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Expanded article
The Decimal, or base 10, system is taught in the early years of school and is second-nature to most people. A digit is a number from 0 through 9. When you write a number, you write it as as series of digits. Essentially, each digit occupies a column. Consider the number "19". The right column contains the digit "9", and the left column contains "1". The "9" is in the "one's column", and the "1" is in the "ten's column". Thus, the "1" really represents the number "10". "19" is merely 10+9. The rightmost column is the least significant digit, because it is always smaller than any column to the left. The leftmost column is the most significant digit, because it is the largest value.
Number Systems are means by which numeric values can be written, read, and upon which arithmetic operations can be performed. The type of number system in most common use is positional. Different glyphs have been used historically, but the most widespread system uses Hindu-Arabic glyphs: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Some ancient languages, such as Hebrew, had no separate glyphs for numerals, using either words for specific values or reusing letters to indicate numbers.
=Positional Number Systems=
Positional number systems organize numerical values in columns which determine the value for the value in that column.
The Decimal, or base 10, system is taught in the early years of school and is second-nature to most people. A digit is a number from 0 through 9. When you write a number, you write it as as series of digits. Each digit occupies a column. Consider the number "19". The right column contains the digit "9", and the left column contains "1". The "9" is in the "one's column", and the "1" is in the "ten's column". Thus, the "1" really represents the number "10". "19" is merely 10+9. The rightmost column is the least significant digit, because it is always smaller than any column to the left. The leftmost column is the most significant digit, because it is the largest value.
The decimal system is called base 10 because there are ten possible digits (0-9). When 1 is added to 0, the result is 1. When 1 is added to 1, the result is 2, and so on. But when 1 is added to 9, there are no digits we can use that are larger than 9. Therefore, we "carry" the 1 over to the next column on the left, and set the current column to 0. This process is repeated for as many digits as are in the number. Thus 19+1 becomes 20. Any digits can be added together, up to 9+9. We still carry the 1, but the amount for the current column is the remainder: 8. Likewise, when subtracting, if the digit is 0, there is no digit less than 0, so we have to "borrow" from the next greater column. If 1 is subtracted, the digit changes from 0 to 9.
The decimal system is called base 10 because there are ten possible digits (0-9). When 1 is added to 0, the result is 1. When 1 is added to 1, the result is 2, and so on. But when 1 is added to 9, there are no digits we can use that are larger than 9. Therefore, we "carry" the 1 over to the next column on the left, and set the current column to 0. This process is repeated for as many digits as are in the number. Thus 19+1 becomes 20. Any digits can be added together, up to 9+9. We still carry the 1, but the amount for the current column is the remainder: 8. Likewise, when subtracting, if the digit is 0, there is no digit less than 0, so we have to "borrow" from the next greater column. If 1 is subtracted, the digit changes from 0 to 9.
<table>
<table>
<tr><th>Value</th><th>Base</th></tr>
<tr><th>Value</th><th>Base</th></tr>
<tr><td>F</td><td>36 (hexatrigesimal)</td></tr>
<tr><td>F</td><td>16 (hexadecimal)</td></tr>
<tr><td>F</td><td>16 (hexadecimal)</td></tr>
<tr><td>15</td><td>10 (decimal)</td></tr>
<tr><td>15</td><td>10 (decimal)</td></tr>
<tr><td>17</td><td>8 (octal)</td></tr>
<tr><td>17</td><td>8 (octal)</td></tr>
<tr><td>120</td><td>3 (trinary)</td></tr>
<tr><td>120</td><td>3 (trinary or ternary)</td></tr>
<tr><td>1111</td><td>2 (binary)</td></tr>
<tr><td>1111</td><td>2 (binary)</td></tr>
</table>
</table>
=Converting a value to decimal=
===Invalid Bases===
Bases must be integer values greater than 1. A negative or fractional base has no meaning. A base of 1 is useless for positional notation since there would only be a single digit (0). However
such a base is essentially what is used for Tally Marks (see below). A base of 0 has no use since it indicates there are no glyphs that can be used to represent a value. Finally, there is no
widely-accepted standard for representing bases greater than 36 in positional systems.
==Converting a value to decimal==
A value in a given base may be converted to its decimal equivalent value by summing the following equation for each column of the number:<br>
A value in a given base may be converted to its decimal equivalent value by summing the following equation for each column of the number:<br>
V x B<sup>C</sup><br>
V x B<sup>C</sup><br>
1x2<sup>2</sup> + 0x2<sup>1</sup> + 1x2<sup>0</sup><br>
1x2<sup>2</sup> + 0x2<sup>1</sup> + 1x2<sup>0</sup><br>
which gives: 4 + 0 + 1 = 5
which gives: 4 + 0 + 1 = 5
=Other Number Systems=
Non-positional number systems do not determine a symbol's value by a columnar position. The most common examples in use today are the [[Roman Numerals]] and Tally Marks.
Tally Marks use a vertical line to indicate 1, they are grouped together in sets of 5, with the fifth line being diagonal and through the other four marks.