2. What are big numbers? IIIImagine the # 1. It represents one object IIIIncrease this by one zero (power of 10) to #10 It represents ten of those objects IIIIncrease this another zero to #100 It represents ten x ten objects (100) NNNNow imagine just how many objects the number 1,000,000 equals? It represents 10,000 times the number of dots above! It would take this computer 27 HOURS just to draw 1 million dots Now Imagine 100,000,000,000 (100 billion) dots! We would have to watch the computer draw that many dots for 317 YEARS In science, 100 billion is a tiny number. For example, your body alone has 1,000 x that number of cells (100,000,000,000,000)

3. Scientific Notation and the world of BIG numbers

4. Scientific Notation is a means of writing BIG numbers in a smaller way  2,461,000,000,000 is a big number.  To use it in math and science is difficult.  Some calculators don’t even accept it.  This number can be written in a shortcut way - by Scientific Notation.  2,461,000,000,000 = 2.46 x 10 12

5. To convert a number into S.N. 1. Place a decimal point after the leftmost whole number > 0 Ex: 5, 3 4 5, 1 0 0, 0 0 0

6. To convert a number into S.N. 2. Count spaces you moved from the right end of the whole number to the decimal point Ex: 5. 3 4 5 1 0 0 0 0 0. equals 9 spaces moved

7. To convert a number into S.N. 3. Round the numbers to the right of the decimal point to 2 or more decimal places (whatever is asked for) Ex: 5. 3 4 5 1 0 0 0 0 0 rounds to 5. 3 5

8. To convert a number into S.N. 4. Finally, insert your count of spaces as an exponent of x 10 ( here ) 5.3 5x10 9 Answer = 5,345,100,000 = 5.35 x 10 9 You counted 9 spaces before, so ….

9. S.N. step Summary 1) Place a decimal point after the leftmost whole number > 0 2) Count spaces from right end of the whole number to the decimal point 3) Round the whole numbers to 2 or more decimal places (whatever is required) 4) Insert your count of spaces as an exponent of x 10 ( here )

10. More examples  Ex: 4,512  1. This becomes 4.512  2. There are 3 spaces in 4.512  3. This rounds to 4.51  4. The number becomes 4.51 x 10 3

11. More examples Ex: 8,867,890,000,000 ( to 3 decimal places) 1.This becomes 8.867890000000 2. There are 12 spaces in 8.867890000000 3. This rounds to 8.868 4. The number becomes 8.868 x 10 12

12. More examples Ex: 5,564,563,440 (to 4 decimal places) 1.This becomes 5. 564563440 2. There are 9 spaces in 5.564563440 3. This rounds to 5.5646 4. The number becomes 5.5646 x 10 9

13. More examples Ex: 8177.023 (to 2 decimal places) 1.This becomes 8. 177023 2. You moved 3 spaces in 8177.023 3. This rounds to 8.18 4. The number becomes 8.18 x 10 3

14. Small fractional numbers have negative exponents Ex: 0.0 0 0 5 4 3 1 Ex: 0.0 0 0 5 4 3 1 (to 2 decimal places) 1. This becomes 5.431 2. You had to move - 4 spaces to get to 5 0.0 0 0 5 4 3 1 = 5.431 0.0 0 0 5 4 3 1 = 5.431 3. This rounds to 5.43 4. The number becomes 5.43 x 10 -4

15. Another fractional example Ex: 0.0 0 8 9 7 2 7 Ex: 0.0 0 8 9 7 2 7 (to 2 decimal places) 1. This becomes 8.9 7 2 7 2. You had to move - 3 spaces to get to 8 0.0 0 8 9 7 2 7 = 8.9727 0.0 0 8 9 7 2 7 = 8.9727 3. This rounds to 8.97 4. The number becomes 8.97 x 10 -3

16. Please Note  The exponent is an exponent of 10, not of the number itself  136,700 = 1.37 x 10 5, NOT 1.37 5  There is a BIG difference between these two numbers

17. To go in reverse, 1111. Write all the numbers you have, including the period (.) 2222. Add the spaces back using zeros 3333. Move the period (.) to the end & reset commas EEx: 6.73 x 10 8 11. Numbers are 6.73 22. Add 8 spaces back = 6.73000000 33. Move (.) and reset commas = 673,000,000.

18. More examples Ex: 2.344 x 10 6 Numbers are 2.344 6 spaces back becomes 2.344000 Move. and reset commas 2.344 x 10 6 = 2,344,000. 2.344 x 10 6 = 2,344,000.

19. More examples Ex: 3.99 x 10 4 Numbers are 3.99 4 spaces back becomes 3.9900 Move. and reset commas = 39,900.

20. More examples Ex: 7.71 x 10 -6 Numbers are 7.71 6 spaces back becomes 000007.71 Add a 0 in front of decimal point, and move (.) Commas are not needed = 0.00000771 0.00000771

21. More examples Ex: 5.5723 x 10 -7 Numbers are 5.5723 7 spaces back becomes.00000055723 Add a 0 in front of decimal point, and move (.) Commas are not needed = 0.00000055723 0.00000055723

22. Your turn!  Complete parts 1 and 2 of Scientific Notation Worksheet

23. To enter big numbers in calculators  Example: Enter 456,344,000,000  Convert to S.N. = 4.56344 x 10 11  Enter 4.56344 EE 11 or  4.56344 EXP 11 or  4.56344 * 10 11 (avoid)

24. More examples  75,679,012,000  10 spaces, so …  Enter 7.5679012 EE 10, or  Enter 7.5679012 Exp 10, or  Enter 7.5679012 * 10 10 (avoid)

25. More examples  0.00387001  3 spaces to right, so …  Enter 3.87001 EE +/- 3, or  Enter 3.87001 EXP +/- 3, or  Enter 3.87001 * 10 -3 (avoid)

26. Complete Parts 3 and 4 using your calculator To additional practice

27. To Multiply Numbers in S. Notation 1. Multiply the numbers you have 2. Add the exponents of 10 (if any) 3. Rewrite and convert the number into Scientific Notation and round as desired

28. Multiplication examples Ex: (5.2 x 10 3 ) x (2.344 x 10 6 ) 1.Multiply the numbers: 5.2 x 2.344 = 12.1888 2. Add exponents: 10 3 * 10 6 = 10( 3+6 ) = 10 9 3. Rewrite whole # in S. N. and round 12.1888 x 10 9 = 1.22 x 10 10

29. Multiplication examples Ex: ( 1.81 x 10 5 ) * ( 8.7 x 10 4 ) 1.Multiply the numbers: 1.81 * 8.7 = 15.747 2. Add exponents : 10 5 * 10 4 = 10 ( 5 + 4 ) = 10 9 3. Rewrite whole # in S. N. and round 15.747 x 10 9 = 1.57 x 10 10

30. Multiplication examples Ex: 74.4 * ( 6.6 x 10 3 ) 1.Multiply the numbers: 74.4 * 6.6 = 491.04 2. Add exponents: _ + 10 3 = 10 3 3. Rewrite whole # in S. N. and round 491.04 x 10 3 = 4.91 x 10 5

31. Multiplication examples Ex: 227.451 * (16.4 x 10 8 ) * ( 4.6 x 10 5 ) 1. Multiply the numbers: 227.451 * 16.4 * 4.6 = 17158.90344 2. Add exponents : _ + 10 8 +10 5 = 10 13 3. Rewrite whole # in S. N. and round 17158.90344 x 10 13 = 1.72 x 10 17

32. To Divide Numbers in S. Notation 1. Divide the numbers 2. Subtract the exponents of 10 (if any) 3. Rewrite # in S.N. and round if needed

33. Division examples Ex: ( 8.7 x 10 5 ) / ( 2.2 x 10 3 ) or 8.7 x 10 5 2.2 x 10 3 Divide the numbers: 8.7 / 2.2 = 3.9545 Subtract exponents: 10 5 /10 3 = 10 ( 5-3 ) =10 2 Rewrite # and round: 3.9545 x 10 2 = 3.95 x 10 2

34. Division examples Ex: ( 2.344 x 10 6 ) / 52 or 2.344 x 10 6 52 52 Divide the numbers: 2.344 / 52 = 0.04507 Subtract exponents: 10 6 - _ = 10 6 Rewrite # and round: 0.04507 x 10 6 = 4.51 x 10 4

35. Division examples Ex: ( 9.23 x 10 3 ) / 4.55 x 10 7 Divide the numbers: 9.23 / 4.55 = 2.028 Subtract exponents: 10 ( 3 - 7 ) = 10 -4 Rewrite # and round: 2.028 x 10 -4 = 2.03 x 10 -4

36. Division examples Ex: ( 4.4596 x 10 4 ) / 9.32 x 10 8 Divide the numbers: 4.4596/ 9.32 = 0.4785 Subtract exponents: 10 ( 4 - 8 ) = 10 -4 0.4785 x 10 -4 = 4.785 x 10 -5 Rewrite # and round: 0.4785 x 10 -4 = 4.785 x 10 -5

37. To Add Numbers in S. Notation 1. 1. Convert both numbers to Scientific Notation first. 2. 2. Raise exponent of the smaller number by moving spaces to the left to match the exponent of the larger #. 3. 3. Add the numbers & carry exponents 4. 4. Round as needed

38. Addition examples Ex: ( 2.344 x 10 6 ) + 8,640 8,640 = 8.64 x 10 3 Convert all to S.N: 8,640 = 8.64 x 10 3 8.64 x 10 3 = 0.008640 x 10 6 Raise small # to big one by moving spaces left: 8.64 x 10 3 = 0.008640 x 10 6 0.00864 + 2.344= 2.35264 Add the numbers: 0.00864 + 2.344= 2.35264 2.35264 x 10 6 = 2.35 x 10 6 Round: 2.35264 x 10 6 = 2.35 x 10 6

39. Addition examples Ex: ( 3.59 x 10 3 ) + ( 9.10 x 10 9 ) Convert all to S.N: already done 3.59 x 10 3 = 0.00000359 x 10 9 Raise small # to big one by moving spaces left: 3.59 x 10 3 = 0.00000359 x 10 9 Add the numbers: 0.00000359 + 9.10 = 9.10000359 0.00000359 + 9.10 = 9.10000359 9.10000359 x 10 9 = 9.10 x 10 9 Round: 9.10000359 x 10 9 = 9.10 x 10 9 Notice that this number is the same as the original. It did not change

40. To Subtract Numbers in S. Notation 1. Convert both numbers to Scientific Notation first. 2. Adjust the exponent of the smaller number to match the exponent of the bigger # by moving spaces left. 3. Subtract the numbers & carry the exponents 4. Round.

41. Subtraction examples Ex : ( 5.324 x 10 6 ) - 128,310 (to 3 places) 128310 = 1.2831 x 10 5 Convert all to S.N: 128310 = 1.2831 x 10 5 1.2831 x 10 5 = 0.12831 x 10 6 Raise small # to big one by moving spaces left: 1.2831 x 10 5 = 0.12831 x 10 6 5.324 – 0.12831 = 5.19569 Subtract the numbers: 5.324 – 0.12831 = 5.19569 Round 5.19569 x 10 6 = 5.196 x 10 6 5.19569 x 10 6 = 5.196 x 10 6

42. Subtraction examples Ex: 2,334,561,000 - 4.2 x 10 11 ( to 2 places) 2,334,561,000 = 2.33 x 10 9 Convert to S.N: 2,334,561,000 = 2.33 x 10 9 2.33 x 10 9 = 0.0233 x 10 11 Raise small # to big one by moving spaces left: 2.33 x 10 9 = 0.0233 x 10 11 0.0233 – 4.2 = - 4.1767 Subtract the numbers: 0.0233 – 4.2 = - 4.1767 Round: - watch the signs -4.1767 x 10 11 = -4.18 x 10 11 -4.1767 x 10 11 = -4.18 x 10 11

43. Multiplication practice (show full answer, then round to 2 places) Multiplication practice (show full answer, then round to 2 places) (complete on page 2 of worksheet) Ex: 3446.78 x 8.7 x 10 7 29986.986 x 10 7 = 3.00 x 10 11 29986.986 x 10 7 = 3.00 x 10 11

44. Ex: 4.53 x 10 4 x 2.91 x 10 10 13.1823 x 10 14 = 1.32 x 10 15 13.1823 x 10 14 = 1.32 x 10 15 Multiplication practice (show full answer, then round to 2 places) Multiplication practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

45. Ex: 6.44 x 10 4 / 3.72 x 10 2 1.7311 x 10 2 = 1.73 x 10 2 1.7311 x 10 2 = 1.73 x 10 2 Division practice (show full answer, then round to 2 places) Division practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

46. Ex: 5.45 x 10 3 / 6.79 x 10 7 0.8026 x 10 -4 = 8.03 x 10 -5 0.8026 x 10 -4 = 8.03 x 10 -5 Division practice (show full answer, then round to 2 places) Division practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

47. Ex: 1.273 x 10 5 + 5.54 x 10 8 5.541273 x 10 8 = 5.54 x 10 8 Addition practice (show full answer, then round to 2 places) Addition practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

48. Ex: 6.632 x 10 4 + 4,322,678.91 4.38932 x 10 6 = 4.39 x 10 6 Addition practice (show full answer, then round to 2 places) Addition practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

49. Ex: 6.632 x 10 4 - 4,322,678.91 -4.25668 x 10 6 = -4.26 x 10 6 -4.25668 x 10 6 = -4.26 x 10 6 Subtraction practice (show full answer, then round to 2 places) Subtraction practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

50. Ex: 8.93 x 10 7 - 8.11 x 10 3 8.929189 x 10 7 = 8.93 x 10 7 8.929189 x 10 7 = 8.93 x 10 7 Subtraction practice (show full answer, then round to 2 places) Subtraction practice (show full answer, then round to 2 places) (complete on page 3 of worksheet)

51. Your turn! Complete Parts 3 and 4 of worksheet