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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 <center><math>\displaystyle 2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math></center>

Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are.  The first digit to the left of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six <math>10^0</math>'s, the second digit tells us there are four <math>10^1</math>'s, the third digit tells us there are seven <math>10^2</math>'s, and the fourth digit tells us there are two <math>\displaystyle 10^3</math>'s.

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. in our example above, <math>2746_{10}</math>, 10 is the radix).


== Base Number Topics ==
* [[base numbers/Common bases | Common bases]]
* [[base numbers/Conversion | Converting between bases]]


== Common bases ==
Commonly used bases are 2, 8, 10 (duh!) and 16. The base doesn't necesarily have to be an integer. There are [[complex base | complex]], [[irrational base | irrational]], [[negative base | negative]], [[improper fractional base | fractional]], and many other kinds of bases. The best known one is [[phinary]], which is base [[phi]]; others include "Fibonacci base" and base negative two.

=== Binary ===
Binary is base 2.  It's a favorite among computer programmers. It has just two digits: <math>0</math> and <math>1</math>.

===Octal ===
Octal is base 8. It was also quite liked by programmers because the octal representation of numbers is 3 times shorter than the binary one and the conversion from octal to binary and back is very easy (can you guess why?). Besides, 8 is quite close to 10 and less than 10, so to learn doing addition and multiplication in base 8 is not very hard: you can basically count in base 10 with partial conversions to base 8 on the way. Let's multiply <math>12345_8</math> by <math>7_8</math>. <math>5\cdot 7=35_{10}=43_8</math> (to get the last result, just divide <math>35</math> by <math>8</math> with remainder). As usual, we write  the last digit <math>3</math> down and keep <math>4</math> in mind. Now, <math>4\cdot 7+4=32_{10}=40_8</math>, so we write down <math>0</math>, getting <math>03</math>, and keeping <math>4</math> in mind. 
And so on. The time needed to get the answer <math>111103_8</math> only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).

=== Decimal ===
Decimal is base 10.  It's the base that everyone knows and loves.  Most numbers in the world are written without a specified radix and  usually it can just be assumed that they are in base 10.  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. Hexadecimal numbers are also used by programmers in the same way as octal numbers, but to learn to count in hexadecimal is harder than in octal.


== History ==

Base-10 is an apparently obvious counting system because people have 10 fingers.  Historically, different societies utilized other systems.  The  Native American cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) 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, 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 MDII!  That's why the introduction of the '''Arabic numeral system''', base-10, revolutionized math and science in Europe.


== Example Problems ==
=== Intermediate ===
* [[1977_Canadian_MO_Problems/Problem_3 1977 | Canadian Mathematics Olympiad Problem 3]]

== See Also ==
*[[Number theory]]
*[[Modular arithmetic]]
*[http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=92951 Richard Rusczyk's Base Number Article]