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Squares that are bigger  Squares that are smaller 

2^{2} is 4  1/2=0·5 and 0·5^{2} is 0·25=1/4 
3^{2} is 9  1/5=0·2 and 0·2^{2} is 0·04=1/25 
10^{2} is 100  1/10=0·1 and 0·1^{2} is 0·01=1/100 
Here is a mathematical derivation (or proof) of the two values. You can skip over this to the answers at the foot of this paragraph if you like.Phi^{2} = Phi + 1 or, subtracting Phi + 1 from both sides:
Phi^{2} – Phi – 1 = 0We can solve this quadratic equation to find two possible values for Phi as follows:
 First note that (Phi – ^{1}/_{2})^{2} = Phi^{2} – Phi + ^{1}/_{4}
 Using this we can write Phi^{2} – Phi – 1 as (Phi – ^{1}/_{2})^{2} – ^{5}/_{4}
and since Phi^{2} – Phi – 1 = 0 then (Phi – ^{1}/_{2})^{2} must equal ^{5}/_{4} Taking squareroots gives (Phi – ^{1}/_{2}) = +^{√5}/_{2} or –^{√5}/_{2}.
 so Phi = ^{1}/_{2} + ^{√5}/_{2} or ^{1}/_{2} – ^{√5}/_{2}.
The larger value 1·618... we denote using the capital Greek letter Phi written as Φ 
the smaller value –0·618... is written as –φ using the small Greek letter phi 
the large P indicates the larger positive value 1·618...  the small p denotes the smaller positive value 0·618.... 
Φ = (√5 + 1) / 2  φ = (√5 – 1) / 2 
Other names used for these values are the Golden Ratio and
the Golden number. We will use
the two Greek letters Phi
(Φ) and phi (φ)
in these pages,
although some mathematicians
use another Greek letters such as tau (τ) or else
alpha (α) and beta (β).
Here, Phi (large P) is the larger value, 1.618033.... and phi (small p)
is the smaller positive value 0.618033... which is also just Phi – 1.
As a little practice at algebra, use the expressions above to show that φ × Φ = 1.
Here is a summary of what we have found already
that we will find useful in what follows:
Phi phi = 1 Phi  phi = 1 Phi + phi = √5  
Phi = 1.6180339..  phi = 0.6180339.. 
Phi = 1 + phi  phi = Phi – 1 
Phi = 1/phi  phi = 1/Phi 
Phi^{2} = Phi + 1  (–phi)^{2} = –phi + 1 or phi^{2} = 1 – phi 
Phi = (√5 + 1)/2  phi = (√5 – 1)/2 
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It starts from basic definitions called axioms or "postulates" (selfevident starting points). An example is the fifth axiom that
In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call "finding the golden section G point on the line". He describes this geometrically.
< 1 > A G B g 1–gEuclid used this phrase to mean the ratio of the smaller part of this line, GB to the larger part AG (ie the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (i.e. is the same as the ratio AG/AB). We can see that this is indeed the golden section point if we let the line AB have unit length and AG have length g (so that GB is then just 1–g) then the definition means that
GB  =  AG  i.e. using the lengths of the sections  1g  =  g 
AG  AB  g  1 
g =  –1 +5  or g =  –1 – 5  
2 
2 
It seems that this ratio had been of interest to earlier Greek mathematicians, especially Pythagoras (580BC  500BC) and his "school". There is an interesting article on The Golden ratio at the St Andrew's MacTutor History of Mathematics site.
A G B x 1so that AB is now has length 1+x. If Euclid's "division of AB into mean and extreme ratio" still applies to point G, what quadratic equation do you now get for x? What is the value of x?
The ratio of the length of a face of the Great Pyramid
(from centre of the bottom of a face to the apex of the pyramid)
to the distance from the same point
to the exact centre of the
pyramid's base square is about 1·6.
It is a matter of debate whether
this was "intended" to be the golden section number or not.
According to Elmer Robinson (see the
reference below), using the average of eight sets of data,
says that "the theory that the perimeter of the pyramid divided by twice its vertical
height is the value of pi" fits the data much better than the theory above about Phi.
The following references
will explain circumstantial evidence for and against:
There are no extant records of the Greek architects' plans for their most famous temples and buildings (such as the Parthenon). So we do not know if they deliberately used the golden section in their architectural plans. The American mathematician Mark Barr used the Greek letter phi (φ) to represent the golden ratio, using the initial letter of the Greek Phidias who used the golden ratio in his sculptures.
Luca Pacioli (sometimes written as Paccioli), 14451517, wrote a book called De Divina Proportione (The Divine Proportion) in 1509. It contains drawings made by Leonardo da Vinci of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called it the sectio aurea (Latin for the golden section).
Today, some mathematicians use phi (φ) for the golden ratio as on these web pages and others use the Greek letters alpha (α) or tau (τ), the initial letter of tome which is the Greek work for "cut".
A Mathematical History of the Golden Number R HerzFischler, Dover (1998) paperback. This is an informative book, densely packed with historical references to the golden mean and its other names.
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Φ has the value  √5 + 1  and φ is  √5 – 1  . 
2  2 
Here is the decimal value of Φ
to 2000 places grouped in blocks of 5 decimal
digits. The value of φ is identical but begins with 0·6.. instead of
1·6.. .
Read this as ordinary text, in lines across,
so Φ is 1·61803398874...)
Dps: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 50 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 100 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 200 06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 500 13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 1000 10767 38937 64556 06060 59216 58946 67595 51900 40055 59089 50229 53094 23124 82355 21221 24154 44006 47034 05657 34797 66397 23949 49946 58457 88730 39623 09037 50339 93856 21024 23690 25138 68041 45779 95698 12244 57471 78034 17312 64532 20416 39723 21340 44449 48730 23154 17676 89375 21030 68737 88034 41700 93954 40962 79558 98678 72320 95124 26893 55730 97045 09595 68440 17555 19881 92180 20640 52905 51893 49475 92600 73485 22821 01088 19464 45442 22318 89131 92946 89622 00230 14437 70269 92300 78030 85261 18075 45192 88770 50210 96842 49362 71359 25187 60777 88466 58361 50238 91349 33331 22310 53392 32136 24319 26372 89106 70503 39928 22652 63556 20902 97986 42472 75977 25655 08615 48754 35748 26471 81414 51270 00602 38901 62077 73224 49943 53088 99909 50168 03281 12194 32048 19643 87675 86331 47985 71911 39781 53978 07476 15077 22117 50826 94586 39320 45652 09896 98555 67814 10696 83728 84058 74610 33781 05444 39094 36835 83581 38113 11689 93855 57697 54841 49144 53415 09129 54070 05019 47754 86163 07542 26417 29394 68036 73198 05861 83391 83285 99130 39607 20144 55950 44977 92120 76124 78564 59161 60837 05949 87860 06970 18940 98864 00764 43617 09334 17270 91914 33650 13715 2000 
1·10011 11000 11011 10111 10011 01110 01011 11111 01001 01001 11110 00001 01011 11100 11100 11100 11000 00001 10000 00101 100 11001 11011 01110 01000 00110 10000 01000 01000 00100 01001 11011 01011 11110 01110 10001 00111 00100 10100 01111 11000 200 01101 10001 10101 00001 00011 10100 00110 00001 10001 11010 01010 10010 01110 11001 11111 10000 10110 00101 01001 11101 300 00100 11110 11011 11111 00000 01101 00011 10000 01000 10110 11010 11011 11110 00110 00001 00111 11110 00000 01100 01000 400 01101 11100 00100 10010 10000 10000 00001 10000 00000 01011 00000 11101 01100 10010 11101 00100 00001 11100 11001 10101 500 
Neither the decimal form of Phi, nor the binary one nor
any other base have any ultimate repeating pattern in their
digits.
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Here is the connection the other way round, where we can discover the Fibonacci numbers arising from the number Φ.
The graph on the right shows a line whose gradient is Φ, that is the line
Let's call these the Fibonacci points. Notice that the ratio y/x for each Fibonacci point (x,y) gets closer and closer to Phi = 1·618... but the interesting point that we see on this graph is that
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The basic Fibonacci relationship is
F(i+2) = F(i+1) + F(i) The Fibonacci relationship 
The graph shows that the ratio F(i+1)/F(i)
seems to get closer and closer to a particular value, which
for now we will call X.
If we take three neighbouring Fibonacci numbers, F(i), F(i+1) and
F(i+2) then,
for very large values of i, the ratio of F(i) and
F(i+1) will be almost the same as the ratio of
F(i+1) and F(i+2),
so let's see what happens if both of these are the same value: X.

(A) 

= 
 
= 
 
= 
 (B) 
X =  F(i+1)  = 1 +  F(i) 
F(i)  F(i+1) 
X =  F(i+1)  = 1 +  F(i)  = 1 +  1 
F(i)  F(i+1)  X 
 
X^{2} = X + 1 
Remember, this supposed that the ratio of two pairs of neighbours in the Fibonacci series was the same value. This only happens "in the limit" as mathematicians say. So what happens is that, as the series progresses, the ratios get closer and closer to this limiting value, or, in other words, the ratios get closer and closer to Φ the further down the series that we go.
Did you notice that we have not used the two starting values in this proof? No matter what two values we start with, if we apply the Fibonacci relationship to continue the series, the ratio of two terms will (in the limit) always be Phi!
i  ...  –10  –9  –8  –7  –6  –5  –4  –3  –2  –1  0  1  2  3  4  5  6  7  8  9  10  ... 

Fib(i)  ...  –55  34  –21  13  –8  5  –3  2  –1  1  0  1  1  2  3  5  8  13  21  34  55  ... 
1  = –1,  –1  = –0.5,  2  = –0.666..,  –3  = –0.6,  5  = –0.625, ... 
–1  2  –3  5  –8 
Experiment with a spreadsheet (or calculator) and you will soon find that F(n)/F(n2) gets closer and closer to 2.618... as n gets larger. This is Φ+1 and also Φ^{2}.
If is easy to verify this using a small variation on the method above (where we showed F(i)/F(i+1) gets closer to Φ as i gets larger), or using Binet's Formula that we met earlier on the page called A Formula For Fib(n).
Phi = 1 +  1  
1 +  1  
1 +  1  
1 + .. 
This continued fraction has a big surprise in store for us....
Mathematicians call all these fractional (and whole) numbers rational numbers because they are the ratio of two whole numbers and it is these number fractions that we will mean by fraction in this section.
It may seem as if all number can be written as fractions  but this is, in fact, false.
There are numbers which are not
the ratio of any two whole numbers, e.g.
√2 = 1.41421356... , π = 3.14159...,
e = 2.71828... . Such values are called irrational since they cannot be
represented as a ratio of two whole numbers (i.e. a fraction). A simple consequence of this
is that their decimal fraction expansions go on for ever and never repeat at any stage!
Any and every fraction has a decimal fraction expansion that either
Can we write Phi as a fraction?
The answer is "No!" and there is a surprisingly simple proof of this. Here it is. [with thanks to Prof Shigeki Matsumoto of Konan University, Japan]
First we suppose that Φ can be written as a fraction and then show this leads to a contradiction, so we are forced to the conclusion Φ cannot be written as a fraction:
Suppose Φ = ^{a}/_{b} and that this fraction is in its lowest terms which means that:
But we know from the alternative definitions of Φ and φ that
 a and b are whole numbers
 a and b are the smallest whole numbers to represent the fraction for Phi
 Since Φ > 1 then a > b
Φ = ^{1}/_{φ} and φ = Φ – 1 which we put in the fraction for Φ:
Φ = ^{1}/_{(Φ –1)}.
Now we substitute ^{a}/_{b} for Φ:
^{a}/_{b} = ^{1}/(_{a/b –1)} = ^{b}/_{(a – b)}
But here we have another fraction for Φ that has a smaller numerator, b since a > b
which is a contradiction because we said we had chosen a and b were chosen to be the smallest whole numbers.
So we have a logical impossibility if we assume Φ can be written as a proper fraction
and the only possibility that logic allows is that Φ cannot be written as a proper fraction  Φ is irrational.
The answer lies in the continued fraction for Φ that we saw earlier on this page.
If we stop the
continued fraction for Φ at various points,
we get values which approximate to Φ:







The proper mathematical term for these fractions which are formed from stopping a continued fraction for
Φ at various points
is the convergents to Φ.
The series of convergents is
1  ,  2  ,  3  ,  5  ,  8  ,  13  ,  21  , ... 
1  1  2  3  5  8  13 
The convergents start with 1/1 = F(1)/F(0)
where F(n) represents the nth Fibonacci number.
To get from one fraction to the next,
we saw that we just take the reciprocal of the
fraction and add 1:
so the next one after F(1)/F(0) is
1 + 

= 1 + 

= 

But the Fibonacci numbers have the property that two successive numbers add to give the next, so F(1) + F(0) = F(2) and our next fraction can be written as

= 

So starting with the ratio of the first two Fibonacci numbers the next convergent to Φ is the ratio of the next two Fibonacci numbers.
This always happens:
if we have F(n)/F(n – 1) as a convergent to Φ, then the
next convergent is F(n+1)/F(n).
We will get all the ratios of successive Fibonacci numbers as values which get closer and closer to Φ.
You can find out more about continued fractions and how they relate to splitting a rectangle into squares and also to Euclid's algorithm on the Introduction to Continued Fractions page at this site.
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In fact, you can start with many values but not all (for instance 0 or 1 will cause problems) and it will still converge to the same value: Phi.
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Φ has the value  √5 + 1 
2 
We want to find a formula for the numbers (x, say) "that have the same decimal part as their squares".
So, if we subtract x from x^{2},
the result will be a whole number because the
decimal parts were identical.
Let's call this difference N, remembering that it is a whole number.
So
x =  1 ± √ 
2 
N:  1  2  3  4  5  ... 

1+4N:  5  9  13  17  21  ... 
For example: if we choose N = 5, then the number x (that increases by exactly 5 when squared) is
x =  1 ± √  =  1 ± 21  = 2.791287847.. and x^{2} = 7.791287847... = 5 + x 
2  2 
Another example: take Φ, which is
(1 + √5)/2 or
(1 + √(1+4×1))/2
so that N = 1.
Thus we can "predict" that Φ squared will be >1 more than
Φ itself
and, indeed, Phi = 1.618033.. and Phi^{2} = 2.618033.. .
We can do the same for other whole number values for N.
More generally: There is nothing in the maths of this section that prevents N from being any number, for instance 0·5 or π. Suppose N is &pi = 3.1415926535... . We can find the number x that, when squared, increases by exactly π! It is
x =  1 ± √  =  1 ± √12.566370614...  = 2·3416277185... 
2  2 
The Mathematical Magic of the Fibonacci Numbers 
Fibonacci Home Page
This is the first page on this topic. Where to now?
The next page on this Topic is... 
The Golden String 
© 19962016 Dr Ron Knott