Number Systems - Part II

CS 215 Lecture # 6

Conversion from Decimal

Let N be the number to be converted. We can write N as

N = q*b + r

whereq = quotient b = the base into which N is to be

convertedr = remainder

Example 6.1

Convert 8010 to binary (base 2)80 = 40*2 + 0

40 = 20*2 + 0

20 = 10*2 + 0

10 = 5*2 + 0

5 = 2*2 + 1

2 = 1*2 + 0

1 = 0*2 + 1

Answer: 10100002

Example 6.2

Convert 8010 to octal(base 8)

80 = 10*8 + 0

10 = 1*8 + 2

1 = 0*8 + 1

Answer: 8010 = 1208

Also, from binary:8010 = 10100002 = 1 010 000 = 120

Example 6.3

Convert 8010 to hexadecimal(base 16)

80 = 5*16 + 0

5 = 0*16 + 5

Answer: 8010 = 5016

Also, from binary:8010 = 10100002 = 101 0000 = 50

Decimal Binary or HexadecimalRepeated division by 2 or 16

Example: Convert 292 to hexadecimal.

292/16 = 18 R 418/16 = 1 R 21/16 = 0 R 1

292 = 12416

Conversion from Decimal

When the quotient is less than 16, the process ends

Conversion from Decimal

Convert 42 to hexadecimal 42/16 = 2 R 10 (but 10 = a16) 2/16 = 0 R 2 42 = 2a16

Convert 109 to binary 109/2 = 54 R 1 54/2 = 27 R 0 27/2 = 13 R 1 13/2 = 6 R 1 6/2 = 3 R 0 3/2 = 1 R 1 1/2 = 0 R 1 109 = 11011012

Shortcut: convert to hexadecimal

It is sometimes easier to convert from binary to hexadecimal, then to decimal, instead of converting directly to decimal 11101010110110 = 0011 1010 1011 0110 = 3 a b 6 = 3*16^3 + 10*16^2 + 11*16 + 6 = 3*4096 + 10*256 + 176 + 6 = 12198 + 2560 + 182 = 12198 + 2742 = 14940

Shortcut: convert to hexadecimal

The converse is also true: converting from decimal to hexadecimal, then to binary generally requires far fewer steps than converting directly to binary 278/16 = 17 R 6 17/16 = 1 R 1 1/16 = 0 R 1 278 = 11616 = 0001 0001 01102 =

1000101102

Example 6.4

Convert the following:5010 = ____16

5010 = ____8

5010 = ____2

5010 = ____5

Conversion of Fractions

A decimal improper fraction a/b is converted into a different base b by converting it into a mixed fraction a/b = d e/f

e.g. 13/8 = 1 5/8 convert d into the required base using the

techniques we have already discussed convert e/f (now a proper fraction) into the

required base by following the steps below the answer is in the format ...f1f0.f-1f-2...

Example 6.5

Convert 2 5/8 to binary

2*(5/8) = 1 + 1/4

2*(1/4) = 0 + 1/2

2*(1/2) = 1 + 0

Answer: 10.101

Example 6.6

Convert 3.6 to binary

2*.6 = 1 + .2

2*.2 = 0 + .4

2*.4 = 0 + .8

2*.8 = 1 + .6

2*.6 = 1 + .2

2*.2 = 0 + .4

2*.4 = 0 + .8

2*.8 = 1 + .6

Answer: 11.10011001...

Converting back into decimal

10.101

fraction part

= 1*(1/2)1 + 0*(1/2)2 + 1*(1/2)3

= .625 or 5/8

integer part

= 1*(21) + 0*(20)

= 2

Answer: 2 5/8

Scientific Notation

Given a number N, we can express N as

N = s*rx

wheres = mantissa or significand

r = radix

x = exponent

The mantissa can be made an integer by multiplying it with a fixed power of the radix and subtracting the appropriate integer constant from the exponent

For example,2.358 * 101 = 2358 * 10-2

Normalization

Standard scientific notation defines the normal form to meet the following rule.

1 s < rwhere s = significand

r = radix A representation is in normal form if

the radix point occurs just to the right of the first significant symbol

Example 6.7

2358 * 10-2 = 2.358 * 101

1525 * 10-1 = 1.525 * 102

Representing Numbers

Choosing an appropriate representation is a critical decision a computer designer has to make

The chosen representation must allow for efficient execution of primitive operations

For general-purpose computers, the representation must allow efficient algorithms for addition of two integers determination of additive inverse

Representing Numbers

With a sequence of N bits, there are 2N unique representations

Each memory cell can hold N bits The size of the memory cell

determines the number of unique values that can be represented

The cost of performing operations also increases as the size of the memory cell increases

It is reasonable to select a memory cell size such that numbers that are frequently used are represented

Binary Representation

The binary, weighted positional notation is the natural way to represent non-negative numbers.

MAL numbers the bits from right to left, beginning with 0 as the least significant digit.

31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Modulo Arithmetic

Consider the set of number {0, … ,7}

Suppose all arithmetic operations were finished by taking the result modulo 8

3 + 6 = 9, 9 mod 8 = 1 3 + 6 = 1

3*5 = 15, 15 mod 8 = 7 3 * 5 = 7

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0

Modulo Arithmetic: Additive Inverse

What is the additive inverse of 7?

7 + x = 0 7 + 1 = 0 0 and 4 are their own

additive inverses

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0