root of 2, which itself is more random than the
square root of 3.
Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:
0 occurs 599963005 times
1 occurs 600033260 times
2 occurs 599999169 times
3 occurs 600000243 times
4 occurs 599957439 times
5 occurs 600017176 times
6 occurs 600016588 times
7 occurs 600009044 times
8 occurs 599987038 times
9 occurs 600017038 times
This shows NO unusual deviation from expected 'random' behaviour.
If you take 10 million random digits, statistically on average you would expect
200 cases where you get 5 digits in a row the same. If you take 10 million
digits of Pi - guess what... you get exactly 200
There is no zero in the first 31 digits of Pi
The fraction (22 / 7) is a well used approximation of Pi. It is accurate to
0.04025%
Decimal places 7, 22, 113 and 355 of Pi are all the number 2. (22/7 and 355/113
are good approximations of Pi)
Another fraction used as an approximation to Pi is (355 / 113) which is
accurate to 0.00000849%
A more accurate but less easily remembered fraction is (104348 / 33215) as an
approximation to Pi. This is accurate to 0.00000001056%
If you have a uniform grid of parallel lines, unit distance apart and you drop
a needle length k < 1 onto the grid, then the probability the needle falls
across a line is 2k / Pi
The first 39 digits of Pi suffice for any application imaginable. i.e:
Calculations involving circles the size of the Universe will have an accuracy
to the size of a PROTON if Pi is taken to 39 decimal places.
I think it this refers to the visible universe, and you need 47 decimal places.
Pi is irrational. An irrational number is a number that cannot be expressed in
the form (a / b) where a and b are integers.
It is not known if Pi is 'normal'. That is if all numbers appear equally often
forever. For all we know after a Billion Billion decimal places we get a random
series of 1's and 0's (00101011101011000010111010....) No one has proved that
Pi is not normal so people generally assume that it is.
There are MANY irrational numbers very close to integers of the form en0.5
where n is an integer. For example e1630.5 is seriously close to
262,537,412,640,768,744
The probability of 2 large random numbers having no common factor (coprime) is
6/(2)
Indiana State Legislature House Bill No. 246, 1897, set the value of Pi to a to
3. Duh !
Actually the wording of the bill is really unclear and can interpreted in
various ways. Other interpretations include 3.2, 4 and 9.2376. The bill goes on
in a confused way to contradict other areas of elementary geometry, and to
contradict itself. It got passed all the way up to Senate level, then a
Mathematics Professor noticed it by chance, informed the Senate, and it was
postponed indefinitely at its second reading.
Pi was calculated to 2,260,321,363 decimal places in 1991 by the Chudnovsky
brothers in New York
The Babylonians found the first known value for Pi in around 2000BC; they used
(25/8)
The Egyptians used Pi = 3 but improved this to (22 / 7). They also used
(256/81). If you imagine a circle in the Great Pyramid at Giza in Egypt like
this:
Then the circle's circumference is twice the base length of the pyramid, and
the circle's area is equal to the pyramid's vertical sectional area through the
peak. The ratio of the perimeter of the base of the Great Pyramid to its height
is twice Pi. The same ratio for the Pyramid of the Sun in Mexico is four times
Pi. Both are built to an accuracy of a few inches.
The Bible uses a value of Pi of 3. Here is a verse from I Kings 7,23: And he
made a molten sea, ten cubits from one brim to the other
it was round all about, and his height was five cubits
and a line of thirty cubits did compass it about.
In around 200 BC Archimedes found that Pi was between (223 / 71) and (22 / 7).
His error was no more than 0.008227%. He did this by approximating a circle as
a 96 sided polygon.
The D&M Pyramid (a geological feature on Mars) is situated at 40.868 degrees
North, which is exactly equal to arcTan (e/
p
). Many of its internal angles also share trigonometric configurations of e and
Pi.
The first person to use the Greek letter was Welshman William Jones in 1706. He
used it as an abbreviation for the 'periphery' of a circle with unit diameter.
Euler adopted the symbol and it quickly became a standard notation.
The Pi memory champion is Hiroyoki Gotu (21 years old) who memorised an amazing
42,000 digits. !!!
A.C. Aitken from Edimbourg University is able, for the first 2000 decimals of
Pi, to tell you what number appears in position x !!!
There are many formulae for Pi. They are completely arithmetical despite the
fact that Pi arises from Geometry:
Wallis: Pi = (8 / 3) . (4.4.6.6.8.8.10.10.12...) / (3.5.5.7.7.9.9.11.11...)
This is very slow to converge - After 7000 terms it is accurate to 7 decimal
places (averaging terms 7000 and 7001)
Gregory (also attributed to Leibniz): Pi = 4 - (4/3) + (4/5) - (4/7) + (4/9) -
(4/11) + ...
This is about the same - After 7000 terms it is also accurate to 7 decimal
places
Pi = Sqrt ( 12 - (12/22) + (12/32) - (12/42) + ...
This is slightly faster - After just 1100 terms it is accurate to at least 9
decimal places There are many other such formulas
Ramanujan developed a formula for Pi which adds 8 decimal places each term!
This is the formula generally used by supercomputers to calculate Pi
David Bailey, Peter Borwein and Simon Plouffe recently computed the ten
billionth digit in the hexadecimal expansion of Pi. They used an astonishing
formula which enables one to calculate the n-th digit of Pi without being
forced to calculate all the preceding n -1 digits. No one had previously even
conjectured that such a digit-extraction algorithm for Pi was possible
Choose 2 random numbers x and y between 1 and -1. Do this N times. If for M of
those N times x2 + y2 < 1 then as N approaches infinity, Pi = 4(M/N)
The circle is the shape with the least perimeter length to area ratio (for a
given shape area). Centuries ago mathematicians were also philosophers. They
considered the circle to be the 'perfect' shape because of this. The sphere is
the 3D shape with the least surface area to volume ratio (for a given volume)
Most people would say that a circle has no corners - but it is more accurate to
say that it has an infinite number of corners
Pi is approximately 6/5 PHI^2 where PHI is the GOLDEN MEAN - another
interesting number that appears all over nature and in art.
Read all about phi on
www.vashti.net
Pi is of course the ratio of a circle's circumference to its diameter. If we
bring everything up one dimension to get a '3D value for Pi'... The ratio of a
sphere's surface area to the area of the circle seen if you cut the sphere in
half is EXACTLY 4
Euler showed that
p
ei + 1 = 0 Where i is the square-root of -1
Landau showed that
p
/2 is the value of x between 1 and 2 for which cos x vanishes.
After saying (correctly) that Pi / 2 is the value of x between 1 and 2 for
which Cos x vanishes, Edmund Landau was dismissed from his position in 1934 for
teaching in an 'un-German' style.
p
= n sin(180/n). Try it with different values of n. Your calculator will need
to be in degrees (not radians) mode. This equation gets more accurate with
larger values of n
The following are all NEARLY Pi:
101/2
Cube root of 31
666/212
10/
p
(97 + 9/22)1/4
9/5 + (9/5)1/2
(19 (7)1/2) / 16
(2)1/2 + (3)1/2
1.1 x 1.2 x 1.4 x 1.7
(296/167) 2
Kochansky found that Pi is NEARLY a root of the equation 9x4 - 240x2 + 1492
A year is about
p
x 107 seconds
Ludolph Van Ceulen (1540 - 1610) spent most of his life working out Pi to 35
decimal places. Pi is sometimes known as
Ludolph's Constant
It is not known if the following are 'irrational':
p
+ e
p
/ e
ln
p
If you approximate the circle with a radius of 1 as a 100 sided polygon, then
its area is only accurate to 1 decimal place or 0.0658%
At position 762 there are six nines in a row. This is known as the Feynman Point
Pi in base Pi is 10
p
= 4(1/2)!2
All permutations of 3 arbitrary digits appear somewhere in Pi
In 1931 a Cleveland businessman published a book announcing that Pi is exactly
256/81
Starting with the conventional 5-by-5 magic square, and then substituting the
nth digit of pi for each number n in the square, we obtain a new array of
numbers. The sum of the numbers in every column is duplicated by a sum of
numbers in every row.
Write the letters of the English alphabet, in capitals, clockwise around a
circle, and cross-out the letters that have right-left symmetry, A, H, I, M,
etc. The letters that remain group themselves in sets of 3, 1, 4, 1, 6"
At one time it was thought there was an illness attached to trying to 'square a
cirle' called Morbus Cyclometricus.
In the following series of natural numbers, constructed by taking successively
larger strings of digits from the beginning of the decimal expansion of the
number Pi: 3, 31, 314, 31415, 314159, 3141592, etc. The first thousand numbers
of the series include only 4 primes.
The earliest known reference to Pi is on a Middle Kingdom papyrus scroll,
written around 1650 BC by Ahmes the scribe.
The sequence 314159 re-appears in the decimal expansion of Pi at place 176451.
This sequence appears 7 times in the first 10 million places (not including
right at the start)
(
p
+ 20)i is almost equal to -1 (where i is the square-root of minus one)
If you approximate the circle as a square then the value you get for Pi is
about 10% out. It just goes to show that you shouldn't approximate the circle
as a square. Well you wouldn't make square wheels would you?
Here's a Pi limerick:
Three point one four one five nine two
Its been around forever - its not new
It appears everywhere
In here and in there
Its irrational I know but its true !
Here is a mnemonic for remembering 22 decimal places of Pi - Each digit is the
number of letters in the corresponding word:
How I love a black Morrissey in Smiths songs, and Sally Cinnamon overcomes
'Spreads', 'Waterfall', and in the 'Elephant' Reni sounds so Stoney
It refers to two groups The Smiths and The Stone Roses.