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Let us try a=7 and b=3, so a+b=10:
	7/3 = 2.333…, but 10/7 = 1.429…, so that won’t work

Let us try a=6 and b=4, so a+b=10:
	6/4 = 1.5, but 10/6 = 1.666…, closer but not there yet!

Let us try a=6.18 and b=3.82, so a+b=10:
	6.18/3.82 = 1.6178…, and 10/6.18 = 1.6181…, getting very close!

In fact the value is:

1.61803398874989484820… (keeps going, without any pattern)

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 would take a long time to get there, 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.

Beauty

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn’t it?

Do you think it is the “most pleasing rectangle”? Maybe you do or don’t, that is up to you!

Many buildings and works of art have the Golden Ratio in them,

such as the Parthenon in Greece.

but it is not known if it was designed that way.

Fibonacci Sequence

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…
…	…	…

This also works if you pick two random whole numbers to begin the sequence, such as 192 and 16 (you would get 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 …

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.