Binary Numbers

1. Binary Numbers

2. Why Binary? • Maximal distinction among values  minimal corruption from noise • Imagine taking the same physical attribute of a circuit, e.g. a voltage lying between 0 and 5 volts, to represent a number • The overall range can be divided into any number of regions

3. Don’t sweat the small stuff • For decimal numbers, fluctuations must be less than 0.25 volts • For binary numbers, fluctuations must be less than 1.25 volts 5 volts 0 volts Binary Decimal

4. It doesn’t matter …. • Recall the power supply voltage measurements from lab 1 • Ideally they should be 5.00 volts and 12.00 volts • Typically they were 5.14 volts or 12.22 volts • So what, who cares

5. How to represent big integers • Use positional weighting, same as with decimal numbers • 205 = 2102 + 0101 + 5100 • 11001101 = 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 = 128 + 64 + 8 + 4 + 1 = 205

6. Converting 205 to Binary • 205/2 = 102 with a remainder of 1, place the 1 in the least significant digit position • Repeat 102/2 = 51, remainder 0

7. Iterate • 51/2 = 25, remainder 1 • 25/2 = 12, remainder 1 • 12/2 = 6, remainder 0

8. Iterate • 6/2 = 3, remainder 0 • 3/2 = 1, remainder 1 • 1/2 = 0, remainder 1

9. Recap 127 + 126 + 025 + 024 + 123 + 122 + 021 + 120 205

10. Adding Binary Numbers • Same as decimal; if sum of digits in a given position exceeds the base (10 for decimal, 2 for binary) then there is a carry into the next higher position

11. Adding Binary Numbers

12. Uh oh, overflow • What if you use a byte (8 bits) to represent an integer • A byte may not be enough to represent the sum of two such numbers

13. Bigger Numbers • You can represent larger numbers by using more words • You just have to keep track of the overflows to know how the lower numbers (less significant words) are affecting the larger numbers (more significant words)

14. Negative numbers • Negative x is that number when added to x gives zero • Ignoring overflow the two eight-bit numbers above sum to zero

15. Two’s Complement • Step 1: exchange 1’s and 0’s • Step 2: add 1

16. Riddle • Is it 214? • Or is it – 42? • Or is it …? • It’s a matter of interpretation • How was it declared?

17. Hexadecimal Numbers • Even moderately sized decimal numbers end up as long strings in binary • Hexadecimal numbers (base 16) are often used because the strings are shorter and the conversion to binary is easier • There are 16 digits: 0-9 and A-F

18. 0  0000  0 1  0001  1 2  0010  2 3  0011  3 4  0100  4 5  0101  5 6  0110  6 7  0111  7 8  1000  8 9  1001  9 10  1010  A 11  1011  B 12  1100  C 13  1101  D 14  1110  E 15  1111  F Decimal  Binary  Hex

19. Binary to Hex • Break a binary string into groups of four bits (nibbles) • Convert each nibble separately

20. Addresses • With user friendly computers, one rarely encounters binary, but we sometimes see hex, especially with addresses • To enable the computer to distinguish various parts, each is assigned an address, a number • Distinguish among computers on a network • Distinguish keyboard and mouse • Distinguish among files • Distinguish among statements in a program • Distinguish among characters in a string

21. How many? • One bit can have two states and thus distinguish between two things • Two bits can be in four states and … • Three bits can be in eight states, … • N bits can be in 2N states

22. IP Addresses • An IP address is used to identify a network and a host on the Internet • It is 32 bits long • How many distinct IP addresses are there?

23. Characters • We need to represent characters using numbers • ASCII (American Standard Code for Information Interchange) is a common way • A string of eight bits (a byte) is used to correspond to a character • Thus 28=256 possible characters can be represented • Actually ASCII only uses 7 bits, which is 128 characters; the other 128 characters are not “standard”

24. Unicode • Unicode uses 16 bits, how many characters can be represented? • Enough for English, Chinese, Arabic and then some.

25. ASCII • 0  00110000 • 1  00110001 • … • A  01000001 • B  01000010 • … • a  01100001 • b  01100010 • …

26. Booleans • A Boolean variable is something that is true or false • Booleans have two states and could be represented by a single bit (1 for true and 0 for false) • Booleans appearing in a program will take up a whole word in memory

28. Fractions • Similar to what we’re used to with decimal numbers

29. Converting decimal to binary II • 98.6 • Integer part • 98 / 2 = 49 remainder 0 • 49 / 2 = 24 remainder 1 • 24 / 2 = 12 remainder 0 • 12 / 2 = 6 remainder 0 • 6 / 2 = 3 remainder 0 • 3 / 2 = 1 remainder 1 • 1 / 2 = 0 remainder 1 • 1100010

30. Converting decimal to binary III • 98.6 • Fractional part • 0.6  2 = 1.2 • 0.2  2 = 0.4 • 0.4  2 = 0.8 • 0.8  2 = 1.6 • 0.6  2 = 1.2 • 0.2  2 = 0.4 • REPEATS • .100110

31. Converting decimal to binary IV • Put together the integral and fractional parts • 98.6  1100010.1001100110011001

32. Scientific notation • Used to represent very large and very small numbers • Ex. Avogadro’s number •  6.0221367  1023 particles •  602213670000000000000000 • Ex. Fundamental charge e •  1.60217733  10-19 C •  0.000000000000000000160217733 C

33. Floats • SHIFT expression so it is just under 1 and keep track of the number of shifts • 1100010.1001100110011001 • .11000101001100110011001  27 • Express the number of shifts in binary • .11000101001100110011001  200000111

34. Mantissa and Exponent • .11000101001100110011001  200000111 • Mantissa • .11000101001100110011001  200000111 • Exponent