Puzzle reference pages » Bases
assembled by Quincunx
Look at your hands and count your fingers. Yes, on both hands, and yes, include your thumbs. Unless you've met with a bit of misfortune or have an unusual genetic condition, you've got 10 fingers. Undoubtedly, sometimes you've used them to count. Probably you start with one outside finger on your left hand, cross that hand for five, then do the same on your right hand to get to 10. If you need to count higher than 10, you start on the left hand again.
So ingrained is the number 10 in our counting processes that, over time, our society has used that 10 in the way we form our numbers. (Except for those wacky Babylonians, who apparently thought 60 was a better number to use. My question: If this is true, why did they use the thumb on one hand, but not the other? I would think either 48 [neither thumb, 12x4] or 70 [both thumbs, 14x5] would have been the result.)
Think back to gradeschool math. If you're in my generation (born between Woodstock and "Y.M.C.A" by the Village People), you probably learned the decimal system with little tiny plastic or wooden blocks (ones), the rows of ten blocks stuck together (tens), the squares (hundreds), and the big block (thousands). So the number 1,234 was represented by 1 thousands block, 2 hundreds squares, 3 tens rows and 4 ones blocks.
And if all this math isn't scaring you off yet, this could be written as 1x1,000 + 2x100 + 3x10 +4x1, or 1x10³ + 2x10² + 3x10¹ + 4x10º. (Any positive number to the 0th power equals 1.) This is how base 10 works.
Now what happens if we use a number other than 10? Let's look at base 5. First of all, counting in base 5, we go 0, 1, 2, 3, 4... and what's next? Since 5 is 1x5 + 0 (or, if you prefer, 1x5¹ + 0x5º), we write the number we call 5 in base 10 as 10 in base 5. The number we call 6 in base 10 is 1x5 + 1 (i.e., 1x5¹ + 1x5º), so we write it as 11 in base 5.
We continue to count: 12, 13, 14. The next number (the one we call 10 in base 10) is not 15, but 2x5 + 0 (aka 2x5¹ + 0x5º), so we write it as 20. By now, you've noticed we don't use the digits 5, 6, 7, 8 or 9 in base 5; we only use 0, 1, 2, 3 and 4, a total of five digits. (In base 10, we use 10 different digits, 0 through 9, so this makes sense.)
Keeping up the count: 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44. (If you've lost count, 44 in base 5 is 24 in base 10.) So how to we represent, in base 5, the number we call 25 in base 10? 1x5² + 0x5¹ +0x5º... so we write 100.
Just to be sure you've grasped this concept, here's a look at base 3. If it helps you to visualize things, imagine those unit cubes, rows, flats and blocks from grade school, except make the rows 3 units long, the flats 3x3 units and the cubes 3x3x3, like a Rubik's Cube.
number in base 3 
number in base 10 

0  0  
1  1  
2  2  
10  3  
11  4  
12  5  
20  6  
21  7  
22  8  
100  9  
101  10  
102  11  
110  12  
111  13  
112  14  
120  15  
121  16  
122  17  
200  18  
201  19  
202  20  
210  21  
211  22  
212  23  
220  24  
221  25  
222  26  
1000  27  
1001  28  
(and...)  
1202  47  
2000  54  
2222  80 
And you may already be familiar with base 2, also known as binary... the 1s and 0s a computer works with. It's the same as the other bases, only with powers of 2 (1, 2, 4, 8, 16, 32, 64, 128...) instead of base 10's powers of 10 (1, 10, 100, 1,000...) or base 3's powers of 3 (1, 3, 9, 27...)
11010011 (base 2) = 1x2^{7}
+ 1x2^{6} + 0x2^{5}
+ 1x2^{4} + 0x2^{3}
+ 0x2^{2} + 1x2^{1}
+ 1x2^{0}, or
1x128 + 1x64 + 0x32
+ 1x16 + 0x8 + 0x4
+ 1x2 + 1x1, which equals 211
in base 10.
Are there bases higher than 10? You bet! One of the most common with computers is base 16, also known as hexadecimal. But what do you use for the digits greater than 9?
base 16  0  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F 
base 10  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 
So BE (base 16) = 11x16^{1} + 14x16^{0}, or 11x16 + 14x1, which equals 190 in base 10.
I once saw a very complex puzzle that used base 36.
base 36  0  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z 
base 10  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35 
A base 10 number, when converted to base 36, spelled a word, similar to how 190 in base 10 becomes BE in base 16. Of course, once you start multiplying powers of 36 (1, 36, 1296...), you're going to need a big calculator fairly quickly. <grin>
So how does this all apply to puzzles? Well, assuming you're not facing down that base 36 puzzle, you may find a baseoriented puzzle combined with the A=1, B=2 ... Z=26 cipher to hide a word or phrase. More info can be found on that page.
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