Class notes for the week of November 12
th
Number representations:
We are all very familiar with the decimal system, it this lecture we will expand on this
knowledge and learn how to express integer quantities in other number bases that offer
advantages for use in digital systems.
The decimal integer quantity 3581 can be written as:
This last form is the most germane for our purpose. Each column of numbers is
multiplied by a power of 10 which starts at 10
0
in the right hand column, then 10
1
for the
next column to the left, and so on increasing by a factor of ten for each column. These
concepts are deeply embedded in our language, and we call the columns the
ones
column,
the
tens
column, the
hundreds
column and so on.
Other number bases work the same way, except that the base for the exponent is not 10
but some other number. In the
binary
system the base is 2. the rightmost column is
therefore 2
0
, then the next column to the left is 2
1
, then 2
2
, and so on. The table below
shows some powers of 2 and their decimal equivalents.
Power of two
2
7
2
6
2
5
2
4
2
3
2
2
2
1
2
0
Decimal equivalent
12
8
6
4
3
2
1
6
8
4
2
1
We construct a binary number in the same way as a decimal number. The binary number
01101101 is equivalent to 0*2
7
+ 1*2
6
+ 1*2
5
+ 0*2
4
+ 1*2
3
+ 1*2
2
+ 0*2
1
+ 1*2
0
. We can
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 Fall '07
 Westerfield
 Binary numeral system, four bits

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