The document discusses binary numbering systems and how to convert between binary and base 10 numbering. It explains that binary uses only two digits, 0 and 1, and that each digit represents a place value that is a power of 2. It provides examples of converting specific numbers between binary and base 10. It also discusses how bytes are used to represent larger numbers than can be stored in a single byte.

1. BINARY : COUNTING TO 1 Computer Sciences NRHS

2. Simple Math First … ONES TENS HUNDREDS THOUSANDS X 10 X 10 X 10 BASE 10 3 6 5 9

3. BASE 10 = TEN DIGITS 1 2 3 4 5 6 7 8 9 0

5. Binary… 1 4 8 X 2 X 2 X 2 BASE 2 2 BIT 0 0 0 0

6. BINARY IS LIKE A LIGHT SWITCH 0 = OFF 1 = ON

7. What number is this? 1 4 8 Answer: 0 2 0 0 0 0

8. What number is this? 1 4 8 Answer: 1 2 0 0 0 1

9. What number is this? 1 4 8 Answer: 2 2 0 0 1 0

10. What number is this? 1 4 8 Answer: 4 2 0 1 0 0

11. What number is this? 1 4 8 Answer: 8 2 1 0 0 0

12. What number is this? 1 4 8 Answer: 9 2 1 0 0 1

13. What number is this? 1 4 8 Answer: 10 2 1 0 1 0

14. What number is this? 1 4 8 Answer: 6 2 0 1 1 0

15. 8 Bits = 1 Byte 16 32 64 128 What is the highest possible number in one Byte? 255 8 4 2 1

16. What number is this? 16 32 64 128 Answer: 170 8 4 2 1 1 0 1 0 1 0 1 0

17. What number is this? 16 32 64 128 Answer: 85 8 4 2 1 0 1 0 1 0 1 0 1

18. What number is this? 16 32 64 128 Answer: 217 8 4 2 1 1 1 0 1 1 0 0 1

19. How do Bytes add up?

20. CD ROM = up to 900 MB DVD ROM = up to 4.7 GB Average MP3 = 1.5 MB

21. Converting Binary to Base10

22. Let’s Start With ½ a Byte (aka Nibble) How do you know that this number is 13? 8 4 2 1 1 1 0 1

23. Determine which are “on” and which are “off” First, you ignore the numbers that are turned off. 8 4 2 1 1 1 0 1

24. Create the Equation Then, you add together the numbers that are turned on. + + + 8 4 2 1 1 1 0 1

25. Do the Equation The equation is … 8+4+1 = 13 + + + 8 4 2 1 1 1 0 1

26. Convert from Base10 to Binary How do you find the binary for 14? 8 4 2 1

27. Start with the Largest Number on the Left Ask: Does 8 go into 14? If YES … put a 1 in that box to turn it “on”. 1 8 4 2 1

28. Subtract to get the Remainder 14-8=6 1 8 4 2 1

29. Move to the next largest number Ask: Does 4 go into 6? If YES … put a 1 in that box to turn it “on”. 1 1 8 4 2 1

30. Subtract to get the Remainder 6-4=2 1 1 8 4 2 1

31. Move to the next largest number Ask: Does 2 go into 2? If YES … put a 1 in that box to turn it “on”. 1 1 1 8 4 2 1

32. Subtract to get the Remainder 2-2=0 1 1 1 8 4 2 1

33. Move to the next largest number Ask: Does 1 go into 0? If NO… put a 0 in that box to turn it “off”. 1 1 1 0 8 4 2 1

34. Now Let’s Do a Byte 16 32 64 128 What is the first step to find out what binary is for 235? 8 4 2 1

35. Start with the Largest Number on the Left 16 32 64 128 Does 128 go into 235? 235-128 = 107 YES! Turn 128 “on” 1 8 4 2 1

36. Move one number to the right 16 32 64 128 Does 64 go into 107? 107-64 = 43 YES! Turn 64“on” 1 1 8 4 2 1

37. Move one number to the right 16 32 64 128 Does 32 go into 43? 43-32= 11 YES! Turn 32“on” 1 1 1 8 4 2 1

38. Move one number to the right 16 32 64 128 Does 16 go into 11? 11-16= NO NO! Turn 16 “off” 1 1 1 0 8 4 2 1

39. Move one number to the right 16 32 64 128 Does 8 go into 11? 11-8= 3 YES! Turn 8 “on” 1 1 1 0 1 8 4 2 1

40. Move one number to the right 16 32 64 128 Does 4 go into 3? 3-4= NO NO! Turn 4 “off” 1 1 1 0 1 0 8 4 2 1

41. Move one number to the right 16 32 64 128 Does 2 go into 3? 3-2= 1 YES! Turn 2 “on” 1 1 1 0 1 0 1 8 4 2 1

42. Move one number to the right 16 32 64 128 Does 1 go into 1? 1-1= 0 YES! Turn 1 “on” 1 1 1 0 1 0 1 1 8 4 2 1

43. Move one number to the right 16 32 64 128 235 in Binary! 1 1 1 0 1 0 1 1 8 4 2 1

44. Numbers Larger than 255

46. Make a Table 1 32768 16384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2

47. Get Rid of All Numbers Larger than the One You Are Trying To Convert (they become 0s) 1 256 128 64 32 16 8 4 2 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0