Sci Not & Sig Dig

1. COMMUNICATING IN
SCIENCE
Scientific Notation
Significant Digits/Figures

2. Scientific Notation
A quick way to write
REALLY, REALLY BIG
or REALLY, REALLY SMALL numbers.

3. Mathematicians are lazy!!!
• They decided that by using powers of 10, they can create short
versions of long numbers.

4. Rules for Scientific Notation
• To be in proper scientific notation the number must be:
• First number must between 1 and 9, then a number after the decimal
place – if this is required!
• multiplied by a power of ten
• Correct:
• 2 x 103
• 5.7 x 108
• Incorrect: (why?)
• 57 X 108
• 2 x 103.5

5. Practice: Write the width of the universe in
scientific notation
210,000,000,000,000,000,000,000 miles
Step 1. Locate the decimal in the above number.
After the last zero.
Step 2: Where would you put the decimal to make this
number be between 1 and 10?
Between the 2 and the 1
Your number now looks like this:
2.10,000,000,000,000,000,000,000

6. Practice: Write the width of the universe in
scientific notation
2.10,000,000,000,000,000,000,000
•Step 3: How many decimal places did you move the decimal?
• 23 (this becomes the exponent but…)
•Step 4: Determine the sign of the exponent. When the original
number is more than 1, the exponent is positive.
• In this case it is positive 23
•The answer in scientific notation is:
2.1 x 1023

7. Learning Checks
Using scientific notation, rewrite the following numbers.
A.347,000
3.47 X 105
A.902,000,000
9.02 X 108
A.61,400
6.14 X 104

8. Learning Check
• In the United States, 15,000,000 households use private wells
for their water supply. Write this number in scientific notation.
• 1.5 X 107

9. Practice: Express 0.0000000902 in
scientific notation.0.0000000902
Step 1: Where would the decimal go to make the number
be between 1 and 10?
9.02
Step 2: The decimal was moved how many places?
8
Step 3: Determine the sign of the exponent. When the
original number is less than 1, the exponent is negative.
9.02 x 10-8

10. Why does a Negative Exponent give us a
small number?
Do you see a pattern?
10000 = 10 x 10 x 10 x 10 = 104
1000 = 10 x 10 x 10 = 103
100 = 10 x 10 = 102
10 = 101
1 = 100

11. Why does a Negative Exponent give us a
small number?
Continue the pattern:
= 10-1
= = 10-2
= = 10-3
= = 10-4

12. Learning Checks
Write the following in
scientific notation:
A.0.000882
8.82 X 10-4
A.0.00000059
5.9 X 10-7
A.0.00004
4 X 10-5
D. 0.0004
4 X 10-4
D. 0.00000306
3.06 X 10-6
D. 0.000892
8.92 X 10-4

13. Learning Checks
• A ribosome, is about 0.000000003 of a meter in diameter.
Write the length in scientific notation.
• 3 X 10-9

14. Converting Scientific Notation to Standard
Form
• Move the decimal place by the number of steps indicated by
the exponent placing zeros to fill in the gaps
• A positive exponent moves the decimal place to the right to
make the number larger
• Example: 3.4 x 105
• 340,000 in standard form
• A negative exponent moves the decimal place to the left to
make the number smaller
• Example: 1.8 x 10-4
• 0.00018

15. Learning Checks
Convert from scientific notation to standard form:
A.1.23 X 105
123,000
A.6.806 X 106
6,806,000
A.6.27 x 103
6,27
A.9.01 x 104
90100

16. Learning Check
• The U.S. has a total of 1.2916 X 107
acres of land reserved for
state parks. Write this in standard form.
• 12,916,000 acres

17. Learning Checks
Convert from scientific notation to standard form.
A.1.248 X 10-6
0.000001248
A.6.123 X 10-5
0.00006123
A.1.23 X 10-4
0.000123
A.6.806 X 10-3
0.006806

18. Learning Checks
• The nucleus of a human cell is about 7 X 10-6
meters in
diameter. What is the length in standard notation?
• 0.000007

19. Learning Checks
• Convert to scientific notation or standard form as needed:
A.0.004
• 4 X 10-3
A.2.48 X 105
• 248,000
A.6.123 X 10-4
• 0.0006123
A.306,000,000
• 3.06 X 108

20. Scientific Notation in Calculators
• Express 4.58 x 106
in standard notation.
• On the graphing calculator, scientific notation is done with the
button.
• 4.58 x 106
is typed 4.58 6

21. Practice Questions
• W/b
• Page 8 a, c, e,… for all questions
• Page 9 a, c, e for all questions

22. Significant Digits in Measurement
• The numbers reported in a measurement are limited by the measuring tool
• Significant digits in a measurement include the known digits plus one
estimated digit from the measuring device
• On an analogue measuring device (where you must interpret
measurement, ex: triple beam balance to measure mass), the significant
digits are the digits you can read off the instrument plus one estimated
digit
• Ex: What temperatures can you make off the following analogue
thermometres?

23. Significant Digits in Measurement
• On a digital measuring device (where the measurement is displayed for
you, ex: digital clock), the significant digits are all the digits you see. The
final digit in the display is considered to be the estimated digit
• Ex: How many significant digits are displayed on the two digital clocks?
What is the estimated digit for each clock?
A)
B)

24. Counting Significant Digits
• Many times, you will not be making the measurements, they
will simply be given to you. In this case, you need to know
what is and is not considered to be a significant digit.
• All non-zero digits in a measured number are: significant or
not significant?
Number of Significant Figures
38.15 cm 4
5.6 ft 2
65.6 lb ?
122.55 m ?

25. Counting Significant Digits
• All non-zero digits in a measured number are significant.
Number of Significant Figures
38.15 cm 4
5.6 ft 2
65.6 lb 3
122.55 m 5

26. Leading Zeros
Leading zeros in decimal numbers are: significant or not
significant?
Number of Significant Figures
0.008 mm 1
0.0156 oz 3
0.0042 lb ?
0.000262 mL ?

27. Leading Zeros
• Leading zeros in decimal numbers are not significant.
Number of Significant Figures
0.008 mm 1
0.0156 oz 3
0.0042 lb 2
0.000262 mL 3

28. Sandwiched Zeros
Zeros between nonzero numbers are: significant or not
significant?
Number of Significant Figures
50.8 mm 3
2001 min 4
0.702 lb ?
0.00405 m ?

29. Sandwiched Zeros
• Zeros between nonzero numbers are significant.
Number of Significant Figures
50.8 mm 3
2001 min 4
0.702 lb 3
0.00405 m 3

30. Trailing Zeros (place holders)
Trailing zeros in numbers that serve as place holders are:
significant or not significant?
Number of Significant Figures
25,000 in. 2
200 yr 1
48,600 gal 3
25,005,000 g ?

31. Trailing Zeros (place holders)
• Trailing zeros in numbers that serve as place holders are not
significant.
Number of Significant Figures
25,000 in. 2
200 yr 1
48,600 gal 3
25,005,000 g 5

32. Trailing Zeros (not place holders)
Trailing zeros in numbers that are not place holders (after
numbers in decimals) are: significant or not significant?
Number of Significant Figures
2500.0 in. 5
20.0 yr 3
4860.0 gal 5
25,005,000.0 g ?

33. Trailing Zeros (not place holders)
• Trailing zeros in numbers that are not place holders (after
numbers in decimals) are significant.
Number of Significant Figures
2500.0 in. 5
20.0 yr 3
4860.0 gal 5
25,005,000.0 g 9

34. Review: When Is A Number
Significant?
• Some numbers are exact such as counted quantities (ex: 29 students
in this class); exact numbers have “infinite” number of significant
digits.
• In Science, we tend to use measurements, we say that
measurements are not exact, measurements are estimates,
therefore they have a distinct number of significant digits.
• When given a measured value, non-zero digits are always significant;
it’s zeroes that can be ambiguous;
• Zeroes inbetween significant digits are significant
• Leading zeroes are always insignificant
• Trailing zeroes are:
• Significant when located after a decimal
• Insignificant when no decimal in the number

35. Learning Checks
• Which answers contain 3 significant digits?
a) 0.4760 b) 0.00476 c) 4760
• All the zeros are significant in:
a) 0.00307 b) 25.300 c) 2.050 x 103
• 534,675 rounded to 3 significant digits is:
a) 535 b) 535,000 c) 5.35 x 105

36. Solution
• Which answers contain 3 significant digits?
a) 0.4760 b) 0.00476 c) 4760
• All the zeros are significant in:
a) 0.00307 b) 25.300 c) 2.050 x 103
• 534,675 rounded to 3 significant digits is:
a) 535 b) 535,000 c) 5.35 x 105

37. Learning Check
How many significant digits are there in each of the following?
A. 0.030 m 1 2 3
B. 4.050 L 2 3 4
C. 0.0008 g 1 2 4
D. 3.00 m 1 2 3
E. 2,080,000 bees 3 5 7

38. Solution
How many significant digits are there in each of the following?
A. 0.030 m 1 2 3
B. 4.050 L 2 3 4
C. 0.0008 g 1 2 4
D. 3.00 m 1 2 3
E. 2,080,000 bees 3 5 7

39. Significant Numbers in Calculations
• A calculated answer cannot be more precise than the
measuring tool.
• A calculated answer must match the least precise
measurement. In other words:
• your final calculated value can only be as precise as the least precise
value in your calculation
• OR you are giving the values in your calculation more precision than
they actually have if you do not round your final answer off to the correct
number of significant digits
• Significant figures are needed for final answers from:
• adding or subtracting
• multiplying or dividing

40. Adding and Subtracting
• The answer has the same number of decimal places as the
measurement with the fewest decimal places.
25.2 one decimal place
+ 1.34 two decimal places
26.54
answer 26.5 one decimal place

41. Learning Check
In each calculation, round the answer to the correct number of
significant digits.
• 235.05 + 19.6 + 2.1 =
a) 256.75 b) 256.8 c) 257
• 58.925 - 18.2 =
a) 40.725 b) 40.73 c) 40.7

42. Solution
In each calculation, round the answer to the correct number of
significant digits.
• 235.05 + 19.6 + 2.1 =
a) 256.75 b) 256.8 c) 257
• 58.925 - 18.2 =
a) 40.725 b) 40.73 c) 40.7

43. Multiplying and Dividing
• Round (or add zeros) to the calculated answer until you have
the same number of significant digits as the measurement
with the fewest significant figures.

44. Learning Check
• 2.19 X 4.2 =
a) 9 b) 9.2 c) 9.198
• 4.311 ÷ 0.07 =
a) 61.58 b) 62 c) 60
• 2.54 X 0.0028 =
0.0105 X 0.060
a) 11.3 b) 11 c) 11.29

45. Solution
• 2.19 X 4.2 =
a) 9 b) 9.2 c) 9.198
• 4.311 ÷ 0.07 =
a) 61.58 b) 62 c) 60
• 2.54 X 0.0028 =
0.0105 X 0.060
a) 11.3 b) 11 c) 0.041