Base numbers
To understand the notion of base numbers, we look at our own number system. We use the decimal, or base-10, number system. To help explain what this means, consider the number 2746. This number can be rewritten as 
. Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. Also, each place in a number represents a value 10 times the place value to its right. Base-10 uses digits 0-9. Usually, the base, or radix, of a number is denoted as a subscript written at the right end of the number; e.g. our example above may be written 
.
What if we want to know how much 
is in base-10? (Notice how base 2 only uses digits 0 and 1. For this reason, binary has many applications where only binary decisions, such as yes or no, are made, such as in computers.) From above, we see that each place value in binary is twice that of the place to its right. Hence, 
.
Commonly used bases are 2, 8, 10 (duh!) and 16. The base doesn't necesarily have to be an integer. There are complex, irrational, negative, and many other kinds of bases. The best known one is phinary, which is base phi.
Common bases
Binary
Binary is base 2. It's a favorite among computer programmers.
Decimal
Decimal is base 10. It's the base that everyone knows and loves. The most commonly used explanation for the origin of base 10 for our number system is the number of fingers we have.
Hexadecimal
Hexadecimal is base 16. The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. One of its common uses is for color charts.
History
Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Babylonians and some Native American cultures (Incas?) are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this coming one) why we count 60 minutes in an hour and 60 seconds in a minute. The Roman system (internal link w/explanation?), which didn't have any base system at all, but rather used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000). Imagine how difficult it would be to multiply LXV by IIMD! That's why the introduction of the Arab numberal system, base-10, revolutionized math and science in Europe.
