slide04 numsys ops part2

8/3/2019 Slide04 NumSys Ops Part2

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Number Systems &

Operations

Part II


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Arithmetic Operations with

Signed NumbersBecause the 2s complement form for representing

signed numbers is the most widely used in computer

systems. Well limit to 2s complement arithmetic on:

Addition

Subtraction Multiplication

Division


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Addition

4 cases that can occur when 2 signed numbers are

added:

Both numbers positive

Positive number with magnitude larger than negative

number

Negative number with magnitude larger than positivenumber

Both numbers negative


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Addition

Both numbers positive:

ex: 00000111! ! 7! ! ! ! !! ! !

00001011! !

11

The sum is positive and is therefore in true

(uncomplemented) binary.


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Addition

Positive number with magnitude larger than negative

number:

ex: 00001111! ! 15! ! ! ! !! ! 1 00001001!! 9

The final carry bit is discarded. The sum is positive and is

therefore in true (uncomplemented) binary.

Discardcarry


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Addition

Negative number with magnitude larger than positive

number:

ex: 00010000! ! 16! ! ! !! ! ! ! 11111000! ! -8

The sum is negative and therefore in 2s complement

form.


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Addition

Both numbers negative:

ex: 11111011!! -5! ! ! !!! ! 1 11110010! ! -14

The final carry bit is discarded. The sum is negative and

therefore in 2s complement form.

Discardcarry


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Addition

Remark: The negative numbers are stored in 2s complement

form so, as you can see, the addition process is very

simple:Add the two numbers and discard any

carry bit.


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Addition

Overflow condition:

When two numbers are added andthe number of bits required to

represent the sum exceeds the

number of bits in the two numbers,

an overflowresults as indicated by

an incorrect sign bit.

An overflow can occur only when

both numbers are + or -.

ex: 01111101 125

10110111 183

Magnitudeincorrect

Sign

incorrect


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AdditionNumbers are added two at a time:

Computer add strings of numbers two numbers at a

time.

ex: add the signed numbers: 01000100, 00011011, 00001110, and00010010

! ! ! 68!! 01000100! ! ! !! !Add 1st two numbers! ! ! 95!! 01011111! 1st sum! ! ! !! !Add 3rd number! ! ! 109!! 01101101! 2nd sum! ! ! !! !Add 4th number! ! ! 127!! 01111111! Final sum


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Subtraction

Subtraction is of addition.

Subtraction is addition with the sign of the subtrahendchanged.

The result of a subtraction is called the difference.

The sign of a positive or negative binary is changed bytaking its 2s complement.


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Subtraction

Since subtraction is simply an addition with the sign of

the subtrahend changed, the process is stated asfollows:

To subtract two signed numbers, take the 2s

complement of the subtrahend and add.

.


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Subtraction

ex: Perform each of the following subtraction of the signed numbers:

(a) 00001000 00000011!! (b) 00001100 11110111!

(c) 11100111 00010011!!

(d) 10001000 - 11100010

! (a) 00001000! 8! ! (b) 00001100! 12!! ! ! ! ! !! 100000101! 5! ! 00010101! 21! (c) 11100111! -25! ! (d) 10001000 -120! ! ! ! ! 111010100! -44! ! 10100110 -90


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Hexadecimal and Octal

Numbers


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Hexadecimal 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 writingbinary numbers since it is very easy to convert betweenbin and hex.

Hex is widely used in computer and microprocessorapplications.


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Hexadecimal Numbers

Decimal Binary Hexadecimal

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 910 1010 A

11 1011 B

12 1100 C

13 1101 D

14 1110 E

15 1111 F


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Hexadecimal Numbers

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 isnot one of A-F using in hex).

For example

16h = 000101102

0Dh = 000011012


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Hexadecimal Numbers

Bin-to-Hex Conversion

Simply break the binary number into 4-bit groups,

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

(a) 1100101001010111! ! (b) 111111000101101001 1100101001010111!! 00111111000101101001

! C A 5 7! ! 3 F 1 6 9! ! = CA5716! ! ! ! = 3F16916


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Hexadecimal Numbers

Hex-to-Bin Conversion

Reverse the process (of bin-to-hex) and replace eachhex symbol with the appropriate four bits.

ex: Determine the binary numbers for the following hex numbers:

(a) 10A4h! (b) CF8Eh!! (c) 9742h1 0 A 4! C F 8 E! 9 7 4 2

0001000010100100! 1100111110001110 1001011101000010


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Hexadecimal Numbers

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.


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Hexadecimal Numbers


Hex-to-Bin first and then Bin-to-Dec

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


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Hexadecimal Numbers


Multiply the decimal values of each hex digits by itsweight 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


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Hexadecimal Numbers

Dec-to-Hex conversion

Repeated division of a dec number by 16

ex: Convert the dec number 650 to hex

650/16 = 40.625 0.625x16 = 10 = A

40/16 = 2.5 0.5x16 = 8 = 8

2/16 = 0.125 0.125x16 = 2 = 2Stop when whole

number quotient isZERO.

MSD

LSD

Hence 65010= 28Ah


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Octal Numbers

Like the hex, the oct provides a convenient wayto express binary numbers and codes. (btw, itsnot 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 hexso far work the

same way on oct just mark this: Hex = 4 binary bits

Oct = 3 binary bits

Now, lets crack the following exercises


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Octal NumbersBin-to-Oct Conversion

(a) 110101!! ! (b) 101111001(c) 100110011010 !(d) 11010000100

Oct-to-Bin Conversion(a) 13

8(b) 25

8(c) 140

8(d) 7526

8

Oct-to-Dec Conversion

(a) 23748Dec-to-Oct Conversion

(a) 35910

slide04 numsys ops part2

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