In the Decimal system, when we write a number like 5837, we know it
means "five thousand, eight hundred, and thirty-seven" The first digit is
the number of thousands (103), the second is the number of hundreds
(102), the third is the number of tens (101), the last is the
number of units (100). We could put our number 5837 in a table with
each digit in a column under its power of ten. See the table below.
The powers of ten indicate the 'weight' of a column.
We could add to this table the numbers 0,1,2,3 to number the columns
according to their power, and we could also add the value of the number base
raised to that power. Again, see the table below.
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So the value of a number expressed as a sequence of digits is dependent
upon the number base being used, the symbols used in that system and the
base raised to the power of the digit's position number.
When we count in the Decimal system we count 0,1,2,3.....9, then we
reset the units value to 0 and carry 1 to the tens column.
In the Octal system we count 0,1,2,.....7, then we reset to 0 and carry 1.
In the Binary system we count 0,1 then we reset to 0 and carry 1. In the
Hexadecimal system we count 0,1,2 ..... 9,A,B,C,D,E,F, then we reset and
carry 1. Here the letters allow us to count up to 15 in our first column
without having to carry.
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