2D Scientific Notation

Numbers in science and statistics classes are sometimes very small (like the size of an atom) or very large (like the distance from Earth to the Sun). Numbers like that can have a huge amount of zeros and are very difficult to read, let alone interpret. Scientific Notation is a way of writing numbers that makes them more manageable, especially very small and very large quantities.

One number that is very important in statistics is “P-value”. P-values lie from zero to one; this supplied example is very close to zero. For example you may see a P-value = 0.000000000173. This is a long decimal representation. Let’s talk about scientific notation and see if there is an easier way of writing this number.

To understand scientific notation we need to first review multiplying and dividing decimals by powers of ten. These concepts were covered in our last two sections. Notice a power of 10 like ${10}^4$ is a 1 with four zeros or 10000. When we multiply a decimal by ${10}^4$, we move the decimal 4 places to the right (making it larger). When we divide a decimal by ${10}^4$, we move the decimal 4 places to the left (making it smaller).

Dividing a decimal by ${10}^4$ is the same as multiplying by ${10}^{-4}$. The negative exponent means it is in the denominator of a fraction (division). Let’s review this idea. So if I multiply 3.5 x ${10}^4$ = 3.5 x 10000 = 35000. If I multiply 3.5 x ${10}^{-4}$= $3.5÷10000$ = 0.00035. Multiplying by a negative power of 10 makes the decimal small. Multiplying by a positive power of 10 makes the number large. OK, now we are ready for scientific notation.

Scientific Notation

For the positive numbers used here, a number is in scientific notation if it is a number between 1 and 10 $(1\le \text{number}<10)$, times a power of 10. For example we saw above that 35000 = 3.5 x ${10}^4$. This is scientific notation. The first number is between 1 and 10 and we are multiplying by a power of 10. 35000 also equals 35 x ${10}^3$. This is not scientific notation since the first number is not between 1 and 10.

How do we convert a large or small number into scientific notation? Let’s look at an example. Suppose we are looking at a distance in astronomy of 83,000,000 miles. To write this number in scientific notation, ask yourself where the decimal point would be to make it a number between 1 and 10. Right now the decimal point is after the last zero. To be a number between 1 and 10 the decimal point needs to be between the 8 and 3. So our scientific notation number should be 8.3 x some power of 10. What power of 10 should we use? We moved the decimal point 7 places so the power of 10 should be ${10}^7$ or ${10}^{-7}$, but which one? Do we need 8.3 to get larger or smaller so it will equal 83,000,000 miles? Larger, of course. So we should use the positive exponent since that makes numbers larger. So the scientific notation for 83,000,000 miles is 8.3 x ${10}^7$.

Now you try the next couple of examples with your instructor.

Practice Problems Section 2D

Convert the following numbers into scientific notation.

1. 2,450

   Check answer

   $$2.45\times {10}^3$$

2. 470,000

3. 13,600

   Check answer

   $$1.36\times {10}^4$$

4. 0.00031

5. 0.0087

   Check answer

   $$8.7\times {10}^{-3}$$

6. 0.00047

7. 570,000

   Check answer

   $$5.7\times {10}^5$$

8. 36,000

9. 0.000000647

   Check answer

   $$6.47\times {10}^{-7}$$

10. 7,400,000

11. 0.000132

    Check answer

    $$1.32\times {10}^{-4}$$

12. 0.00004152

13. 5,200,000

    Check answer

    $$5.2\times {10}^6$$

14. 1,450,000

15. 0.0000089

    Check answer

    $$8.9\times {10}^{-6}$$

16. 0.0003016

17. 0.00000132

    Check answer

    $$1.32\times {10}^{-6}$$

18. 647,000,000

19. 52,600,000

    Check answer

    $$5.26\times {10}^7$$

20. 6,111,000,000

21. 0.0000993

    Check answer

    $$9.93\times {10}^{-5}$$

22. 0.008144

23. 0.0000306

    Check answer

    $$3.06\times {10}^{-5}$$

24. 189,000

25. 13,800

    Check answer

    $$1.38\times {10}^4$$

26. 6,458,000

27. 440,000

    Check answer

    $$4.4\times {10}^5$$

28. 0.0004881

29. 0.00000055

    Check answer

    $$5.5\times {10}^{-7}$$

30. 0.0261

31. 246,670

    Check answer

    $$2.4667\times {10}^5$$

32. 0.00000132

33. 0.0000011

    Check answer

    $$1.1\times {10}^{-6}$$

34. 0.0000816

35. 672,100

    Check answer

    $$6.721\times {10}^5$$

36. 94,000

The following numbers are in scientific notation. Write the actual number they represent.

37. 1.48 x ${10}^3$

    Check answer

    1,480

38. 3.9 x ${10}^{-1}$

39. 4.975 x ${10}^9$

    Check answer

    4,975,000,000

40. 6.42 x ${10}^{-6}$

41. 9.25 x ${10}^{-2}$

    Check answer

    0.0925

42. 7.113 x ${10}^3$

43. 7.6 x ${10}^4$

    Check answer

    76,000

44. 4.81 x ${10}^{-7}$

45. 2.59 x ${10}^6$

    Check answer

    2,590,000

46. 8.94 x ${10}^{-8}$

47. 3.15 x ${10}^{-2}$

    Check answer

    0.0315

48. 1.45 x ${10}^7$

49. How can we tell if a number in scientific notation represents a large or small number?

    Check answer

    Small numbers have a negative exponent. Large numbers have a positive exponent.

    Publisher clarification: for a positive normalized coefficient at least 1 and less than 10, a negative exponent gives a value between zero and one; exponent zero gives a value at least 1 and less than 10; a positive exponent gives a value at least 10. Large and small are contextual descriptions.

50. Let’s see if you are ready to tackle the P-value example from the beginning of the section. Remember the P-value was 0.000000000173. Write this number in scientific notation. We often compare P-values to 0.05, so was this P-value larger or smaller than 0.05?

51. Statistics computer programs can also calculate P-values, but they are often given in scientific notation. Here is a printout from the stat program. (P-value = 3.578 x ${10}^{-6}$)

    What number is this scientific notation representing? Is it larger or smaller than 0.05?

    Check answer

    0.000003578, smaller

52. A science journal stated that the distance from the Earth to Pluto is approximately 4.67 billion (4,670,000,000) miles. Write this number in scientific notation.

53. An astronomy book stated that the distance between Mars and the Sun is approximately 2.279 x ${10}^8$ kilometers. Since this is scientific notation, what is the actual number?

    Check answer

    227,900,000 km

54. A chemistry book estimates that the diameter of a particular atom is about 0.0000000107 centimeter. Write this number in scientific notation.

55. We estimate that a human hair has a diameter of 0.00008 meters. Write this number in scientific notation.

    Check answer

    $$8\times {10}^{-5}$$

    meters

56. A student said that the scientific notation for 6,800 feet is 68 x${10}^2$. Why is this not scientific notation?

57. A student said that the scientific notation for the diameter of a human hair is $0.08\times {10}^{-3}$. Why is this not scientific notation?

    Check answer

    Publisher clarification: for the positive values used here, the first number must be at least 1 and less than 10: 1 is included and 10 is excluded. The coefficient 0.08 is less than 1.

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.