Class 2: Numbering Why different Systems numbering systems?
• Numbering systems – Decimal – Binary – Hexadecimal • Hexadecimal to Binary Conversion • Data Organization • Homework 1 -> due a week from today!
Different ways of Decimal System representing the same (a review) value • Commonly used by: people! • Each digit represents a power of 10 • Digits go from 0 to 9 • Also known as Base 10 • Example:
123 = • How many frogs? 10 If moving to the right of the decimal point, powers decrease:
2.4510 =
1 Binary System, Binary System continued
• Used by: computer internal • For convenience, a numeric representation. value can be assigned to each • In binary, each digit represents bit position:
a power of 2. x7 x6 x5 x4 x3 x2 x1 x0 • Binary numbers are made entirely of 0 and 1. • Also known as Base 2. • Binary digits are also known as bits.
Binary to Decimal Decimal to Binary Conversion Conversion: Method 1
• multiply the digit by 2 to the • Divide by Decreasing Powers of power of the bit position 2
• Examples:
1012 =
Example: Converting 7610 to Binary
110010102 =
2 Decimal to Binary Hexadecimal System Conversion: Method 2
• Disadvantage: need to know • Used by: Debug utility. the powers of two • Also used as a more compact • Alternative: repeatedly divide way to represent numbers rather by 2 than binary. • In hexadecimal (or hex), each -- picture from Irvine, p 575 digit represents a power of 16. • Digits are 0 to 9 and A to F. • Hexadecimal numbers are denoted by the number, followed by an H or h.
Decimal to Hex Hex to Decimal Conversion Conversion
• A good method is to repeatedly • Convert each digit to decimal. divide by 16. • Multiply the digit by 16 to the
• Example: Converting 15,26810 power of the bit position. to hexadecimal • Example: 3BA4h =
3 Converting between Hex Binary/Hexadecimal and Binary Conversion Table
• Each hexadecimal digit • from AoA represents four binary digits. • Lookup tables (or, better yet, memorized bit patterns) can be used to do the conversions. • No multiplication or addition required!
Binary to Hexadecimal Hexadecimal to Binary
• Break the number up into four • Convert each hex digit into the digit (bit) pieces, starting from four digit (bit) binary the least significant (right-most) representation bit • Example: • Convert each four digit segment 8A2640 into the corresponding hexadecimal digit • Example: 10101011100101111000011011100101
4 Data Organization Bytes
Bit numbering in a byte: • Computers work with groups of bits (binary digits) 7 6 5 4 3 2 1 0 • Common groups of bits are: – Single bits – Groups of 4 bits – nibbles High Order Low Order – Groups of 8 bits – bytes Nibble Nibble – Groups of 16 bits – words – Groups of 32 bits – double words A byte can represent values: 0..255 (unsigned) -128..+127 (signed)
or, special data types that require no more than 256 different values
Words
Bit numbering in a word:
151413 12 11 10 9 8 7 6 5 4 3 2 1 0
High Order Low Order Byte Byte
A word can represent values: 0..65,535 (unsigned) -32,768..+32,767 (signed)
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