You are on page 1of 25

# Number Systems &

Operations
Part II

## Arithmetic Operations with

Signed Numbers
Because the 2s complement form for
representing signed numbers is the most
widely used in computer systems. Well
limit to 2s complement arithmetic on:

Subtraction
Multiplication
Division

4 cases that can occur when 2 signed

## Both numbers positive

Positive number with magnitude larger than
negative number
Negative number with magnitude larger than
positive number
Both numbers negative

Both numbers positive:
ex:

00000111
+00000100
00001011

7
+4
11

## The sum is positive and is therefore in true

(uncomplemented) binary.

Positive number with magnitude larger
than negative number:
ex:

00001111
15
+11111010
+6
00001001
9
carry

## The final carry bit is discarded. The sum is

positive and is therefore in true
(uncomplemented) binary.

Negative number with magnitude larger
than positive number:
ex:

00010000
+11101000
11111000

16
+24
8

## The sum is negative and therefore in 2s

complement form.

Both numbers negative:
ex:

11111011
5
+11110111
+9
11110010
14
carry

## The final carry bit is discarded. The sum is

negative and therefore in 2s complement
form.

Remark:

## The negative numbers are stored in 2s

complement form so, as you can see, the
numbers and discard any final carry bit.

Overflow condition:

## When two numbers are added

and the number of bits
required to represent the sum
exceeds the number of bits in
the two numbers, an overflow
results as indicated by an
incorrect sign bit.
An overflow can occur only
when both numbers are + or -.

ex: 01111101
+00111010
10110111

Magnitude
incorrect
Sign
incorrect

125
+58
183

Numbers are added two at a time:

## Computer add strings of numbers two

numbers at a time.

## ex: add the signed numbers: 01000100, 00011011, 00001110, and

00010010
68
01000100
+27
+00011011
95
01011111
1stsum
+14
+00001110
109
01101101
2ndsum
+18
+00010010
127
01111111
Finalsum

Subtraction
Subtraction is a special case of addition.

## Subtraction is addition with the sign of the

subtrahend changed.
The result of a subtraction is called the
difference.

## The sign of a positive or negative binary is

changed by taking its 2s complement.

_______________________________

Subtrahend = , Minuend =

Subtraction
Since subtraction is simply an addition
with the sign of the subtrahend changed,
the process is stated as follows:

## To subtract two signed numbers, take the 2s

complement of the subtrahend and add.

Subtraction
ex: Perform each of the following subtraction of the signed numbers:
(a)0000100000000011

(b)0000110011110111

(c)1110011100010011

(d)1000100011100010

(a)00001000
+11111101
100000101

8
+3
5

(b)00001100
+00001001
00010101

(c)11100111
+11101101
111010100

25
+19
44

(d)10001000120
+00011110+30
1010011090

12
+9
21

Numbers

We will call it for short as hex.
It has 16 characters. Digits 0-9 and letters
A-F.
It used primarily as a compact way of
displaying or writing binary numbers since
it is very easy to convert between bin and
hex.

## Hex is widely used in computer and

microprocessor applications.

Decimal

Binary

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

10

1010

11

1011

12

1100

13

1101

14

1110

15

1111

If you see h mixing in numbers (in the
context of computer systems), please note
that its most likely that the numbers are
hexadecimal numbers. (Be careful. h is
not one of A-F using in hex).

For example
16h=000101102
0Dh=000011012

Bin-to-Hex Conversion

## Simply break the binary number into 4-bit

groups, starting at the right-most bit and
replace each 4-bit group with the equivalent
hex symbol.

(a)1100101001010111

(b)111111000101101001

1100101001010111

00111111000101101001

CA57
=CA5716

3F169

=3F16916

Hex-to-Bin Conversion

## Reverse the process (of bin-to-hex) and

replace each hex symbol with the appropriate
four bits.

ex: Determine the binary numbers for the following hex numbers:
(a)10A4h
(b)CF8Eh
10A4 CF8E

(c)9742h
9742

0001000010100100 11001111100011101001011101000010

Hex-to-Dec Conversion

2 methods:
Hex-to-Bin first and then Bin-to-Dec.
Multiply the decimal values of each hex digits by its
weight and then take the sum of these products.

Hex-to-Dec Conversion

## ex: Convert the following hex numbers to decimal:

(a)1Ch
1Ch=00011100=16+8+4=2810
(b)A85h
A85h=101010000101=2048+512+128+4+1=269310

Hex-to-Dec Conversion

## Multiply the decimal values of each hex digits by its

weight and then take the sum of these products.

## ex: Convert the following hex numbers to decimal:

(a)E5h
E5h=(Ex16)+(5x1)=(14x16)+5=224+5=22910
(b)B2F8h
B2F8h=(Bx4096)+(2x256)+(Fx16)+(8x1)
=(11x4096)+(2x256)+(15x16)+(8x1)
=45,056+512+240+8=45,81610

Dec-to-Hex conversion

## ex: Convert the dec number 650 to hex

LSD

650/16 = 40.625

0.625x16 = 10 = A

40/16 = 2.5

0.5x16

2/16 = 0.125

0.125x16 = 2 = 2

## Stop when whole

number quotient is
ZERO.

=8 =8

## Hence 65010 = 28Ah

MSD

Octal Numbers
Like the hex, the oct provides a convenient
way to express binary numbers and codes. (but,
its not as commonly used as hex).
8 digits: 0-7

0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,

## Operations we learn about hex so far work the

same way on oct just mark this:

## Hex = 4 binary bits

Oct = 3 binary bits

## Now, lets crack the following exercises

Octal Numbers
Bin-to-Oct Conversion
(a)110101 (b)101111001
(c)100110011010 (d)11010000100

Oct-to-Bin Conversion
(a)138(b)258(c)1408(d)75268

Oct-to-Dec Conversion
(a)23748

Dec-to-Oct Conversion
(a)35910
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}