36
Chapter 2
Binary Values and Number Systems
Let’s express this idea formally. If a number in the base
R
number system
has
n
digits, it is represented as follows, where
d
i
represents the digit in the
i
th position in the number.
d
n
*
R
n
±
1
+
d
n
±
1
*
R
n
±
2
+
...
+
d
2
*
R
+
d
1
Look complicated? Let’s look at a concrete example:
63578 in base 10.
n
is 5 (the number has 5 digits), and
R
is 10 (the base). The formula says that the fifth digit
(last digit on the left) is multiplied by the base to the
fourth power; the fourth digit is multiplied by the base
to the third power; the third digit is multiplied by the
base to the second power; the second digit is multiplied
by the base to the first power; and the first digit is not
multiplied by anything.
6 * 10
4
+ 3 * 10
3
+ 5 * 10
2
+ 7 * 10
1
+ 8
In the previous calculation, we have assumed that
the number base is ten. This is a logical assumption
since our number system
is
base ten. However, there
is nothing about the number 943 that says it couldn’t
be representing a value in base 13. If so, to determine
the number of ones, we would have to convert it to
base 10.
9* 1
3
2
= 9 * 169 = 1521
+4* 1
3
1
=4* 1
3=
5
2
+3* 1
3
0
=3*
1=
3
1576
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This note was uploaded on 01/13/2011 for the course CSE 1550 taught by Professor Marianakant during the Fall '10 term at York University.
 Fall '10
 MARIANAKANT

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