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# 4 2scientific notation

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### 4 2scientific notation

1. 1. Scientific Notation Back to 123a-Home
2. 2. Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers.
3. 3. 100 = 1 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with
4. 4. 100 = 1 101 = 10 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with
5. 5. 100 = 1 101 = 10 102 = 100 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with
6. 6. 100 = 1 101 = 10 102 = 100 103 = 1000 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with
7. 7. 100 = 1 101 = 10 102 = 100 103 = 1000 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger
8. 8. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger
9. 9. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger
10. 10. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger
11. 11. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger pack 0’s to the left for negative exponents so they get smaller
12. 12. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger pack 0’s to the left for negative exponents so they get smaller If r is a number then r x 10k = shifting the decimal point of r,
13. 13. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger pack 0’s to the left for negative exponents so they get smaller If r is a number then r x 10k = shifting the decimal point of r, if k is positive (+), shift the point right, if k is negative (–), shift left.
14. 14. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger pack 0’s to the left for negative exponents so they get smaller If r is a number then r x 10k = shifting the decimal point of r, if k is positive (+), shift the point right, if k is negative (–), shift left. In particular, every number x can be written in the form r x 10K with 1 ≤ r < 10. This form is called the scientific notation of x.
15. 15. 100 = 1 101 = 10 102 = 100 103 = 1000 10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001 Scientific Notation An important application for exponents is the usage of the powers of 10 in calculation of very large or very small numbers. Powers of 10: starting with pack 0’s to the right for positive exponents so they get larger pack 0’s to the left for negative exponents so they get smaller If r is a number then r x 10k = shifting the decimal point of r, if k is positive (+), shift the point right, if k is negative (–), shift left. In particular, every number x can be written in the form r x 10K with 1 ≤ r < 10. This form is called the scientific notation of x. For example, 2500 = 2.5 x 103 and that 0.25 = 2.5 x 10–1.
16. 16. Scientific Notation Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N.
17. 17. Scientific Notation Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number.
18. 18. Scientific Notation Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number.
19. 19. Scientific Notation Move right 4 places, Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number.
20. 20. Scientific Notation Move right 4 places, Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number.
21. 21. Scientific Notation Move right 4 places, Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number.
22. 22. Scientific Notation Move right 4 places, Move left 3 places Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 = 0. 001 23 Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number.
23. 23. Scientific Notation Move right 4 places, Move left 3 places Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123 Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number.
24. 24. Scientific Notation Move right 4 places, Move left 3 places Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123 To represent a number x with scientific notation as r x 10N, first identify the r using x, Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number.
25. 25. Scientific Notation Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number. Move right 4 places, Move left 3 places Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123 To represent a number x with scientific notation as r x 10N, first identify the r using x, then multiply r by 10N to adjust the decimal point of r to get back the x.
26. 26. Scientific Notation Let's change a number in scientific notation r x 10K back to the standard form by moving the decimal point of r according to N. i. If N is positive, move the decimal point of r to the right, i.e. make r into a larger number. ii. If N is negative, move the decimal point of r to the left, i.e. make r into a smaller number. Move right 4 places, Move left 3 places Example B. Write the following numbers in the standard form. a. 1. 23 x 10 +4 = 1 2300 . = 12300. b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123 To represent a number x with scientific notation as r x 10N, first identify the r using x, then multiply r by 10N to adjust the decimal point of r to get back the x. To find N, we count.
27. 27. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r.
28. 28. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. Example B. Write the following numbers in scientific notation. a. 12300. .
29. 29. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x
30. 30. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x r
31. 31. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x r
32. 32. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x r
33. 33. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 r
34. 34. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 b. 0.00123. r
35. 35. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. ii. If the point moved to the right N spaces so r is more than x, Move left 4 places. Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 b. 0.00123. r
36. 36. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. ii. If the point moved to the right N spaces so r is more than x, Move left 4 places. Move right 3 places Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 b. 0.00123 = 0. 001 23 = 1. 23 x r r
37. 37. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. ii. If the point moved to the right N spaces so r is more than x, then use negative exponent N to compensate for the change. Move left 4 places. Move right 3 places Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 b. 0.00123 = 0. 001 23 = 1. 23 x 10 –3 r r
38. 38. Scientific Notation To express a given number x with scientific notation as r x 10N, move the decimal point of x to the back of the first nonzero digit, this is r. i. If the point moved left N spaces so the r is smaller than x, then use positive exponent N to compensate for the change. ii. If the point moved to the right N spaces so r is more than x, then use negative exponent N to compensate for the change. Move left 4 places. Move right 3 places Example B. Write the following numbers in scientific notation. a. 12300. = 1 2300 . = 1. 23 x 10 +4 b. 0.00123 = 0. 001 23 = 1. 23 x 10 –3 Scientific notation simplifies complicated calculation of very large and very small numbers. r r
39. 39. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) Scientific Notation
40. 40. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 Scientific Notation
41. 41. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 Scientific Notation
42. 42. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 Scientific Notation
43. 43. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 = 0.000156 Scientific Notation
44. 44. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 = 0.000156 b. 6.3 x 10-2 2.1 x 10-10 Scientific Notation
45. 45. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 = 0.000156 b. 6.3 x 10-2 2.1 x 10-10 = 6.3 2.1 x 10 – 2 – ( – 10) Scientific Notation
46. 46. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 = 0.000156 b. 6.3 x 10-2 2.1 x 10-10 = 6.3 2.1 x 10 – 2 – ( – 10) = 3 x 108 Scientific Notation
47. 47. Example C. Calculate. Give the answer in both scientific notation and the standard notation. a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12 = 1.56 x 108 –12 = 1.56 x 10 –4 = 0.000156 b. 6.3 x 10-2 2.1 x 10-10 = 6.3 2.1 x 10 – 2 – ( – 10) = 3 x 108 = 300,000,000 Scientific Notation
48. 48. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. Scientific Notation
49. 49. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 0.00015 Scientific Notation
50. 50. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 Scientific Notation
51. 51. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 Scientific Notation
52. 52. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 1.5 x 10–4 Scientific Notation
53. 53. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 1.5 x 10–4 = 2.4 x 2.5 x 108 x 10–6 1.5 x 10–4 Scientific Notation
54. 54. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 1.5 x 10–4 = 2.4 x 2.5 1.5 x 10 8 + (–6) – ( – 4) = 2.4 x 2.5 x 108 x 10–6 1.5 x 10–4 Scientific Notation
55. 55. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 1.5 x 10–4 = 2.4 x 2.5 1.5 x 10 8 + (–6) – ( – 4) = 2.4 x 2.5 x 108 x 10–6 1.5 x 10–4 = 4 x 108 – 6 + 4 Scientific Notation
56. 56. Example D. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation. 240,000,000 x 0.0000025 = 0.00015 2.4 x 108 x 2.5 x 10–6 1.5 x 10–4 = 2.4 x 2.5 1.5 x 10 8 + (–6) – ( – 4) = 2.4 x 2.5 x 108 x 10–6 1.5 x 10–4 = 4 x 108 – 6 + 4 = 4 x 106 = 4,000,000 Scientific Notation For calculators, the 10N portion in scientific notation is displayed as E+N or E–N where E means exponents. Hence 2.5 x10–6 is displayed as 2.5 E–6 on the calculators.