14:332:231 Lecture Notes - Lecture 2: Radix, Octal, Negative Number
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18 Jul 2018
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CHAPTER TWO: NUMBER SYSTEMS AND CODES
2.1 Positional Number System
●Positional Number System: number represented by a string of digits
○ Each digit position has an associated weight
■ 1734 = 1*1000 + 7*100 + 3*10 + 4*1
■ Here, 10 is called the base
or radix
■ Radix Point
is where there are p digits to the left and n digits to the right of
a point
●Most Significant Digit (MSD): leftmost digit
●Least Significant Digit (LSD): rightmost digit
●Binary Digits (Bits): used with values 0 and 1 to represent digital signals, has binary
radix of r=2
○ Examples:
● Octal, Hexadecimal, 3-Bit/4-Bit Strings
○ Octal: radix r=8 (needs 8 digits), uses digits 0-7
○ Hexadecimal: radix r=16 (needs 16 digits), uses digits 0-9 and A-F
● Conversions
○Binary-to-Octal
■ Separate bits into groups of 3
■ Start separating from the right and fill in zeros to the left to create the
groups of 3
■ Replace each set of 3 with the corresponding octal digit
■ Example:
○Binary-to-Hexadecimal
■ Separate bits into groups of 4
■ Start separating from the right and fill in zeros to the left to create the
groups of 4
■ Replace each set of 4 with the corresponding hexadecimal digit
■ Example:
○Octal/Hexadecimal-to-Binary
■ Replace each octal or hexadecimal digit with the corresponding 3 or 4 bit
string that is binary
■ Examples:
● Nibble: a 4-bit hexadecimal digit
● 0x Prefix: prefix used to denote a hexadecimal number (Ex: 0xBFC0000)
2.3 General Positional-Number-System Conversions
●Radix-R-to-Decimal Conversion
○ Simply evaluate
○ Take each digit of radix r and define it based on the radix
○ Examples: