# 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: