1. CSE 111 Representing Numeric Data in a Computer Slides adapted from Dr. Kris Schindler

2. Unsigned Binary Numbers Range: 0  2 n -1  where n is the number of bits Positional Notation Example: 101100 two

3. Unsigned Binary Numbers How do we convert from a decimal number to a binary number?  Continue until q=0

4. Unsigned Binary Numbers How do we convert from a decimal number to a binary number?  Example: 39 ten

5. Bit Positions MSB  Most Significant Bit  Leftmost Bit Position LSB  Least Significant Bit  Rightmost Bit Position

6. Signed Binary Numbers The most significant bit (leftmost) represents the sign  Negative (-): 1  Positive (+): 0

7. Signed Binary Numbers Computers represent signed numbers using two’s complement notation

8. 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 2 n - x  This process is called taking the two’s complement of a number  Taking the two’s complement of a number negates it

9. 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 (0  1,1  0) after the first 1 as you continue to move left

10. 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

11. 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

12. 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)

13. Floating Point Very large/small numbers Fractions Example  8.5 x 2 23  100.1 2 x 2 23  Normalized 1.001 2 x 2 27  Exponent Bias = 127 127+26 = 153 = 10011001 2  Significand: 00100000000000000000000  Sign: 0  Number: 01001000100100000000000000000000

14. References J. Glenn Brookshear, Computer Science - An Overview, 11 th 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, 3 rd Edition, Morgan Kaufmann Publishers, Inc., 2005