Binary numbers, bits, and Boolean operations

1. Binary numbers, bits, and Boolean operations CSC 2001

2. Overview • sections 1.1, 1.5 • bits • binary (base 2 numbers) • conversion to and from • addition • Boolean logic

3. “Bits” of information • binary digits • 0 or 1 • why? • not necessarily intuitive, but… • easy (on/off) • powerful (more in a later lecture)

4. Number bases • When we see that number 10, we naturally assume it refers to the value ten. • So, when we read this… • There are 10 kinds of people in this world: those who understand binary, and those who don't. • It might seem a little confusing.

5. Number bases • In the world today, pretty much everyone assumes numbers are written in base ten. • Originated in India • This cultural norm is very useful! • But 10 does not necessarily mean ten. • What it really means is… • (1 x n1) + (0 x n0), where n is our base or our number system.

6. Base ten (decimal) • So in base ten, we’ll set n = ten. • Thus… 10 = (1 x n1) + (0 x n0) = (1 x ten1) + (0 x ten0) = (1 x ten) + (0 x one) = ten

7. Base ten (decimal) 527 = (5 x ten2) + (2 x ten1) + (7 x ten0) = (5 x one hundred) + (2 x ten) + (7 x one) = five hundred and twenty seven

8. Other bases • In computer science, we’ll see that base 2, base 8, and base 16 are all useful. • Do we ever work in something other than base 10 in our everyday life?

9. Other bases • Base twelve • Sumerian? • smallest number divisible by 2, 3, & 4 • time, astrology/calendar, shilling, dozen/gross, foot • 10 (base twelve) = 12 (base ten) [12 + 0] • 527 (base twelve) = 751 (base ten) [(5x144) + (2x12) + (7x1)]

10. Other bases • Base sixty • Babylonians • smallest number divisible by 2, 3, 4, & 5 • time (minutes, seconds), latitude/longitude, angle/trigonometry • 10 (base sixty) = 60 (base ten) [60 + 0] • 527 (base sixty) = 18,127 (base ten) [(5x602) + (2x60) + 7 = (5x3600) + 120 + 7]

11. Base two (binary) • Just like the other bases… • number abc = (a x two2) + (b x two1) + (c x two0) = (a x 4) + (b x 2) + (c x 1) • So.. • There are 10 kinds of people in this world: those who understand binary, and those who don't. • means there are 2 kinds of people (1x2 + 0x1)

12. binary -> decimal practice • 11 • 1010 • 1000001111

13. Answers • 11 = (1x2) + (1x1) = 3 • 1010 = (1x23)+(0x22)+(1x2)+(0x1) = 8 + 0 + 2 + 0 = 10 • 1000001111 = (1x29) + (1x23) + (1x22) + (1x2) + (1x1) = 512 + 8 + 4 + 2 + 1 = 527

14. 20 = 1 21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 26 = 64 27 = 128 28 = 256 29 = 512 210 = 1024 Powers of two

15. decimal -> binary • Algorithm (p. 42) figure 1.17 • Step 1: Divide the value by two and record the remainder • Step 2: As long as the quotient obtained is not zero, continue to divide the newest quotient by two and record the remainder • Step 3: Now that a quotient of zero has been obtained, the binary representation of the original value consists of the remainders written from right to left in the order they were recorded.

16. Example 1: • 13 (base ten) = ?? (base 2) • Step 1: Divide the value by two and record the remainder 13/2 = 6 (remainder of 1) 1

17. Example 1: • 13 (base ten) = ?? (base 2) 13/2 = 6 (remainder of 1) 1 • Step 2: As long as the quotient obtained is not zero, continue to divide the newest quotient by two and record the remainder 6/2 = 3 (remainder of 0) 0 3/2 = 1 (remainder of 1) 1 1/2 = 0 (remainder of 1) 1

18. Example 1: • 13 (base ten) = ?? (base 2) 13/2 = 6 (remainder of 1) 1 6/2 = 3 (remainder of 0) 0 3/2 = 1 (remainder of 1) 1 1/0 = 0 (remainder of 1) 1 • Step 3: Now that a quotient of zero has been obtained, the binary representation of the original value consists of the remainders written from right to left in the order they were recorded. 1 1 0 1

19. Example 2: 527 527/2 = 263 r 1 1 263/2 = 131 r 1 1 131/2 = 65 r 1 1 65/2 = 32 r 1 1 32/2 = 16 r 0 0 16/2 = 8 r 0 0 8/2 = 4 r 0 0 4/2 = 2 r 0 0 2/2 = 1 r 0 0 1/2 = 0 r 1 1 1 0 0 0 0 0 1 1 1 1

20. In-class practice • 37 • 18 • 119

21. Answers • 37: • 37/2=18r1; 18/2=9r0; 9/2=4r1; 4/2=2r0; 2/2=1r0; 1/2=0r1 • 100101 = 1 + 4 + 32 = 37 • 18: • 18/2=9r0; 9/2=4r1; 4/2=2r0; 2/2=1r0; 1/2=0r1 • 10010 = 2 + 16 = 18 • 119: • 119/2=59r1; 59/2=29r1; 29/2=14r1; 14/2=7r0; 7/2=3r1; 3/2=1r1; 1/2=0r1 • 1110111 = 1 + 2 + 4 + 16 + 32 + 64 = 119

22. Binary operations • Basic functions of a computer • Arithmetic • Logic

23. Binary addition • Addition • Useful binary addition facts: • 0 + 0 = 0 • 1 + 0 = 1 • 0 + 1 = 1 • 1 + 1 = 10

24. Example 1 1 1 101011 +011010 1 0 0 0 1 0 1

25. Multiplication and division by 2 • Multiply by 2 • add a zero on the right side • 1 x 10 = 10 • 10 x 10 = 100 • Integer division by 2 (ignore remainder) • drop the rightmost digit • 100/10 = 10 • 1000001111/10 = 100000111 • (527/2 = 263)

26. Binary numbers & logic • As we have seen, 1’s and 0’s can be used to represent numbers • They can also represent logical values as well. • True/False (1/0) • George Boole

27. Logical operations and binary numbers • Boolean operators • AND • OR • XOR (exclusive or) • NOT

28. Truth tables

29. Truth tables (0 = F; 1 = T)

30. In-class practice • (1 AND 0) OR 1 • (1 XOR 0) AND (0 AND 1) • (1 OR ???) • (0 AND ???)

31. Answers • (1 AND 0) OR 1 = • 0 OR 1 = 1 • (1 XOR 0) AND (0 AND 1) = • 1 AND 0 = 0 • (1 OR ???) = • 1 • (0 AND ???) = • 0

32. Summary • Binary representation and arithmetic and Boolean logic are fundamental to the way computers operate. • Am I constantly performing binary conversions when I program? • Absolutely not (actually hardly ever!) • But understanding it makes me a better programmer. • Am I constantly using Boolean logic when I program? • Definitely! • A good foundation in logic is very helpful when working with computers.