notes-number-systems - CSE103 - Number Systems Last...

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CSE103 - Number Systems Prepared by Doug Hogan for 9/10/2002 and 9/12/2002 Last modified: 09/08/02 Intro | Binary Definition | Binary --> Decimal | | Any Base --> Decimal | Decimal --> Other Bases | Binary --> Hex, Octal | Addition | Addition Illustrated Number Systems in General A Brief Evolution We need a way to write down numbers "Tally" marks Roman numerals Arabic numerals What Defines a Number System A number system is defined by two entities: A radix or base A set of symbols The radix essentially defines how many symbols the number system has. In general, a number system with radix n has symbols that could be defined as 0, . .., n-1 . Decimal Numbers Radix 10 Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 As computer scientists we are interested in other number systems. Intro | Binary Definition | Binary --> Decimal | | Any Base --> Decimal | Decimal --> Other Bases | Binary --> Hex, Octal | Addition | Addition Illustrated Binary Number System Defined by Radix 2 Symbols: 0, 1 Each digit is referred to as a bit , short for binary digit The symbols 0 and 1 are often thought of like "true" and "false" or "on" and "off." At the most basic (least abstract) level, computers are built of transistors that can assume the states "on" and "off," so everything we
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do with computers is translated to 0s and 1s in order for the computer to understand it. Thus, binary is important to us! Examples Decimal Binary 0 0 1 1 2 10 we've counted as high as we can in the ones place, so we need to introduce a new column 3 11 4 100 another new place 5 101 The Anatomy of a Binary Number 1 0 1 1 0 0 1 0 msb most significan t bit lsb least significant bit By the way, this number is what we refer to as 8-bit . Intro | Binary Definition | Binary --> Decimal | | Any Base --> Decimal | Decimal --> Other Bases | Binary --> Hex, Octal | Addition | Addition Illustrated ...So What Does That Mean???? Converting a Binary Number to a Decimal Number Let's look at that previous example again. ... Each bit corresponds to a power of 2. The least significant bit corresponds to the power 0 (NOT 1 - here is just one of the many places where you should get in the habit of counting from zero). As we move toward the
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msb, the powers of two increase by one. the number
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notes-number-systems - CSE103 - Number Systems Last...

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