Number Systems for Students

8/8/2019 Number Systems for Students

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Number Systems

Decimal

Binary

Octal

Hexadecimal


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Decimal (base 10) Binary (Base 2) Octal (Base 8) Hexadecimal (Base 16)

00 0000 00 0

01 0001 01 1

02 0010 02 2

03 0011 03 3

04 0100 04 4

05 0101 05 5

06 0110 06 6

07 0111 07 7

08 1000 10 8

09 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F


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Conversion from base r to decimal


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Decimal System

Radix or base 10

10 digits (0,1,2,9) Coefficients are multiplied by powers of 10

Example:


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Octal (base-8) to Decimal

System Radix or base 8

8 digits (0,1,2,3,4,5,6,7) Coefficients are multiplied by powers of 8

Example:


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Hexadecimal (base-16) to decimal

system Radix or base 16

16 digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) Coefficients are multiplied by powers of 16

Example:


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Binary (base-2) to decimal

System Radix or base 2

2 digits (0,1) Coefficients are multiplied by powers of 2

Example:


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Examples: convert to decimal

equivalent(511.4)10

(408.5)10

(46687)10

(4021.2)5 =

(630.4)8 =

(B65F)16 =


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Conversion from decimal to base r


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Convert Decimal to Binary (Integer

Part)Example: 50 (divide by 2)

(0110010)2


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Convert Decimal to binary

(Fraction Part)Example: 0.625 (multiply by 2)

(0.101)2


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Convert Decimal to Binary (Integer

and fraction )Task:

(41)10 to (bbbb)2(0.6875)10 to (bbbb)2

(41.6875)10 to (bbbb)2

(101001.1011) to (ddd)10

(101001)2

(0.1011)2

(101001.1011)2


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Convert from Decimal to

Hexadecimal and backExample:

(450)10 to (hhh)16 divide by 16

(1C2)16 to (ddd)10 multiply by powers of 16

(0.521)10 to (hhh)16 mulitply by 16(0.8560)16 to (ddd)10 multiply by powers of 16


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Why octal and hexadecimal ??

23 = 8 and 24=16

3 digits required for octal

4 digits required for hexadecimal

(10 110 001 101 011.111 100 000 110)2=(26153.7406)

8

2 6 1 5 3 7 4 0 6

(10 1100 0110 1011.1111 0000 0110)2=(2C6B.F06)162 C 6 B F 0 6


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Why octal and hexadecimal ??

Each octal digit converted to its 3 bit

binary equivalent Each hexadecimal digit converted to its 4

bit binary equivalent(673.124)8=(110 111 011.001 010 100)26 7 3 1 2 4

(306.D)16=(0011 0000 0110. 1101)

2

3 0 6 D


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Complements


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Binary Numbers

An N-bit number a = aN-1 aN-2..a2a1 a0

maybe treated as signed or unsignednumber.

Unsigned Number: All the Nbits of anunsigned number are used to express themagnitude of the number.


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Numbers


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Signed-Magnitude

Designate left-most bit as a signbit with no

arithmetic weight 1-> negative, 0-> positive. Easy to determine sign

Positive and negative zero (0000 vs. 1000)

Difficult to add numbers of different sign orsubtract numbers of same sign (comparison)

N-1 bits are available to represent magnitude Range of N-bit signed-magnitude number is:

-(2N-11) to (2N-11)


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Signed-Magnitude

Advantages

intuitive appeal & conceptual simplicity symmetric range

simple negation

Disadvantages

fewer numbers encoded (two encodings for 0)

subtraction is more complicated than in 2s-comp


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2s ComplementExample: 2s complement ve number


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2s Complement

A specific case of radix complement

To negate or complementan Ndigitnumber, subtract it from 2N.

If you then add the number and itscomplement, you get 2N.

If you only keep N-digits (discard finalcarry), you have zero.


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2s Complement

However, we need an extra bit to allow

both positive and negative numbers. 7 is 0111

-7 is computed as 16-7=9 10000 - 0111 = 1001 = -7

Add 7 and 7, discard carry, we get 0 0111 + 1001 = 1_0000


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2s Complement


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2s Complement

2 C l t


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2s ComplementExample: 2s complement ve number

E l U i N


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Example: Unsigned Number


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2s ComplementExample of Equivalent Representation

2s Complement


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2s Complement2s Representation of 4-bit Numbers & Their Unsigned Equivalent Numbers

2s Complements


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2 s ComplementsRepresentation Characteristics

The representation of ve No. facilitates theH/W implementation of many basic arithmetic

operations This representation is widely used for

executing arithmetic operation in specified

algorithms and general purpose architecture In the example, the minimum +ve No. is 0 and

maximum +ve No. is 7

The min ve No. is -8 There is no equal opposite of -2N-1 in Nbits2s complement representation


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2s Complements

Method1

we move from LSB to MSB leaving thefirst Non-Zero bit as it is and flipping all therest of the bits.

This is certainly not the best way of taking2c complement in H/W

2s complement of 0010 is 1110


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2s Complements


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Overflow

overflow occurs when:

POS+POS=NEG or NEG+NEG=POS

Addition/subtraction and


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Addition/subtraction and

Overflow +3 0011 +5 0101 +5 0101

+4 0100 +6 0110 -6 1010

-------- -------- ---

+7 0111 +11 1011 -1 1111

-5 1011 -3 1101 -5 1011

+6 0110 -4 1100 -6 1010

-------- -------- -------

+1 0001 -7 1001 -11 0101


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Multiplication (Unsigned)

Example: 0101 * 0011 (5 * 3)


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2s Complement Sign Extension

Store 1101 (-3) in an 8 bit register

How to fix the problem??

Example: 0101 * 0011 (5 * 3) Answer = 1111 (is it 15 or -1??)

How to Fix the problem??

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Number Systems for Students