
Using "place value" a number is formed from a small number of symbols, 0, 1, 2, etc. up to one less than the "base". In the base 10 number system, which is our standard system, the symbols range from 0 to 9. To write "10", we use the 1 again, and move it one place to the left, so it has a "place value" of 10.
The smallest base that is in ordinary use is 2. In this system, called the "binary" system, the only symbols that are used are 0 and 1. The place value increases as you move left not by powers of 10, but by powers of 2. So 10_{2} = 2, 100_{2} = 4, 1000_{2} = 8, etc.
The rules of addition in other bases are similar to those for base 10, but you have to "think in the base" you are using. That means, for example, if you are using base 7, you have to remember that 4+5=12, so you write the "2" and carry the "1".
Though very foreign to our ordinary way of thinking, there's nothing in theory that prevents a number base from being a number other than an integer, or even irrational.
The golden ratio, represented by the Greek letter, φ (phi), is equal to (sqrt(5)+1)/2. This number makes a very interesting base because of this special property:
φ^{2} = φ + 1, which means 100_{φ} = 011_{φ} and 0200_{φ} = 1001_{φ}. These two rules can be applied in either direction against any sequence of consecutive digits in a baseφ number to put it in "standard form", which means no digits other than 0 or 1, and no two 1's in a row. In addition, the repeating decimal 0.101010... is equal to 1.
This relation can be applied anywhere in a baseφ number to handle most "carries" when adding. So, for example,
111.111
+111.111

222.222
is the result after doing the addition, but the "2" digits in the result aren't considered "normal form", so they must be reorganized into zeros and ones by "carrying" the digits to the left. Unlike integer base arithmetic, in which digits need to be carried only to the next column, in baseφ, we have to carry two nonzero digits two places to the left using the identity 100_{φ} = 011_{φ}. Here's how that looks:
111.111
+111.111

222.222, and then process the "carries" this way:
222.311
223.201
232.101
321.101
1211.101
10111.101
Does this always work? No, we have a problem if we encounter an "02" pattern. One way to handle it is to carry one digit to the left, and one digit two places to the right, so 02.00 becomes 10.01. Then we might have another problem if the "rightbound carry" lands on a digit that's already one, because we might have to carry a digit two more places to the right. Here's an example where this comes up:
10100
+10101

20201, and then process the "carries" this way:
21002
21010.01
110010.01
In standard form, an integer has only 1's and 0's, and no two 1's are consecutive.
In addition, the repeating decimal, 0101010... should be eliminated by rewriting it as 1. To understand why 1 = φ^{1}+φ^{3}+φ^{5}+..., read this:
let s = φ^{1}+φ^{3}+φ^{5}+...
then φ^{2}s = φ+φ^{1}+φ^{3}+φ^{5}+...
so (φ^{2}1)s = φ, and since φ^{2}1 = φ, it follows that s=1.
The "greedy algorithm" for converting a positive integer, n, starts with the highest power of φ that doesn't exceed n, writing a "1" in that place, and continuing down until the number, base φ, is exactly equal to n. Note that using the greedy algorithm, there will never be two consecutive 1's, because the greedy algorithm would have chosen a 1 in the place to the left of the two 1's.
1. To convert an integer x to a baseφ number, note that x = (x + 0φ).
2. Subtract the highest power of φ, which is still smaller than the number we have, to get our new number, and record a "1" in the appropriate place in the resulting baseφ number.
3. Unless our number is 0, go to step 2.
4. Finished.
The "least greedy" algorithm is strikingly similar to the greedy algorithm. In this algorithm, a 1 isn't placed in the number unless absolutely necessary. It's only necessary to put a 1 into the number if 011111... would not be large enough. The number 011111... is φ times 100000..., so this is the criterion for deciding whether to write a 1. The least greedy algorithm works this way:
1. To convert an integer x to a baseφ number, using the least greedy algorithm,
2. Find the highest power of φ, which is still smaller than the number we have, and subtract the nexthigher power of φ to get out new number, and record a "1" in the appropriate place in the resulting baseφ number.
3. Unless our number is 0, go to step 2.
4. Finished.This algorithm, as written, will never finish, because all integers coded with the "least greedy" algorithm end in a never ending sequence of 101010...
n  n, base φ (greedy algorithm)  A105424  n, base φ (least greedy algorithm)  A118240 
1  1  0.101010... 
2  10.01  1.101010... 
3  100.01  10.11101010... 
4  101.01  11.11101010... 
5  1000.1001  101.11101010... 
6  1010.0001  111.01101010... 
7  10000.0001  1010.1011101010... 
8  10001.0001  1011.1011101010... 
9  10010.0101  1101.1011101010... 
10  10100.0101  1110.1111101010... 
11  10101.0101  1111.1111101010... 
12  100000.101001  10101.1111101010... 
13  100010.001001  10111.0111101010... 
14  100100.001001  11010.1101101010... 
15  100101.001001  11011.1101101010... 
16  101000.100001  11101.1101101010... 
17  101010.000001  11111.0101101010... 
18  1000000.000001  101010.101011101010... 
19  1000001.000001  101011.101011101010... 
20  1000010.010001  101101.101011101010... 
21  1000100.010001  101110.111011101010... 
22  1000101.010001  101111.111011101010... 
23  1001000.100101  110101.111011101010... 
24  1001010.000101  110111.011011101010... 
25  1010000.000101  111010.101111101010... 
26  1010001.000101  111011.101111101010... 
27  1010010.010101  111101.101111101010... 
28  1010100.010101  111110.111111101010... 
29  1010101.010101  111111.111111101010... 
30  10000000.10101001  1010101.111111101010... 
31  10000010.00101001  1010111.011111101010... 
32  10000100.00101001  1011010.110111101010... 
33  10000101.00101001  1011011.110111101010... 
34  10001000.10001001  1011101.110111101010... 
35  10001010.00001001  1011111.010111101010... 
36  10010000.00001001  1101010.101101101010... 
37  10010001.00001001  1101011.101101101010... 
38  10010010.01001001  1101101.101101101010... 
39  10010100.01001001  1101110.111101101010... 
40  10010101.01001001  1101111.111101101010... 
41  10100000.10100001  1110101.111101101010... 
42  10100010.00100001  1110111.011101101010... 
43  10100100.00100001  1111010.110101101010... 
44  10100101.00100001  1111011.110101101010... 
45  10101000.10000001  1111101.110101101010... 
46  10101010.00000001  1111111.010101101010... 
47  100000000.00000001  10101010.10101011101010... 
48  100000001.00000001  10101011.10101011101010... 
49  100000010.01000001  10101101.10101011101010... 
50  100000100.01000001  10101110.11101011101010... 
51  100000101.01000001  10101111.11101011101010... 
52  100001000.10010001  10110101.11101011101010... 
53  100001010.00010001  10110111.01101011101010... 
54  100010000.00010001  10111010.10111011101010... 
55  100010001.00010001  10111011.10111011101010... 
56  100010010.01010001  10111101.10111011101010... 
57  100010100.01010001  10111110.11111011101010... 
58  100010101.01010001  10111111.11111011101010... 
59  100100000.10100101  11010101.11111011101010... 
60  100100010.00100101  11010111.01111011101010... 
61  100100100.00100101  11011010.11011011101010... 
62  100100101.00100101  11011011.11011011101010... 
63  100101000.10000101  11011101.11011011101010... 
64  100101010.00000101  11011111.01011011101010... 
65  101000000.00000101  11101010.10101111101010... 
66  101000001.00000101  11101011.10101111101010... 
67  101000010.01000101  11101101.10101111101010... 
68  101000100.01000101  11101110.11101111101010... 
69  101000101.01000101  11101111.11101111101010... 
70  101001000.10010101  11110101.11101111101010... 
71  101001010.00010101  11110111.01101111101010... 
72  101010000.00010101  11111010.10111111101010... 
73  101010001.00010101  11111011.10111111101010... 
74  101010010.01010101  11111101.10111111101010... 
75  101010100.01010101  11111110.11111111101010... 
76  101010101.01010101  11111111.11111111101010... 
77  1000000000.1010101001  101010101.11111111101010... 
78  1000000010.0010101001  101010111.01111111101010... 
79  1000000100.0010101001  101011010.11011111101010... 
80  1000000101.0010101001  101011011.11011111101010... 
81  1000001000.1000101001  101011101.11011111101010... 
82  1000001010.0000101001  101011111.01011111101010... 
83  1000010000.0000101001  101101010.10110111101010... 
84  1000010001.0000101001  101101011.10110111101010... 
85  1000010010.0100101001  101101101.10110111101010... 
86  1000010100.0100101001  101101110.11110111101010... 
87  1000010101.0100101001  101101111.11110111101010... 
88  1000100000.1010001001  101110101.11110111101010... 
89  1000100010.0010001001  101110111.01110111101010... 
90  1000100100.0010001001  101111010.11010111101010... 
91  1000100101.0010001001  101111011.11010111101010... 
92  1000101000.1000001001  101111101.11010111101010... 
93  1000101010.0000001001  101111111.01010111101010... 
94  1001000000.0000001001  110101010.10101101101010... 
95  1001000001.0000001001  110101011.10101101101010... 
96  1001000010.0100001001  110101101.10101101101010... 
97  1001000100.0100001001  110101110.11101101101010... 
98  1001000101.0100001001  110101111.11101101101010... 
99  1001001000.1001001001  110110101.11101101101010... 
100  1001001010.0001001001  110110111.01101101101010... 
Imagine how hard it would have been to teach children to add integers if we had arbitrarily chosen φ as our number base instead of 10. The explanation would go something like this:
"Now, children, pay attention. As we learned yesterday, φ+1=φ*φ, so 100 = 11, and that's true even in the φimal places. So another name for 1.00 is 0.11. I hope you have all memorized that by now because it is very important in today's lesson. Now, to add 1+1, write one of the 1's as 0.11, to avoid having to carry  sorry, 'regroup' [the 'new math' has made it to φland]  so you add 1+1 this way:
1.00
+0.11

1.11 (Two)
We have a special name for this number. Two. Everyone say it with me, Twooooo, that's right. Is this clear to everyone? Now, let's see about the next integer. What's 1+Two? Time to regroup! You remember from yesterday that 11=100, right? So obviously 110=1000, because you can multiply both numbers by φ. Then add 1 to both numbers, and you get 111=1001. Then just rewrite 1+1=1.11 as 1+1=10.01, just another way of writing Two. And then you can add 1 yet again, like this. Here's 1+Two:
1.00
+10.01 (Two)

11.01 (Three)
I have eyes in the back of my head, Jimmy! This is important, now, so look closely. You need to understand regrouping if you ever hope to learn about the integers! Now, Two+Two has another special name, does anyone know it? Four! That's right. Now, since Two is 1.11 and it's also 10.01, we can add them up to make Four. Uhoh. We have a collision in the φ^{Two} place. We have to regroup again using the same trick: 1.11 is the same as 1.1100, which is the same as 1.1011. Do you see how I took the last '100' and turned it into '011'? Let me write it on the board.
1.1011 (Two)
+10.0100 (Two)

11.1111 (Four)
Do you all see how regrouping works in this addition problem? (chirp, chirp...) Anyone? OK, I'll do it again, but watch me this time. We have a lot of numbers to get through before we've completed are addition table! Then, for homework, I want you all to study the number Five. That's 1 plus Four, and it's also Two plus Three. Good luck, children!"
Wikipedia, Golden Ratio Base
Mathworld, Base and Phi Number System
The golden ratio
The webmaster and author of this Math Help site is Graeme McRae.