Lect4 Base Conversion

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TA C162

Lecture 4 Base Conversion


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Todays Agenda

Base Conversion

Binary to Decimal - Recap

Decimal to Binary Decimal to Octal & Octal to Decimal

Octal to Binary & Binary to Octal

Binar to Hexadecimal & Hexadecimal to

Binary

Saturday, January 16, 2010 2Biju K Raveendran@BITS Pilani.


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What is Base? Decimal Number System

Base is 10

All numbers are represented by 0 to 9

Binary Number System

ase s

All numbers are represented by 0 and 1

Base is 8

Inference

-



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Converting Binary (2s C) to Decimal8-bit 2s complement Number is of the forma7 a6 a5 a4 a3 a2 a1 a0

n 2n

0 1

ea ng a7 s one; en e num er s

negative take twos complement to get a

positive number; Otherwise number is positive.

1 2

2 4

3 8

The Magnitude is:

a6.26+a5.25+a4.24+a3.23+a2.22+a1.21+a0.20

4 16

5 32

Add powers of 2 that have 1 in the7 128

8 256

. If original number was negative, affix a minus

sign in front.

9 512

10 1024

ssum ng - s comp emen num ers.



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Addition & SubtractionExample 1: 5 + 4 ? (in 5 bit notation)5 001014 00100

Example 2: 9 + - 12 ? (in 5 bit notation)9 01001

-12 101009 + (-12) 11101 (-3)Example 3: -3 + -7 ? (in 5 bit notation)

3 00011- 7 00111-7 11001

- + - -Saturday, January 16, 2010 6Biju K Raveendran@BITS Pilani.


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Converting Decimal to Binary (2s C)

First Method: Division

+

Divide by two remainder is least significant bit. Keep dividing by two until answer is zero, writing

remainders from right to left.

Append a zero as the MS bit.

If original number was negative, take twos

. Example: X = 105ten



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Example:

X = 105ten

=

52/2 = 26 r 0 bit 1

= r

13/2 = 06 r 1 bit 36/2 = 03 r 0 bit 4

3/2 = 01 r 1 bit 5

1/2 = 0 r 1 bit 6

X = 01101001



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Converting Decimal to Binary (2s C) Second Method: Subtract Powers of Two

n

Subtract largest power of two less than or equalto number.

0 1

1 2

2 4

Put a 1 in the corresponding bit position.

Keep subtracting until result is zero.

3 8

4 16

5 32

ppen a zero as ; or g na was

negative, take twos complement.

Put a 0 in the other bit ositions

6 64

7 128

8 256

10 1024



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Example:

= = en

40 32 = 8 bit 6 is 1

X = 01101000two



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Algorithm

-

Step 1:

Step 2:

,

If N is negative, then form negative of this 2s



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Decimal to Octal1. Find magnitude of decimal number. (Always +ve)

2. Divide by Eight remainder is least significant bit.

. eep v ng y g un answer s zero, wr ng

remainders from right to left.

Example: Convert 87410 to its octal equivalent.

Octal to Decimal

Sum the multiplication of each digit with its position value

Example: Convert 6248 to its decimal equivalent.6 x 82 + 2 x 81 + 4 x 80 = 40410



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Hexadecimal Notation It is more convenient to write binar base-2 numbers as

hexadecimal (base-16) numbers.

Why?

ewer g s -- our s per ex g se u or ea ng w ong s r ngs

of binary digits)

Less error prone -- easy to corrupt long string of 1s and 0s

Binary Hex Decimal

0000 0 0

Binary Hex Decimal

1000 8 8

0010 2 2

0011 3 3

1010 A 10

1011 B 11

0101 5 5

0110 6 6

1101 D 13

1110 E 14



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Hexadecimal NotationA 16-bit binary string takes the forma15 a14 a13 a12 a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0

uns gne n eger s va ue can e compu e as

a15.215 + a14.2

14 + a13.213 + a12.2

12 + a11.211 + a10.2

10 + a9.29 +

8 7 6 5 4 3 2 1 08. 7. 6. 5. 4. 3. 2. 1. 0.

Can be represented as Observe212 [a15.2

3 + a14.22 + a13.2

1 + a12.20] +

28 [a11.23 + a10.2

2 + a9.21 + a8.2

0] +

Value inside the

square bracket

a7.

a6.

a5.

a4. 20 [a3.2

3 + a2.22 + a1.2

1 + a0.20]

= 3 2 1 0

0 to 15

. . . .



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Documents

Lect4 Base Conversion