Transcript CPS120: Introduction to Computer Science

CPS120: Introduction to
Computer Science
Computer Math: Signed Numbers
Representing Signed Numbers

Remember, all numeric data is represented inside
the computer as 1s and 0s
– Arithmetic operations, particularly subtraction raise the
possibility that the result might be negative

Any numerical convention needs to differentiate
two basic elements of any given number, its sign
and its magnitude
– Conventions
•
•
•
•
Sign-magnitude
Ten's complement
Two’s complement
One’s complement
Representing Negatives

It is necessary to choose one of the bits of the
“basic unit” as a sign bit
– Usually the leftmost bit
– By convention, 0 is positive and 1 is negative


Positive values have the same representation in all
conventions
However, in order to interpret the content of any
memory location correctly, it necessary to know
the convention being used used for negative
numbers
Comparing the Conventions
Bit Pattern
000
001
010
011
100
101
110
111
Unsigned
0
1
2
3
4
5
6
7
Sign1's
2's
Magnitude Complement Complement
0
0
0
1
1
1
2
2
2
3
3
3
-0
-3
-4
-1
-2
-3
-2
-1
-2
-3
-0
-1
Representing Negative Values

You have used the signed-magnitude
representation of numbers since grade
school. The sign represents the ordering,
and the digits represent the magnitude of the
number.
Representing Negative Values
(Cont’d)

There is a problem with the sign-magnitude
representation:
– There are two representations of zero.
– There is plus zero and minus zero. Two
representations of zero within a computer can
cause unnecessary complexity.
Sign-Magnitude
For a basic unit of N bits, the leftmost bit is
used exclusively to represent the sign
 The remaining (N-1) bits are used for the
magnitude
 The range of number represented in this
convention is –2 N+1 to +2 N-1 -1

Sign-magnitude Operations

Addition of two numbers in sign-magnitude is
carried out using the usual conventions of binary
arithmetic
– If both numbers are the same sign, we add their
magnitude and copy the same sign
– If different signs, determine which number has the
larger magnitude and subtract the other from it. The
sign of the result is the sign of the operand with the
larger magnitude
– If the result is outside the bounds of –2 n+1 to +2 n-1 –1,
an overflow results
Other Ways of Representing
Negative Values
• If we allow only a fixed number of values, we can
represent numbers as just integer values, where half
of them represent negative numbers.
• For example, if the maximum number of decimal
digits we can represent is two, we can let 1 through
49 be the positive numbers 1 through 49 and let 50
through 99 represent the negative numbers -50
through -1.
Representing Negative Values
(Cont’d)

To perform addition within this scheme, you
just add the numbers together and discard
any carry.
Representing Negative Values
(Cont’d)

A-B=A+(-B). We can subtract one number
from another by adding the negative of the
second to the first.
Representing Negative Values
(Cont’d)

There is a formula that you can use to
compute the negative representation:

This representation of negative numbers is
called the ten’s complement.
Representing Negative Values
(Cont’d)
Two’s Complement:
To make it easier to look at
long binary numbers, we
make the number line
vertical.
Representing Negative Values
(Cont’d)

Addition and subtraction are accomplished
the same way as in 10’s complement
arithmetic:
-127
+ 1
-126

10000001
00000001
10000010
Notice that with this representation, the
leftmost bit in a negative number is
always a 1.
Two’s Complement Convention


A positive number is represented using a
procedure similar to sign-magnitude
To express a negative number
1.
2.
3.
–
–
Express the absolute value of the number in binary
Change all the zeros to ones and all the ones to zeros (called
“complementing the bits”)
Add one to the number obtained in Step 2
The range of negative numbers is one larger than
the range of positive numbers
Given a negative number, to find its positive
counterpart, use steps 2 & 3 above
One’s Complement
Devised to make the addition of two numbers
with different signs the same as two numbers
with the same sign
 Positive numbers are represented in the usual
way
 For negatives

– STEP 1: Start with the binary representation of the
absolute value
– STEP 2: Complement all of its bits