Pi

Draw a circle with a radius of 1.
The distance half way around the edge of the circle
will be 3.14159265... a number known as Pi

Or you could draw a circle with a diameter of 1.
Then the circumference (the distance all the way
around the edge of the circle) will be Pi

Pi (the symbol is the Greek letter π) is:

The ratio of the Circumference
to the Diameter
of a Circle.

In other words, if you measure the circumference, and then divide by the diameter of the circle you get the number π
It is approximately equal to:

3.14159265358979323846…

The digits go on and on with no pattern. In fact, π has been calculated to over one trillion decimal places and still there is no pattern.

Example: You walk around a circle which has a diameter of 100m, how far have you walked?

Distance walked = Circumference = π × 100m = 314.159...m

= 314m (to the nearest m)

Approximation

A quick and easy approximation for π is 22/7

22/7 = 3.1428571...

But as you can see, 22/7 is not exactly right. In fact π is not equal to the ratio of any two numbers, which makes it an irrational number.
A better approximation (but stll not exact) is:

355/113 = 3.1415929...
(think "113355", then divide the "355" by the "113")

Remembering

I usually just remember "3.14159", but you can also count the letters of:

"May I have a large container of butter today"
3 1 4 1 5 9 2 6 5

Here is π with the first 100 decimal places:
3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679...

e

The number e is a famous irrational number, and is one of the most important numbers in mathematics.
The first few digits are:

2.7182818284590452353602874713527 (and more ...)

It is often called Euler's number after Leonhard Euler

e is the base of the Natural Logarithms (invented by John Napier). On the other hand Common Logarithms have 10 as their base.
e is found in many interesting areas, so it is worth learning about.

Calculating

There are many ways of calculating the value of e, but none of them ever give an exact answer, because e is irrational (not the ratio of two integers). But it is known to over 1 trillion digits of accuracy!
For example, the value of (1 + 1/n)ⁿ approaches e as n gets bigger and bigger:

n	(1 + 1/n)ⁿ
1	2.00000
2	2.25000
5	2.48832
10	2.59374
100	2.70481
1,000	2.71692
10,000	2.71815
100,000	2.71827

The value of e is also equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

(Note: "!" means factorial)

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556

Remembering

To remember the value of e (to 10 places) just remember this saying (count the letters!):

To
express
e
remember
to
memorize
a
sentence
to
simplify
this

Or you can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

2.7 1828 1828

And following THAT is the angles in a Right-Angled Isosceles (two equal angles) Triangle of 45°, 90°, 45°:

2.7 1828 1828 45 90 45

Golden Ratio

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618
It appears many times in geometry, art, architecture and other areas.

The Idea Behind It

If you divide a line into two parts so that:

the longer part divided by the smaller part

is also equal to

the whole length divided by the longer part

then you will have the golden ratio.

Have a try yourself (use the slider):

Beauty

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?
Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!

Many buildings and artworks have the Golden Ratio in them,
such as the Parthenon in Greece,
but it is not really known if it was designed that way.

The Actual Value

The Golden Ratio is equal to:

1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational Number, and I will tell you more about it later.

Calculating It

You can calculate it yourself by starting with any number and following these steps:

A) divide 1 by your number (=1/number)
B) add 1
C) that is your new number, start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with 2 and got this:

Number	1/Number	Add 1
2	1/2=0.5	0.5+1=1.5
1.5	1/1.5 = 0.666...	0.666... + 1 = 1.666...
1.666...	1/1.666... = 0.6	0.6 + 1 = 1.6
1.6	1/1.6 = 0.625	0.625 + 1 = 1.625
1.625	1/1.625 = 0.6154...	0.6154... + 1 = 1.6154...
1.6154...

It is getting closer and closer!
But it takes a long time to get even close, however there are better ways and it can be calculated to thousands of decimal places quite quickly.

Drawing It

Here is one way to draw a rectangle with the Golden Ratio:

Draw a square (of size "1")
Place a dot half way along one side
Draw a line from that point to an opposite corner (it will be √5/2 in length)
Turn that line so that it runs along the square's side

Then you can extend the square to be a rectangle with the Golden Ratio.

The Formula

Looking at the rectangle we just drew, you can see that there is a simple formula for it. If one side is 1, the other side will be:

The square root of 5 is approximately 2.236068, so The Golden Ratio is approximately (1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the two numbers before it.)

And here is a surprise: if you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio.
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A	B	B/A
2	3	1.5
3	5	1.666666666...
5	8	1.6
8	13	1.625
...	...	...
144	233	1.618055556...
233	377	1.618025751...
...	...	...

You don't even have to start with 2 and 3, here I chose 192 and 16 (and got the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

A	B	B / A
192	16	0.08333333...
16	208	13
208	224	1.07692308...
224	432	1.92857143...
...	...	...
7408	11984	1.61771058...
11984	19392	1.61815754...
...	...	...

The Most Irrational ...

I believe the Golden Ratio is the most irrational number. Here is why ...

One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this:

	(In numbers: 1.61803... = 1 + 1/1.61803...)

That can be expanded into this fraction that goes on for ever (called a "continued fraction"):

So, it neatly slips in between simple fractions.

Whereas many other irrational numbers are reasonably close to rational numbers (for example Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

Other Names

The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.

Feigenbaum Constant

The Feigenbaum constant

is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function

(1)

and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter

is increased for fixed

. The plot above is made by iterating equation (1) with

several hundred times for a series of discrete but closely spaced values of

, discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.
FeigenbaumConstantIteration

A similar plot that more directly shows the cycle may be constructed by plotting

as a function of

. The plot above (Trott, pers. comm.) shows the resulting curves for

, 2, and 4.
Let

be the point at which a period

-cycle appears, and denote the converged value by

. Assuming geometric convergence, the difference between this value and

is denoted

(2)

where

is a constant and

is a constant now known as the Feigenbaum constant. Solving for

gives

(3)

(Rasband 1990, p. 23; Briggs 1991). An additional constant

, defined as the separation of adjacent elements of period doubled attractors from one double to the next, has a value

(4)

where

is the value of the nearest cycle element to 0 in the

cycle (Rasband 1990, p. 37; Briggs 1991).
For equation (1) with

, the onsets of bifurcations occur at

, 1.25, 1.368099, 1.39405, 1.399631, ..., giving convergents to

for

, 2, 3, ... of 4.23374, 4.5515, 4.64617, ....
For the logistic map,

			(5)
			(6)
			(7)
			(8)

(Sloane's A006890, A098587, and A006891; Broadhurst 1999; Wolfram 2002, p. 920), where

is known as the Feigenbaum constant and

is the associated "reduction parameter."
Briggs (1991) calculated

to 84 digits, Briggs (1997) to 576 decimal places (of which 344 were correct), and Broadhurst (1999) to 1018 decimal places. It is not known if the Feigenbaum constant

is algebraic, or if it can be expressed in terms of other mathematical constants (Borwein and Bailey 2003, p. 53).
Briggs (1991) calculated

to 107 digits, Briggs (1997) to 576 decimal places (of which 346 were correct), and Broadhurst (1999) to 1018 decimal places.
Amazingly, the Feigenbaum constant

and associated reduction parameter

are "universal" for all one-dimensional maps

has a single locally quadratic maximum. This was conjecture by Feigenbaum, and demonstrated rigorously by Lanford (1982) for the case

, and by Epstein (1985) for all

.
More specifically, the Feigenbaum constant is universal for one-dimensional maps if the Schwarzian derivative

(9)

is negative in the bounded interval (Tabor 1989, p. 220). Examples of maps which are universal include the Hénon map, logistic map, Lorenz attractor, Navier-Stokes truncations, and sine map

. The value of the Feigenbaum constant can be computed explicitly using functional group renormalization theory. The universal constant also occurs in phase transitions in physics.
The value of