What is scientific Notation?

1. What is scientific Notation? • Scientific notation is a way of expressing really big numbers or really small numbers. • It is most often used in “scientific” calculations where the analysis must be very precise. • IT IS AN ABBREVIATION!

2. Scientific Notation • A number is expressed in scientific notation when it is in the form • a x 10n • where a is between 1 and 10 • and n is an integer • FOR EXAMPLE: 2.898 x 108

3. When using Scientific Notation, there are two kinds of exponents: positive and negative-- Positive Exponent: REALLY BIG NUMBERS 2.35 x 108 Negative Exponent: REALLY SMALL NUMBERS 3.97 x 10-7

4. When using Scientific Notation, there are two kinds of exponents: positive and negative-- Positive Exponent: REALLY BIG NUMBERS 2.35 x 108 Negative Exponent: REALLY SMALL NUMBERS 3.97 x 10-7

5. Converting from standard notation to scientific notation. 210,000,000,000,000,000,000,000 Where is the decimal point now? Where would you put the decimal to make this number be between 1 and 10? *Between the 2 and the 1 So, 2.1 becomes the coefficient. And, because the decimal moved 23 places, the answer is 2.1 x 10²³ *The exponent is positive because our original number was greater than one.

6. Converting from standard notation to scientific notation. Example • Given: 289,800,000 • Where would you put the decimal to make the number be between 1 and 10? • Answer: 2.898 x 10⁸ *The exponent is positive because our original number was greater than one.

7. Converting from standard notation to scientific notation. Example • Given: 0.000567 • Where would you put the decimal to make the number be between 1 and 10? • Answer: 5.67 x 10⁻⁴ *The exponent is negative because our original number was less than one.

8. When changing from scientific notation to standard notation, the exponent tells how many spaces to move the decimal, and in what direction: 4.08 x 103 = 4 0 8 In this problem, the exponent is +3, so the decimal moves 3 spaces to the right. Don’t forget to fill in your zeroes! Answer: 4,080

9. When changing scientific notation to standard notation, the exponent tells you if you should move the decimal: With a negative exponent, move the decimal to the left: 4.08 x 10-3 = Don’t forget to fill in your zeroes! Answer 0.00408

10. An easy way to remember this is: • If an exponent is positive, the number gets larger, so move the decimal to the right. • If an exponent is negative, the number gets smaller, so move the decimal to the left.

11. Try changing these numbers to and from Standard Notation to Scientific Notation: • .00008376 • 56730000 • 3.45 x 10-5 • 9.872 x 106

12. Comparing Numbers in Scientific Notation Which number is smaller? 1.23 x 1012 or 8.75 x 1010 Which number is greater? 6.25 X 10-6 or 7.43 x 10⁻⁴ *Remember, negative numbers are the opposite from positive numbers. So, -6 is smaller than -4!

13. What is the Pattern? 32000 x 10¯³ = 32 3200 x 10¯² = 32 320 x 10¯¹ = 32 32 x 10° = 32 3.2 x 10¹ = 32 .32 x 10² = 32 .032 x 10³ = 32 • Notice how as the coefficient decreases, and the exponent increases, but the answer is always 32! • Even though only one these numbers is correctly written in Scientific Notation, it is necessary to be able to manipulate our decimal point for using operations with Scientific Notation.

14. Multiplying with Scientific Notation- Use multiplication rules and laws of exponents to evaluate: Problem: (2.8 x 103)(5.1 x 10-7) • multiply the coefficient • Keep your base and add the exponents • You will get: 14.28 x 10-4 • To put the answer in scientific notation, you have to move the decimal one place LEFT.Answer: 1.428 x 10-3

15. Let’s Try One! • (3 x 103)(8 x 109) • Step one? • Step two? • Step three? • Are we done?

16. Dividing with Scientific Notation-Use division rules and laws of exponents to evaluate: 4.5 x 1051.5 x 102 • Divide the coefficient • Keep your base and subtract the exponents Write your answer in scientific notation. 4.5= 3.00 105= 1031.5 102 3.00 x 103

17. Let’s try one! • 7.2 x 10-31.2 x 102 • Step one? • Step two? • Step three? • Are we done? Which is scientifically notated as 6 x 10 -5 The answer in decimal notation is 0.00006

18. What is the Pattern? 32000 x 10¯³ = 32 3200 x 10¯² = 32 320 x 10¯¹ = 32 32 x 10° = 32 3.2 x 10¹ = 32 .32 x 10² = 32 .032 x 10³ = 32 • Notice how as the coefficient decreases, and the exponent increases, but the answer is always 32! • Even though only one these numbers is correctly written in Scientific Notation, it is necessary to be able to manipulate our decimal point for using operations with Scientific Notation.

19. Adding and Subtracting with Scientific Notation • When adding or subtracting numbers in scientific notation, the exponents must be the same. • If they are different, you must convert one of the numbers so that they will have the same exponent.

20. Adding With the Same Exponent • (3.45 x 103) + (6.11 x 103) • 3.45 + 6.11 = 9.56 • 9.56 x 103

21. Adding With Different Exponents • (4.12 x 106) + (3.94 x 104) • (412 x 104) + (3.94 x 104) • 412 + 3.94 = 415.94 • 415.94 x 104 • Express in proper form: 4.15 x 106 To make the exponents the same, manipulate your number by moving your decimal and changing your exponent by equal amounts (levels).

22. Example 2 You can also expand your numbers into standard form and then convert them back into Scientific Notation. • 2.46 X 106 + 3.4 X 103 • 246000 + 3400 = 249400 • 2.494 x 10⁵ Put your answer in Scientific Notation!

23. Subtracting With the Same Exponent • (8.96 x 107) – (3.41 x 107) • 8.96 – 3.41 = 5.55 • 5.55 x 107

24. Subtracting With Different Exponents • (4.23 x 103) – (9.56 x 102) • (42.3 x 102) – (9.56 x 102) • 42.3 – 9.56 = 32.74 • 32.74 x 102 • Express in proper form: 3.27 x 103 To make the exponents the same, manipulate your number by moving your decimal and changing your exponent by equal amounts (levels).

25. Example 2 You can also expand your numbers into standard form and then convert them back into Scientific Notation. • 5.762 X 103 – 2.61 X 102 • 5762 – 261 = 5501 • Answer: 5.501 X 103 Put your answer in Scientific Notation!