Lecture Three concerns NUMBERING SYSTEM.pptx

It's a power point presentation about numbering system

1. NUMBERING SYSTEM A number system is a method of writing numbers that is a mathematical way of representing the numbers of a given set by mathematically using numbers or symbols. A number system is defined as a writing system for denoting numbers logically using digits or symbols .

2. Types of Number System The four common types of Number System are:  Decimal Number System Binary Number System Octal Number System Hexadecimal Number System

3. Here, units has the place value as 10 , tens has the place value as 10 1 , hundreds as 10 2 , thousands as 10 3 , and so on.  (1 × 10 4 ) + (0 × 10 3 ) + (2 × 10 2 ) + (8 × 10 1 ) + (5 × 10 ) 1 × 10000 + 0 × 1000 + 2 × 100 + 8 × 10 + 5 × 1 10000 + 0 + 200 + 80 + 5 10285

4. Binary Number System Number System with base value 2 is termed as Binary number system. It uses 2 digits i.e. 0 and 1 for the creation of numbers. The numbers formed using these two digits are termed as Binary Numbers. Binary number system is very useful in electronic devices and computer systems because it can be easily performed using just two states ON and OFF i.e. 0 and 1.  Decimal Numbers 0-9 are represented in binary as: 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001 In a binary number system, each digit is called the “bit”.

5. Binary to Decimal Conversion A binary number is converted into a decimal number by multiplying each digit of the binary number by the power of either 1 or 0 to the corresponding power of 2. Convert (10011) 2 to a decimal number. (10011) 2 = (1 × 2 4 ) + (0 × 2 3 ) + (0 × 2 2 ) + (1 × 2 1 ) + (1 × 2 ) = 16 + 0 + 0 + 2 + 1 = (19) 10 Hence, the binary number (10011) 2 is expressed as (19) 10 .

6. Decimal to Binary Conversion A decimal number is converted into a binary number by dividing the given decimal number by 2 continuously until we get the quotient as 1, and we write the numbers from downwards to upwards. Example: Convert (28) 10 into a binary number Exercise Convert the number 25 10 to the binary system

7. Octal Number System Octal Number System is one in which the base value is 8. It uses 8 digits i.e. 0-7 for creation of Octal Numbers. Octal Numbers can be converted to Decimal value by multiplying each digit with the place value and then adding the result. Here the place values are 8 , 8 1 , and 8 2 . Example on Decimal to Octal Suppose 560 is a decimal number, convert it into an octal number. Solution : If 560 is a decimal number, then, 560/8 = 70 and the remainder is 0 70/8 = 8 and the remainder is 6 8/8 = 1 and the remainder is 0 And 1/8 = 0 and the remainder is 1 So the octal number starts from MSD to LSD, i.e. 1060 Therefore, 560 10  = 1060 8 Exercise Convert 85 10 to the octal number system

8. Octal to Decimal To convert an octal number to a decimal number we need to multiply each digit of the given octal with the reducing power of 8.  Example 1: Suppose 215 8  is an octal number, then it’s decimal form will be, 215 8 = 2 × 8 2 + 1 × 8 1 + 5 × 8           = 2 × 64+ 1 × 8 + 5 × 1 = 128 + 8 + 5           =  141 10 Example 2:  Let 125 is an octal number denoted by 125 8 . Find the decimal number. 125 8  = 1× 8 2  + 2 × 8 1  + 5 × 8         = 1 × 64 + 2 × 8 + 5 × 1 = 64+16+5         =85 10

9. Binary To Octal Number Binary To Octal Number A binary number can be converted into an octal number, with the help of the below-given table. Octal Number Equivalent Binary Number 1 1 2 10 3 11 4 100 5 101 6 110 7 111

10. Binary to Octal Example: Convert (100010) 2 to an octal number. Solution: With the help of the table we can write, 100→4 and 010→2 Therefore, (100010) 2 = 42 Similarly, we can convert an octal number to a binary number with the help of the table.

11. Hexadecimal Number System Number System with base value 16 is termed as Hexadecimal Number System. It uses 16 digits for the creation of its numbers. Digits from 0-9 are taken like the digits in the decimal number system but the digits from 10-15 are represented as A-F i.e. 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. Hexadecimal Numbers are useful for handling memory address locations.  The hexadecimal number system is sometimes also represented as, ‘hex’.

12. Hexadecimal is a number system with a base value of 16. Hexadecimal numbers use 16 symbols or digital values: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A, B, C, D, E, and F represent 10, 11, 12, 13, 14, and 15 in single-bit form. If you see an “0x” as Prefix, it indicates the number is in Hexadecimal. For example, 0x3A The position of each digit in a Hexadecimal number has a weight of 16 to the power of its position.

13. Hexadecimal Number System Table Hexadecimal Decimal Binary 0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111

14. 8 8 1000 9 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111

15. Hexadecimal to Decimal Conversion Converting hexadecimal to decimal follows a similar process as before, where each digit is multiplied by the respective power of 16. Example: Convert (A7B) 16 to decimal. A7B) 16 = A × 16 3 + 7 × 16 2 + B × 16 1 ⇒ (A7B) 16 = 10 × 4096 + 7 × 256 + 11 × 16 (convert symbols A and B to their decimal equivalents; A = 10, B = 11) ⇒ (A7B) 16 = 40960 + 1792 + 176 ⇒ (A7B) 16 = 42828 Therefore, the decimal equivalent of (A7B) 16 is (42828) 10 .

16. Decimal to Hexadecimal Conversion