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Binary Number System: Conversions and Representation

Explore the fundamentals of binary numbers, conversions to decimal, octal, and hexadecimal systems, and how computers utilize the binary system. Learn about decimal, binary, octal, and hexadecimal number systems, including the significance of each system's base and positional values.

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Binary Number System: Conversions and Representation

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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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