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بسم الله الرحمن الرحيم |
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Conventional number systems: Conventional number systems are also called fixed-radix systems. We use the symbol r to represent the radix. Properties of conventional numbers: ¨ Non-redundant: * Every number has a unique representation, thus * No two sequences have the same numerical value * Digit Set: 0 £ xi £ r-1, where r is the radix * xi ³ r introduces redundancy into the fixed-radix number system ¨ Weighted: * A sequence of weights wn-1,wn-2,...,w1,w0 determines the value of the n-tuple xn-1,xn-2,...,x1,x0 by
* wi is the weight assigned to xi, digit in ith position ¨ Positional: * The weight wi depends only on the position i of digit xi Examples:
The Decimal Number Base Systems The Decimal Number System uses base 10. It includes the digits from 0 through 9. The weighted values for each position are as follows:
You have been using the decimal (base 10) numbering system for so long that you often take it for granted. When you see a number like "123", you don't think about the value 123. Instead, you generate a mental image of how many items this value represents. In reality, however, the number 123 represents: 1 * 10^2 + 2 * 10^1 + 3 * 10^0 = 1 * 100 + 2 * 10 + 3 * 1 = 100 + 20 + 3 = 123 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. For example, the value 725.194 is represented as 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
The Binary Number Base Systems Most modern computer systems (including the IBM PC) operate using binary logic. The computer represents values using two voltage levels (usually 0V for logic 0 and either +3.3 V or +5V for logic 1). With two levels we can represent exactly two different values. These could be any two different values, but by convention, we use the values zero and one. These two values, coincidentally, correspond to the two digits used by the binary number system. The binary number system works like the decimal number system except the Binary Number System: uses base
2 The weighted values for each position is determined as follows:
To make binary numbers more readable, we will add a space every four digits starting from the least significant digit on the left of the decimal point. For example, the binary value 1010111110110010 will be written 1010 1111 1011 0010.
The Octal Number Base System Although this was once a popular number base, especially in the Digital Equipment Corporation PDP/8 and other old computer systems, it is rarely used today. The Octal system is based on the binary system with a 3-bit boundary. The Octal Number System: uses base
8 The weighted values for each position is as follows:
The Hexadecimal Number Base System When dealing with large values, binary numbers quickly become too unwieldy. The hexadecimal (base 16) numbering system solves these problems. Hexadecimal numbers offer the two features: o hex numbers are very compact o it is easy to convert from hex to binary and binary to hex. The Hexadecimal system is based on the binary system using a Nibble or 4-bit boundary. We place an H at the end of the number to denote the number base. The Hexadecimal Number System: uses base
16 To convert a hexadecimal number into a binary number, simply break the binary number into 4-bit groups beginning with the LSB and substitute the corresponding four bits in binary for each hexadecimal digit in the number.
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- Unconventional Number System - Additive & Non-additive Multiplier - Counters |
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