NUMBER SYSTEM

Natural Numbers

The natural numbers can be represented on a number line as shown below.

4 + 2 = 6, again a natural number;

6 + 21 = 27, again a natural number;

22 – 6 = 16, again a natural number, but

2 – 6 is not defined in natural numbers.

Similarly, 4 × 3 = 12, again a natural number

12 × 3 = 36, again a natural number

Two natural numbers can be added and multiplied in any order and the result obtained is always same. This does not hold for subtraction and division of natural numbers.

Whole Numbers

When a natural number is subtracted from itself we can

not say what is the left out number. To remove this

difficulty, the natural numbers were extended by the

number zero (0), to get what is called the system of whole

numbers

Thus, the whole numbers are 0, 1, 2, 3, ...........

The number 0 has the following properties:

a + 0 = a = 0 + a

a – 0 = a but (0 – a) is not defined in whole numbers

a × 0 = 0 = 0 × a

Division by zero (0) is not defined.

Integers

While dealing with natural numbers and whole numbers we

found that it is not always possible to subtract a number from

another.

For example, (2 – 3), (3 – 7), (9 – 20) etc. are all not

possible in the system of natural

numbers and whole numbers. Thus, it needed another

extension of numbers which allow such subtractions.

Natural Numbers

Whole Numbers

Integers

Numeral System:-

b - numeral system base

dn - the n-th digit

n - can start from negative number if the number has a fraction part.

N+1 - the number of digits

Binary Numeral System - Base-2

Binary numbers uses only 0 and 1 digits.

B denotes binary prefix

Examples:

101012 = 10101B = 1×24+0×23+1×22+0×21+1×20

= 16+4+1= 21

101112 = 10111B = 1×24+0×23+1×22+1×21+1×20

= 16+4+2+1= 23

1000112 = 100011B =

1×25+0×24+0×23+0×22+1×21+1×20=32+2+1= 35

Decimal Numeral System - Base-10

Decimal numbers uses digits from 0..9…. These are the

regular numbers that we use.

Example:

253810 = 2×103+5×102+3×101+8×100

Conversion of binary to decimal ( base

2 to base 10)

Example: convert (1000100)2 to decimal

= 64 + 0 + 0+ 0 + 4 + 0 + 0

= (68)10

Conversion of decimal to binary ( base 10 tobase2)

Example: convert (68)10 to binary

68/¸2 = 34 remainder is 0

34/ 2 = 17 remainder is 0

17 / 2 = 8 remainder is 1

8 / 2 = 4 remainder is 0

4 / 2 = 2 remainder is 0

2 / 2 = 1 remainder is 0

1 / 2 = 0 remainder is 1

Answer = 1 0 0 0 1 0 0

Note: the answer is read from bottom (MSB) totop (LSB) as 10001002

Conversion of decimal fraction to

binary fraction

Instead of division , multiplication by 2 is carried out and the integer part of the result is saved and placed after the decimal point.

The fractional part is again multiplied by 2 and the process repeated.

Example: convert ( 0.68)10 to binary fraction.

0.68 * 2 = 1.36 integer part is 1

0.36 * 2 = 0.72 integer part is 0

0.72 * 2 = 1.44 integer part is 1

0.44 * 2 = 0.88 integer part is 0

Answer = 0. 1 0 1 0…..

Example: convert ( 68.68)10 to binary equivalent.

Answer = 1 0 0 0 1 0 0 . 1 0 1 0….

Octal Number System

Base or radix 8 number system.

1 octal digit is equivalent to 3 bits.

Octal numbers are 0 to7. (see the chart down below)

Numbers are expressed as powers of 8

Conversion of octal to decimal

( base 8 to base 10)

Example: convert (632)8 to decimal

= (6 x 82) + (3 x 81) + (2 x 80)

= (6 x 64) + (3 x 8) + (2 x 1)

= 384 + 24 + 2

= (410)10

Examples:

278 = 2×81+7×80 = 16+7 = 23

308 = 3×81+0×80 = 24

43078 = 4×83+3×82+0×81+7×80= 2247

Conversion of decimal to octal ( base

10 to base 8)

Example: convert (177)10 to octal

177 / 8 = 22 remainder is 1

22 / 8 = 2 remainder is 6

2 / 8 = 0 remainder is 2

Answer = 2 6 1

Note: the answer is read from bottom to top as(261)8, the same

as with the binary case.

Note:-Conversion of decimal fraction to octal fraction is

carried out in the same manner as decimal to binary

except that now the multiplication is carried out by 8.

Hexadecimal Number System

Base or radix 16 number system.

•1 hex digit is equivalent to 4 bits.

•Numbers are 0,1,2…..8,9, A, B, C, D, E, F.

B is 11, E is 14

Numbers are expressed as powers of 16.

160 = 1, 161 = 16, 162 = 256, 163 = 4096, 164 =65536, …

Conversion of hex to decimal ( base

16 to base10)

Example: convert (F4C)16 to decimal

= (F x 162) + (4 x 161) + (C x 160)

= (15 x 256) + (4 x 16) + (12 x 1)

Conversion of decimal to hex (base 10 to base16)

Example: convert (4768)10 to hex.

= 4768 / 16 = 298 remainder 0

= 298 / 16 = 18 remainder 10 (A)

= 18 / 16 = 1 remainder 2

= 1 / 16 = 0 remainder 1

Answer: 1 2 A 0

Conversion of binary to octal and hex

Conversion of binary numbers to octal and hex simply

requires grouping bits in the binary numbers into groups

of three bits for conversion to octal and into groups of four

bits for conversion to hex

Groups are formed beginning with the LSB and progressing

to the MSB.

Thus, 11 100 1112 = 3478

11 100 010 101 010 010 0012 = 30252218

1110 01112 = E716

11000 1010 1000 01112 = 18A8716

Decimal, Binary, Octal, and Hex

Numbers

Decimal Binary Octal Hexadecimal

0 0000 0 0

1 0001 1 1

2 0010 2 2

3 0011 3 3

4 0100 4 4

5 0101 5 5

6 0110 6 6

7 0111 7 7

8 1000 10 8

9 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

Representing signed integer numbers

inside a computer

1. Sign-Magnity Representation Method

2. The two's complement encoding method

A straight forward encoding for signed numbers is

the sign-magnitude encoding method:

First bit of the data represents the sign of the integer

number

The remaining bits of the data represents the magnitude of

the integer number

Sign-Magnity Representation Method

A straight forward encoding for signed numbers is

the sign-magnitude encoding method:

First bit of the data represents the sign of the integer

number

The remaining bits of the data represents the magnitude of

the integer number

Example: - 8 bits representation

Problems with the Sign-Magnity Representation

Method

There are two different representations for ZERO

0000....0000 (+0) and

1000....0000 (-0)

The two's complement encoding

method

Modern Computers use the two's complement

encoding method to represent signed integer numbers.

2s complement presentation for positive values is same as

that used in binary number system.

Example:

2s complement representation

for negative values added to the binary

number for its absolute value is equal to

100000000.

Converting a value v to its 2's

complement code:

If value v is positive, then:

the 2's complement code is the same as its representation in the

binary number system

If value v is negative, then: First, obtain the binary number

representation x for v then do 1’s complement and then add

1 to it.

Example:

v = 7 The value is positive, so:

(1) Binary number representation is: 111

(2) 8 digit 2's complement representation is: 00000111

16 digit 2's complement representation is:

0000000000000111 and so on...

v = -7 The value is negative, so:

(1) Binary number representation for 7 is: 111

(2) 8 digit Binary number representation for 7 is:

: 00000111

(2a) 1's complement representation for 7 is:

: 11111000

(3a) 2's complement representation for -7 is:

: 11111001