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Numbers 1
14  Computer Representation of Integer
and Real Numbers
In everyday life, numbers are almost exclusively represented using the base 10 decimal
system. This means, for example, that an integer
N
written as
N
=
a
n
a
n—1
…
a
1
a
0
(N=231)
Corresponds to a shorthand for
N
= (
a
n
a
n
—1
…
a
1
a
0
)
10
=
a
n
∗
10
n
+
a
n
—1
∗
10
n
—1
+ … +
a
1
∗
10
1
+
a
0
∗
10
0
(N= 2
∗
10
3
+
3
∗
10
1
+ 1
∗
10
0
)
In the above case, 10 is called the
base
(or
radix
) of the system. The
a
’s are integers
between 0 and 9.
Rational numbers, numbers with decimal points, make use of the
negative powers of the base:
N
= (0.
a
1
a
2
…
a
n1
a
n
)
10
=
a
1
∗
10
1
+
a
2
∗
10
2
+ … +
a
n1
∗
10
n+1
+
a
n
∗
10
n
(N= 0.231=2
∗
10
1
+
3
∗
10
2
+ 1
∗
10
3
Computers generally use base 2 which is called the
binary system
. An integer
N
is
represented as
N
= (
a
n
a
n
—1
…
a
1
a
0
)
2
=
a
n
∗
2
n
+
a
n
—1
∗
2
n
—1
+ … +
a
1
∗
2
1
+
a
0
∗
2
0
(N=11100111)
N
= 1*2
7
+1∗
2
6
+ 1
∗
2
5
+ 0*2
4
+ 0
∗
2
3
+ 1
∗
2
2
+ 1
∗
2
1
+ 1
∗
2
0
The
a
’s are only allowed to be 0 and 1. The binary system is quite convenient to
computers, since the basic unit of information that they store is an electrical pulse which
can be either "on" or "off".
A single binary digit (a 0 or a 1) is called a
bit
.
An 8bit
sequence of 0’s and 1’s is usually called a
byte
.
MATLAB makes it easy to convert between base 10 (normal) and binary numbers.
>> A=dec2bin(231)
A=
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11100111
You can convert the other way using
bin2dec
.
Experiment with this command to figure
out how to use it.
Storage of binary numbers
The set of all integers is countably infinite. This means that in order to represent all
integers, computers would need infinite storage capacity. In reality, computers can store
only a finite subset of all integers because each integer is only allowed a certain number
bits.
The number of bits allocated for any integer is sometimes called a
wordlength
.
Using 16bit integer wordlength, the integer
N
= 231 may be stored as
0 000000011100111.
The first of the 16 bits stores the sign of the number (here "positive" is stored as 0), while
the remaining 15 bits store the number itself using the binary system representation.
Real numbers have decimal points.
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This note was uploaded on 02/12/2011 for the course E 7 taught by Professor Patzek during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Patzek

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