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- 1. Scientific Notation Back to 123a-Home
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

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