Computer Organization & Architecture (NUMBER SYSTEMS )

NUMBER SYSTEMS 

1. Decimal Number System

• The Decimal system is a base-10 number system.
• It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
• Each digit’s place value is a power of 10 (e.g., 100, 101, 102).
• It is the standard system for everyday counting and calculations.

2. Binary Number System

• The Binary system is a base-2 number system.
• It uses two digits: 0 and 1.
• Each digit’s place value is a power of 2 (e.g., 20, 21, 22).
• The Binary system is the foundation for data representation in computers and digital electronics.

3. Octal Number System

• The Octal system is a base-8 number system.
• It uses eight digits: 0, 1, 2, 3, 4, 5, 6 and 7.
• Each digit’s place value is a power of 8 (e.g., 80, 81, 82).
• It is often used to simplify the representation of binary numbers by grouping them into sets of three bits.

4. Hexadecimal Number System

• The Hexadecimal system is a base-16 number system.
• It uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F (where A = 10, B = 11, etc.).
• Each digit’s place value is a power of 16 (e.g., 160, 161, 162).
• Hexadecimal simplifies binary by representing every 4 bits as one digit (0-F).

CONVERSIONS AMONG NUMBER SYSTEMS

Number System Conversion Methods

A number N in base or radix b can be written as: 

(N)b = dn-1 dn-2 -- -- -- -- d1 d0 . d-1 d-2 -- -- -- -- d-m

In the above, dn-1 to d0 is the integer part, then follows a radix point and then d-1 to d-m is the fractional part. 

dn-1 = Most significant bit (MSB) 
d-m = Least significant bit (LSB)

1. Decimal to Binary Number System Conversion

For Integer Part:

• Divide the decimal number by 2.
• Record the remainder (0 or 1).
• Continue dividing the quotient by 2 until the quotient is 0.
• The binary equivalent is the remainders read from bottom to top.

For Fractional Part:

• Multiply the fractional part by 2.
• Record the integer part (0 or 1).
• Take the fractional part of the result and repeat the multiplication.
• Continue until the fractional part becomes 0 or reaches the desired precision.
• The binary equivalent is the integer parts recorded in sequence.

2. Binary to Decimal Number System Conversion

For Integer Part:

• Write down the binary number.
• Multiply each digit by 2 raised to the power of its position, starting from 0 (rightmost digit).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the binary integer.

For Fractional Part:

• Write down the binary fraction.
• Multiply each digit by 2 raised to the negative power of its position, starting from -1 (first digit after the decimal point).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the binary fraction.

Example: (1010.01)2 

1x23 + 0x22 + 1x21+ 0x20 + 0x2 -1 + 1x2 -2 = 8+0+2+0+0+0.25 = 10.25 

Thus, (1010.01)2 = (10.25)10 

3. Decimal to Octal Number System Conversion

For Integer Part:

• Divide the decimal number by 8.
• Record the remainder (0 to 7).
• Continue dividing the quotient by 8 until the quotient is 0.
• The octal equivalent is the remainders read from bottom to top.

For Fractional Part:

• Multiply the fractional part by 8.
• Record the integer part (0 to 7).
• Take the fractional part of the result and repeat the multiplication.
• Continue until the fractional part becomes 0 or reaches the desired precision.
• The octal equivalent is the integer parts recorded in sequence.

Example: (10.25)10 

For Integer Part (10):

• Divide 10 by 8 → Quotient = 1, Remainder = 2
• Divide 1 by 8 → Quotient = 0, Remainder = 1

Octal equivalent = 12 (write the remainder, read from bottom to top). So, the octal equivalent of the integer part 10 is 12.

For Fractional Part (0.25):

• Multiply 0.25 by 8 → Result = 2.0, Integer part = 2

The fractional part ends here as the result is now 0. So, the octal equivalent of the fractional part 0.25 is 0.2.

The octal equivalent of (10.25)10 = (12.2)8 

4. Octal to Decimal Number System Conversion

For Integer Part:

• Write down the octal number.
• Multiply each digit by 8 raised to the power of its position, starting from 0 (rightmost digit).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the octal integer.

For Fractional Part:

• Write down the octal fraction.
• Multiply each digit by 8 raised to the negative power of its position, starting from -1 (first digit after the decimal point).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the octal fraction.

Example: (12.2)8

1 x 81 + 2 x 80 +2 x 8-1 = 8+2+0.25 = 10.25 

Thus, (12.2)8 = (10.25)10 

5. Decimal to Hexadecimal Conversion

For Integer Part:

• Divide the decimal number by 16.
• Record the remainder (0-9 or A-F).
• Continue dividing the quotient by 16 until the quotient is 0.
• The hexadecimal equivalent is the remainders read from bottom to top.

For Fractional Part:

• Multiply the fractional part by 16.
• Record the integer part (0-9 or A-F).
• Take the fractional part of the result and repeat the multiplication.
• Continue until the fractional part becomes 0 or reaches the desired precision.
• The hexadecimal equivalent is the integer parts recorded in sequence.

Example: (10.25)10

Integer part:

• 10 ÷ 16 = 0, Remainder = A (10 in decimal is A in hexadecimal)

Hexadecimal equivalent = A

Fractional part:

• 0.25 × 16 = 4, Integer part = 4

Hexadecimal equivalent = 0.4

Thus, (10.25)10 = (A.4)16

6. Hexadecimal to Decimal Conversion

For Integer Part:

• Write down the hexadecimal number.
• Multiply each digit by 16 raised to the power of its position, starting from 0 (rightmost digit).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the hexadecimal integer.

For Fractional Part:

• Write down the hexadecimal fraction.
• Multiply each digit by 16 raised to the negative power of its position, starting from -1 (first digit after the decimal point).
• Add up the results of these multiplications.
• The sum is the decimal equivalent of the hexadecimal fraction.

Example: (A.4)16

(A × 160) + (4 × 16-1) = (10 × 1) + (4 × 0.0625)

Thus, (A.4)16 = (10.25)10

7. Hexadecimal to Binary Number System Conversion

To convert from Hexadecimal to Binary:

• Each hexadecimal digit (0-9 and A-F) is represented by a 4-bit binary number.

• For each digit in the hexadecimal number, find its corresponding 4-bit binary equivalent and write them down sequentially.

Example: (3A)16

• (3)16 = (0011)2
• (A)16 = (1010)2

Thus, (3A)16 = (00111010)2 

8. Binary to Hexadecimal Number System Conversion

To convert from Binary to Hexadecimal:

• Start from the rightmost bit and divide the binary number into groups of 4 bits each.
• If the number of bits isn't a multiple of 4, pad the leftmost group with leading zeros.
• Each 4-bit binary group corresponds to a single hexadecimal digit.
• Replace each 4-bit binary group with the corresponding hexadecimal digit.

Example: (1111011011)2

0011 1101 1011
| | |
3 D B

Thus, (001111011011 )2 = (3DB)16 

9. Binary to Octal Number System

To convert from binary to octal:

• Starting from the rightmost bit, divide the binary number into groups of 3 bits.
• If the number of bits is not a multiple of 3, add leading zeros to the leftmost group.
• Each 3-bit binary group corresponds to a single octal digit.
• Replace each 3-bit binary group with the corresponding octal digit.

Example: (111101101)2

111 101 101
| | |
7 5 5

Thus, (111101101)2 = (755)8

10. Octal to Binary Number System Conversion

To convert from octal to binary:

• Each octal digit (0-7) corresponds to a 3-bit binary number.
• For each octal digit, replace it with its corresponding 3-bit binary equivalent.

Example: (153)8

• Break the octal number into digits: 1, 5, 3
• Convert each digit to binary:
  • 1 in octal = 001 in binary
  • 5 in octal = 101 in binary
  • 3 in octal = 011 in binary

Thus, (153)8 = (001101011)2