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CSE 111 Representing Numeric Data in a Computer Slides adapted from Dr. Kris Schindler Unsigned Binary Numbers Range: 02n-1 where n is the number of bits Positional Notation P o s i t i o n W e i g h t B i t n 1 n 2 5 4 3 2 1 0 n 1 n 2 3 2 1 6 8 4 2 1 2 2 b b n 1 n 2 b b b b b b 5 4 3 2 1 0 Example: 101100two 0 1 2 3 n 2 n 1 b 2 b 2 b 2 b 2 b 2 b 2 0 1 2 3 n 2 n 1 n 2 n 1 b 1 b 2 b 4 b 8 b 2 b 2 0 1 2 3 n 2 n 1 0 1 0 2 1 4 1 8 0 16 1 32 4 8 32 4 Unsigned Binary Numbers How do we convert from a decimal number to a binary number? i / 2 q 0 , r0 b 0 r0 q 0 / 2 q 1 , r1 b 1 r1 q 1 / 2 q 2 , r2 b 2 r2 ... ... where i Decimal Number b k Binary Bit k q x Quotient r Re mainder Binary Number Continue until q=0 b n ... b 2 b 1 b 0 Unsigned Binary Numbers How do we convert from a decimal number to a binary number? Example: 39ten 39 / 2 19 ,1 19 / 2 9 ,1 9 / 2 4 ,1 4 / 2 2 ,0 2 / 2 1, 0 1 / 2 0 ,1 39 ten b0 1 b1 1 b2 1 b3 0 b4 0 b5 1 100111 two Bit Positions MSB Most Significant Bit Leftmost Bit Position LSB Least Significant Bit Rightmost Bit Position Signed Binary Numbers The most significant bit (leftmost) represents the sign Negative (-): 1 Positive (+): 0 Signed Binary Numbers Computers represent signed numbers using two’s complement notation Signed Binary Numbers Two’s Complement Representation of a negative binary number Consider an n-bit number, x The two’s complement of the number is 2n - x This process is called taking the two’s complement of a number Taking the two’s complement of a number negates it Signed Binary Numbers Two’s Complement Shortcut for taking the two’s complement of a number Start at the least significant (rightmost) bit and move left (toward the most significant bit) Keep every bit until you reach the first 1 Keep that 1 Invert every bit (01,1 0) after the first 1 as you continue to move left Signed Binary Numbers Two’s Complement Examples: -4 Take the two’s complement of 4 (00000100) 11111100 = -4 -9 Take the two’s complement of 9 (00001001) 11110111 = -9 Since the above are negative, taking the two’s complement will allow you to determine the magnitude, which is the positive equivalent Signed Binary Numbers Two’s Complement Examples: +6 Since the number is positive, you don’t need to take the two’s complement 000000110 = +6 +18 Since the number is positive, you don’t need to take the two’s complement 000010010 = +18 Signed Binary Numbers Two’s Complement Since taking the two’s complement of a number negates it, taking the two’s complement twice gives you the original number back Example: +12 is represented by 00001100 Taking the two’s complement results in -12 (11110100) Taking the two's complement of -12 results in +12 (00001100) Floating Point Very large/small numbers Fractions Example 8.5 x 223 SE x p o n e n t( E ) S i g n i f i c a n d ( F ) 1 8 2 3 B i t s S ( E b i a s ) ( 1 ) X 1 . F X 2 100.12 x 223 Normalized 1.0012 x 227 Exponent Bias = 127 127+26 = 153 = 100110012 Significand: 00100000000000000000000 Sign: 0 Number: 01001000100100000000000000000000 References J. Glenn Brookshear, Computer Science - An Overview, 11th edition, Addison-Wesley as an imprint of Pearson, 2012 Donald D. Givone, Digital Principles and Design, McGraw-Hill, 2003 John L. Hennessy and David A. Patterson, Computer Organization and Design, The Hardware/Software Interface, 3rd Edition, Morgan Kaufmann Publishers, Inc., 2005