Every numbering system uses symbols to represent "this many" of something. The process of figuring out "how many" is called counting. Since there are limitless things in this universe, systems of counting must use some method of arranging number symbols to represent very large counts of things. To create numbers large enough to represent large quantities, most number systems use columns.
The columns represent a complete count through all the values of the previous column and are thus one order of magnitude larger. Some systems count by twos, so you go to the next column whenever you reach two of anything. Some counting systems go to the next column when you reach eight of anything. Some systems don't move on to the next column until they reach sixteen. Every number system gets it's name from this
This limit on how high you count before moving to the next column is where any numbering system bases it's name. Systems using a maximum value of two before rolling over to the next column are called binary and referred to as a base two counting system. Systems using a maximum value of sixteen before rolling over to the next column are called hexadecimal.
This idea of placing number symbols in columns makes tracking large quantites of things very simple. Let's use your hands as a simple example.
You have six values you can represent using one hand.
0 = Zero fingers raised
1 = One finger raised
(and so on)
5 = All five fingers raised
To count to six, you'd reset your first hand to zero and then raise one finger on the other hand. That finger on your second hand would represent the fact that you hand counted all the way through all the values you can represent on the first hand. You are using the fingers on your second hand to count how many times you've counted from zero to five on the first hand. That's what the columns are for: to count how many times we've run through all the number symbols in the previous column.
Mosth humans think in decimal; we watch numbers roll from zero to nine, add one to the next column for ten, then count to nineteen then add one to the next column for twenty and so on. The same holds true in the other systems, they simply flip to the next column at a different counted quantity.
MULTIPLICATION - A shortcut for counting by columns
Multiplication is nothing more than a shortcut for figuring out how many times to count through a range of values. If we count to ten four times, that's the same quantity as if we'd just multiplied ten items by four.
Powers (exponents)
Powers (exponents) are special cases in multiplication where a number is multiplied by itself. Since we are counting from zero to a value in each of the counting colums, we could think each column as representing the process of multiplying the maximum column value by itself, thus a thousand objects are equivalent to 10x10x10 which is the same thing as 103.
There are different ways to represent what the columns mean in our base ten or decimal counting system.
| Decimal Value | 1000 | 100 | 10 | 0 |
| Multiplication | 10 x 10 x 10 | 10 x 10 | 10 | 1 |
| As a Power | 103 | 102 | 101 | 100 |
So what's the point?
The point here is that we're going to cover binary and hexadecimal in this number systems tutorial and you will encounter both counting systems elsewhere. Everything computers do is in binary. Computers often display their binary values in octal and hexadecimal format.
Network card NIC addresses are in Hexadecimal; IP version 6 addresses are in hexadecimal; core dumps are in either octal or hexadecimal (depends on who did the programming). Unix file permissions can be set using octal values with the chmod command.