Understanding Binary Number Representation in Computer Science

1. Data Representation CS 1428

2. Binary Representation of Information • Detecting Voltage Levels • Why not 10 levels? • Would be unreliable • Not enough difference between states • On/Off • Fully Charged - Fully Discharged • Magnetized - Demagnetized

3. Bits, Bytes, and so on • A bit is one 0 or 1 • Short for “binary digit” • A byte is a collection of 8 bits • They named it “byte” instead of “bite” so you couldn’t easily mess up the spelling and confuse it with “bit”.

4. The Binary Numbering System • A computer’s internal storage techniques are different from the way people represent information in daily lives • We see and type numbers and letters. • The computer sees ones and zeros for everything • All information inside a digital computer is stored as a collection of binary data

5. Binary Representation of Numeric and Textual Information • Binary numbering system • Base-2 • Built from ones and zeros • Each position is a power of 2 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 • Decimal numbering system • Base-10 • Each position is a power of 10 3052 = 3 x 103 + 0 x 102 + 5 x 101 + 2 x 100

6. Binary Numbers • Base 2 - digits 0 and 1 ___ ___ ___ ___ ___ ___ 25 24 23 22 21 20

8. Calculating in binary • First practice counting to 2010 in binary • Practice adding: 10112 + 1012 • Try subtracting: 10112 - 1012

9. Representing Signed Numbers • What about negative numbers? • We can divide the bit patterns into two halves but we need to be careful • What to do for zero? • Need to consider binary arithmetic as well • One approach is to use the sign and magnitude method • Reserve 1 bit (usually the high-order bit) that represents the sign • Rest of the bits make up the magnitude

10. Sign-and-Magnitude • Easy to understand but not efficient • Two representations for zero • Lot of extra overhead to do binary arithmetic • Recall the ALU from Lecture 1 • ALU does binary arithmetic (mainly addition) • We would need to redesign the ALU to do arithmetic with sign-and-magnitude representation • What’s the alternative?

11. 2’s complement • The high-order bit represents the sign but the magnitude is computed differently • Has a single representation for zero • Has an extra negative number • No extra overhead for binary arithmetic • Almost all modern computers use this representation • Example

12. Converting Decimal to 2’s Complement • Get the binary representation for the absolute value of the number • Flip all the bits • Add 1 to the complement • Example -3 00011 // 5-bit binary for absolute value of -3 11100 // all bits flipped 11101 // 1 added to the complement

13. Converting 2’s Complement to Decimal • If the high-order bit is 0 then convert the number in the usual way • Else • Subtract 1 • Flip all bits • Convert to decimal • Affix a minus sign

14. Converting 2’s Complement to Decimal • Example 11010 // 2’s complement binary 11001 // 1 subtracted 00110 // bits flipped -6 // affixed the negative sign

15. 2’s Complement Arithmetic • Binary addition, discard the final carry • Example 0011 1110 -------- 0001 • Be careful of overflow • For a 5 bit 2’s complement representation • 16 is too large! • -17 is too small!

16. More on 2s Complement • http://en.wikipedia.org/wiki/Two's_complement • http://www.hal-pc.org/~clyndes/computer-arithmetic/twoscomplement.html

17. Summary • Things you should be able to do • Convert binary to decimal • Convert decimal to binary • Convert 2’s complement binary to decimal • Convert decimal to 2’s complement binary • Towards the end of the semester perhaps write a program that does this for you