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NUMBER SYSTEM

NUMBER SYSTEM. Objectives. Understand why computers use binary (Base-2) numbering. Understand how to convert Base-2 numbers to Base-10 or Base-8. Understand how to convert Base-8 numbers to Base-10 or Base 2. Understand how to convert Base-16 numbers to Base-10, Base 2 or Base-8.

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NUMBER SYSTEM

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  1. NUMBER SYSTEM

  2. Objectives • Understand why computers use binary (Base-2) numbering. • Understand how to convert Base-2 numbers to Base-10 or Base-8. • Understand how to convert Base-8 numbers to Base-10 or Base 2. • Understand how to convert Base-16 numbers to Base-10, Base 2 or Base-8.

  3. Types Of Numbers • Natural Numbers • The number 0 and any number obtained by repeatedly adding a count of 1 to 0 • Negative Numbers • A value less than 0 • Integer • A natural number, the negative of a natural number, and 0. • So an integer number system is a system for ‘counting’ things in a simple systematic way Example = -5 -4 -3 -2 -1 0 1 2 3 4 5

  4. Number systems include decimal, binary, octal and hexadecimal • Each system have four number base

  5. Why Binary System? • Computers are made of a series of switches • Each switch has two states: ON or OFF • Each state can be represented by a number – 1 for “ON” and 0 for “OFF”

  6. (1 0 0 1 1)2 Converting Base-2 to Base-10 OFF OFF ON/OFF ON ON ON Exponent: 21 24 23 22 20 16 0 0 2 1 = Calculation: + + + + (19)10

  7. 1.2 The Binary Number Base Systems • Most modern computer system using binary logic. The computer represents values(0,1) using two voltage levels (usually 0V for logic 0 and either +3.3 V or +5V for logic 1). • The Binary Number System uses base 2 includes only the digits 0 and 1 • The weighted values for each position are : Base

  8. Position weights 22 21 20 Number digits 1 1 0 0 x20 = 0 + 1 x21 = 2 + 1 x 22 = 4 6 Binary Numbering System • How is a positive integer represented in binary? • Let’s analyze the binary number 110: 110 = (1 x 22) + (1 x 21) + (0 x 20) = (1 x 4) + (1 x 2) + (0 x 1) • So a count of SIX is represented in binary as 110

  9. 1.1 Decimal Number System • The Decimal Number System uses base 10. It includes the digits {0, 1,2,…, 9}. The weighted values for each position are: Base left of the decimal point Rightof decimal point

  10. Each digit appearing to the left of the decimal point represents a value between zero and nine times power of ten represented by its position in the number. • Digits appearing to the right of the decimal point represent a value between zero and nine times an increasing negative power of ten. • Example:the value 725.194 is represented inexpansion formas follows: • 7 *10^2+ 2 * 10^1+ 5 *10^0 + 1 *10^-1+ 9 *10^-2+ 4 * 10^-3 • =7 * 100+ 2 *10+ 5 *1 + 1 *0.1+ 9 *0.01+ 4 *0.001 • =700 + 20 + 5 + 0.1 + 0.09 + 0.004 • =725.194

  11. Position weights 102 101 100 Number digits 3 7 5 5 x100 = 5 + 7 x101 = 70 + 3 x 102 = 300 375 Decimal Numbering System • How is a positive integer represented in decimal? • Let’s analyze the decimal number 375: 375 = (3 x 100) + (7 x 10) + (5 x 1) = (3 x 102) + (7 x 101) + (5 x 100)

  12. Example: convert the number 333 to binary.

  13. OCTAL NUMBER SYSTEM • Octal Number system base of 8 • It has eight digit Numbers 0 1 2 3 4 5 6 & 7 • Octal number value is 0 to 7 • The octal Number system is also a positional number system. Each octal digit its own positional value or weight expressed as a power of 8

  14. Octal Numbering System • Base: 8 • Digits: 0, 1, 2, 3, 4, 5, 6, 7 • Octal number: 3578 = (3 x 82 ) + (5 x 81) + (7 x 80) • To convert to base 10, beginning with the rightmost digit, multiply each nth digit by 8(n-1), and add all of the results together.

  15. Hexadecimal Number System • It’s also called base 16 number system Since it consists number between 0 to 15.But we can represent 0-9 only. Then, how to represent 10,11,12,13,14,15. • In hexadecimal, we will use A to F for remaining 6 numbers i.e. 10 to 15.Like,Hexadecimal representation for 10 is A, 11-B 12-C, 13-D,14-E,15-F.

  16. Hexadecimal (Hex)Numbering System • Base: 16 • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • Hexadecimal number: 1F416 = (1 x 162 ) + (F x 161) + (4 x 160)

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