## Base 2 (Binary)

We use base 10 (also called decimal). Base 10 has 10 different digits (0,1,2,3,4,5,6,7,8,9). The first few numbers are 1,2,3,4,5,6,7,8,9,10,11,12,...,20,21,...,100,101,... We apparently use base 10 because we (most of us) have ten fingers. Base 2 (also known as binary) is what computers use, internally. It has been joked that this is because they have two fingers (two electrical states, actually). In base 2, there are two different digits (0 and 1). And the first few numbers are 1,10,11,100,101,110,111,1000,1001,... Arithmetic with base 2 is actually easier than with base 10. But the numbers are longer, making them harder to read.

Why would we ever want to study base 2? There are a couple fairly obvious reasons, actually. One is that it may prove to be interesting. But more importantly, it may teach us some things about base 10.

Here are the addition and multiplication tables:

```  + |  0   1    x |  0   1
---+--------  ---+--------
0 |  0   1    0 |  0   0
1 |  1  10    1 |  0   1```

There are no carries in multiplication, until we start adding up the partial products. It is educational to practice doing arithmetic in base 2:

```   10110      1101
+1101      x101
-------    ------
100011      1101
1101
---------
1000001```

A base 2 digit represents that digit times a power of 2, just as a base 10 digit represents that digit times a power of 10. In base 10, 375 is 3x100+7x10+5. Similarly, 100011 is 2^5+2^1+2^0 or 32+2+1 or 35. Practicing converting from base 2 to base 10, or vice versa, is educational, as well. 375 is 256+64+32+16+4+2+1, which is 101110111 in base 2.

It is important to label base two numbers (usually with a subscript 2) because they can be mistaken for base 10 numbers. For example 10102=1010. Unfortunately, some readers of this page use browsers which cannot handle subscripts. So, hopefully, the numbers in this article are clear, because of their contexts.

You might be curious what some fractions look like in base 2. Of course 1/2 is 0.1 and 1/4 is 0.01. 1/3 is 0.010101... and 1/10 is 0.000110011... which repeat forever. This repeating 1/10 is one of the reasons for roundoff error in computers. Of course, pi (11.001001000011111101...) and the square root of 2 (1.011010100000100...) do not repeat in base 2.