Math Conversion Made Simple

1. Math ConversionMade Simple Ivy Tech Community College Penny Miguel

2. Mathematical Systems • Just like speaking another language • Any number can be displayed in another system; however, the number may require a longer numerical string to depict its value. • What does the number 101 mean?

3. What is a number? • What does the number 101 mean? • The possibilities are: • 101 (base ten) • 5 (in binary) • 257 (in hexadecimal (Hex)) • How do we know which number it represents?

4. Numbers have a base • We are accustomed to using base 10 • 10110 is really the number we recognize as one hundred and one • How do we know which number is being represented? • We assume base 10 when it is not specified • When another base is used, it is shown as a subscript • How do we determine the value of the number shown?

5. Explaining our grid • We learned base 10 so long ago that most of us have forgotten just how we learned to translate the numbers. • This will seem elementary, but will make learning the new concept much easier!

6. The base of a number controls the number of possible values in each position. Base 10 A range of 10 possible values exists before you need to change position on the grid The values are 0-9 Explaining our grid

7. In the number 10110 there is a 1 in the 100 or the ones place

8. In the number 10110 there is a 1 in the 100 • 0 in the 101 or tens place

9. In the number 10110 there is a 1 in the 100 • 0 in the 101 or tens place • 1 in the 102 or the hundreds place

10. Determining values • We then add the values together to find the value of the number 1 times 1 equals 1 0 times 10 equals 0 1 times 100 equals 100 Total 10110 • 10110 means one hundred and one, base 10 Explaining our grid

11. Only two values before changing position The possible values are 0-1 To find the value of the next place holder on the grid calculate the value. Working with Base 2 (Binary)

12. No need to memorize grids The grids could continue to the left as far as desired, we will use simple grids for our lesson

13. When a value is needed place a one in the placeholder When a value is skipped over, place a zero in the placeholder to retain the correct position in your binary number

14. Calculating the value • Calculate the values 1 x 1 = 1 0 x 2 = 0 1 x 4 = 4 • Add the values 1 0 4 5 1012 = 5

15. Using the grid to find Binary value • Find the largest number that will go into the desired number • Subtract the selected value from the target number • *Place a 1 in the corresponding slot on the grid • Find the largest number that will go into the remainder • Repeat steps from * until remainder is zero • Pad unused slots between ones with zeros

16. The highest number that will go into 43 is 32 Place a 1 in the slot under 32 Subtract 32 from 43 The remainder is 11 The highest number that will go into 11 is 8 Place a 1 in the slot under 8 Subtract 8 from 11 The remainder is 3 2 will go into 3, put 1 in slot under 2 The remainder is 1 1 will go into 1, put 1 in slot under 1 Remainder is 0 Pad the empty slots with zeros Determine the binary value of 43

17. 1010112 = 43 Determine the binary value of 43

18. What is the value in base 10? 1101012

19. Using the grid to find Base 10 value • Step 1 - Place the 1 and 0 s in the respective slots on the grid beginning from the right and moving to the left • Step 2 - List the number values corresponding to the 1s • Step 3 - The base 10 equivalent is the sum of the numbers listed

20. What is the value in base 10? Step 1 • Place the 1 and 0 s in the respective slots on the grid beginning from the right and moving to the left 1101012

21. What is the value in base 10? Step 2 • List the number values corresponding to the 1s 32 16 4 1 53 1101012

22. Finding the value of 1101012 Step 3 • Calculate the values 1 x 1 = 1 0 x 2 = 0 1 x 4 = 4 0 x 8 = 0 1 x 16 = 16 1 x 32 = 32 • Add the values 1 + 0 + 4 + 0 + 16 + 32 = 5310

23. Understanding the grids

24. Understanding the grids

25. 2 + 0 = 2 Understanding the grids

26. 2 + 1 = 3 Understanding the grids

27. 4 + 0 = 4 Understanding the grids

28. 4 + 1 = 5 Understanding the grids

29. 4 + 2 = 6 Understanding the grids

30. 4 + 2 + 1 = 7 Understanding the grids

31. 8 + 0 = 8 Understanding the grids

32. 8 + 1 = 9 Understanding the grids

33. 8 + 2 = 10 Understanding the grids

34. 8 + 2 + 1 = 11 Understanding the grids

35. 8 + 4 = 12 Understanding the grids

36. 8 + 4 + 1= 13 Understanding the grids

37. 8 + 4 + 2 = 14 Understanding the grids

38. 8 + 4 + 2 + 1 = 1510 Understanding the grids

39. Hex is really Binary 4 bits at a time Premise behind Hex is that there are 16 possible combinations of Binary values from 0000 to 1111 Characters are used to represent what we consider to be numbers Each numbering system has their own set of characters Must use single character showing the positional value, can’t use 10 – 15 because they use 2 character spaces. The possible values are: 0 – 9, A, B, C, D, E, F (10 – 15 are represented by the letters) Understanding Hexadecimal

40. Using both grids proves thatHex is really Binary 4 bits at a time Understanding the grids

41. 0 + 0 + 0 + 1 = 1 00012 is the same as 116 Understanding the grids

42. 2 + 0 = 2 00102 is the same as 216 Understanding the grids

43. 2 + 1 = 3 00112 is the same as 316 Understanding the grids

44. 4 + 0 = 4 01002 is the same as 416 Understanding the grids

45. 4 + 1 = 5 01012 is the same as 516 Understanding the grids

46. 4 + 2 = 6 01102 is the same as 616 Understanding the grids

47. 4 + 2 + 1 = 7 01112 is the same as 716 Understanding the grids

48. 8 + 0 = 8 10002 is the same as 816 Understanding the grids

49. 8 + 1 = 9 10012 is the same as 916 Understanding the grids

50. 8 + 2 = 10 10102 is the same as A16 Understanding the grids