Fundamentals of Programming Lecture 2 Hamed Rasifard
Outline • Numerical Systems • Converting Numbers • Summation and Subtraction of Binary Numbers
Numerical Systems • Binary (base 2) • Octal (base 8) • Decimal (Base 10) • Hexadecimal (Base 16)
Digits Binary Octal Decimal Hexadecimal 0 0 0 0 1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 9 9 A B C D E F
Numbers & Notation • Numbers: Binary: 101, 1100, 1001, 10011 Decimal: 101, 676, 245, 1470 Octal: 101, 345, 777 Hexadecimal:101, 234, 547E, A45CF • Notation Mention base: (101) 2 , (101) 10 , (345) 8
Positional Value • (747) 10 = (7*100 + 4*10 + 7*1) 10 • (101) 2 = (1*100 + 0*10 + 1*1) 2 • (457) 8 = (4*100 + 5*10 + 7*1) 8
Bits & Bytes • Memories consists of bits • Each bit could keep 0 or 1 • Every 8 bits make a Byte
Converting Binary Numbers to Octal and Hexadecimal Numbers • Binary to Octal: Separate each 3 Bits and rewrite their equivalents (101011) 2 (53) 8 • Binary to Hexadecimal: Separate each 4 Bits and rewrite their equivalents (101011) 2 (2B) 16
Converting Octal and Hexadecimal Numbers to Binary Numbers • Octal to Binary: write 3 binary digits for every octal digits (653) 8 (110101011) 2 • Hexadecimal to Binary: write 4 binary digits for every octal digits (FAD5) 16 (1111101011010101) 2
Converting Octal and Hexadecimal and Binary to Decimal • Express base positional values in decimal (1*100 + 0*10 + 1*1) 2 =(1*2 2 + 0*2 1 + 1*2 0 ) 10 (1*100 + 0*10 + 1*1) 8 =(1*8 2 + 0*8 1 + 1*8 0 ) 10 (1*100 + 0*10 + 1*1) 16 =(1*16 2 + 0*16 1 + 1*16 0 ) 10
Converting Decimal to Octal and Hexadecimal and Binary • write the positional values of the columns right to left • Find a column whose positional value is greater than the decimal number • From the leftmost column to the right divide number to positional values
Represent Negative Numbers in Computer Systems • Bit-Sign • One’s complement notation • Two’s complement notation
Summation and Subtraction in Computer Systems • Summation: Just ordinary add(consider overflow) • Subtraction: Add first number to two’s complement of second number
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