Rules for Counting Significant Figures 1.
Nonzero Integers. Nonzero integers always count as significant figures. Ex: Ex:
2.
Zeros. There are three classes of zeros: a.
Leading zeros are zeros that precede all the nonzero digits. These do not count as significant figures. Ex: Ex:
b.
c.
The number .00092 has only 2 significant figures. The number 0.0345 has only 3 significant figures.
Captive zeros are zeros between nonzero digits. significant figures. Ex: Ex:
These always count as
The number 905, has 3 significant figures. The number 64,004 has 5 significant figures.
Trailing zeros are zeros at the right end of the number. They are significant only if the number contains a decimal point. Often times a line is placed over the last zero that is significant if a decimal point is not used. Ex: Ex: Ex: Ex:
3.
The number 72,555 has 5 significant figures. The number 343 has 3 significant figures.
The number 7,000 has only 1 significant figure. The number .00350 has 3 significant figures. The number 200. has 3 significant figures. The number 12,000 has 4 significant figures.
Exact numbers. Many times calculations involve numbers that were not obtained using measuring devices but were determined by counting.
Ex:
18 students, 8 apples, 356 molecules
Such numbers are called exact numbers. They can be assumed to have an infinite number of significant figures. Often times they are part of an equation. Ex:
circumference = 2r where 2 is an exact number
Exact numbers also can arise from definitions. Ex:
1 inch = 2.54cm This is a definition so both numbers are exact.
Rules for Rounding
1.
If the digit to be removed a.
is less than 5, the preceding digit stays the same. Ex:
b.
is greater than 5, the preceding digit is increased by 1. Ex:
c.
1.33 rounds to 1.3
1.36 rounds to 1.4
is EXACTLY 5: - if the preceding digit is EVEN (which includes zero), it stays the same - if the preceding digit is ODD, it is increased by 1. Ex:
2.
2.85 rounds to 2.8 1.35 rounds to 1.4
When rounding off, use only the first number to the right of the last significant figure.
Rules for Using Significant Figures in Calculations
1.
2.
For multiplication or division, the number of significant figures in the result is the same as that in the measurement with the smallest number of significant figures. We say this measurement is limiting, because it limits the number of significant figures in the result.
Ex:
4.56 x 1.4 = 6.38
which is rounded to 6.4 because 1.4 has only 2 significant figures.
Ex:
8.315 298 = .0279027
which is rounded to .0279 because 298 has 3 significant figures
For addition or subtraction, the limiting term is the number that has the leftmost estimated digit. Round the final answer to the place value of this estimated digit. If all number have decimal values the limiting term is the one with the smallest number of decimal places. Ex:
12.11 + 18.0 + 1.013 = 31.123
which is rounded to 31.1 b/c 18.0 has only one decimal place
SPECIAL NOTE ABOUT SCIENTIFIC NOTATION: All numbers shown in the base are considered to be significant:
Ex: 3.45 x 10 -5 has 3 significant figures 2.000 x 10 8 has 4 significant figures
SIGNIFICANT DIGITS Summary: associated with every measurement made is some degree of uncertainty. For instance, you might measure the length of the dark line shown in the diagram as 20.7 cm. The 0 5 10 15 20 25 cm digits 2 and 0 are certain - there is no doubt that the length is "20 point something" cm. The 7 is uncertain - it might be a little less or The length of the line is approx. 20.7 cm. a little more. The number of ‘significant digits’ indicates the The 2 and 0 are certain, the 7 is uncertain. certainty of our measurement. There are three significant digits in All three digits are significant. this case (20.7). Thus, significant digits in a measurement or calculation consist of all those digits that are certain, plus one uncertain digit. Although your calculator may give you an answer to eight decimal places or more, you should not include all of these digits in your answer.
Rules For Determining The Number Of Significant Digits If you have trouble determining the number of significant digits, follow these steps. 1. All digits from 1 to 9 (non-zero digits) are considered to be significant. Example Number of significant digits 1.23 g 3 2.
Zeros between non-zero digits are always significant 1.03 g 3
3.
Zeros to the left of non zero digits, serve only to locate the decimal point; they are not significant. 0.00123 g 3; zeros to the left of the 1 simply locate the decimal point. To avoid confusion you can write numbers in scientific notation. I.e. 0.00123 = 1.23 x 10–3
4.
Any zero printed to the right of a non-zero digit is significant if it is also to the right of the decimal point. 2.0 g and 0.020 g 2 for both; all zeros that are right of both a non-zero digit and the decimal point are significant.
5.
Any zero printed to the right of a non-zero digit may or may not be significant if there is no decimal point indicated. For example, if someone tells you that a mountain is 3600 m high they are probably certain of the 3, and uncertain of the 6. In other words, there are likely 2 significant digits. However 3600 m may also have 3 significant digits (if the measurement was taken to the nearest 10 m) or 4 significant digits if the measurement was taken to the nearest 1 m). 100 g 1, 2, or 3; in numbers that do not contain a decimal point, "trailing" zeros may or may not be significant. To eliminate possible confusion, one practice is to underline the last significant digit. Thus, 100 has two significant digits, whereas 100 has three. Ideally, we write the number in scientific notation: for example 1.0 x 10–2 has two significant digits and 1.00 x 10–2 has three significant digits. Notice that for numbers written in scientific notation, all digits are significant.
6.
Any number that is counted instead of measured has an infinite number of significant digits. 3 test tubes Infinite; exact numbers, for example, the number of meters in a kilometer or numbers obtained by counting (4 people, 5 beakers), are said to have an infinite number of significant digits.
A) How many significant digits do the following measured quantities have? i) 2.83 cm iii) 14.0 g v) 0.02 mL vii) 2.350 x 10–2 L ii) 36.77 mm iv) 0.0033 kg vi) 0.2410 km viii) 1.00009 L
ix) 3 fingers x) 0.0056040 g
B) i) 83.25 – 0.1075 ii) 4.02 + 0.001 iii) 0.2983 + 1.52 C) i) 7.255 ÷ 81.334 ii) 1.142 x 0.002 iii) 31.22 x 9.8 D) Solve the following (do one step at a time, according to BEDMAS): i) 6.12 x 3.734 + 16.1 ÷ 2.3 ii) 0.0030 + 0.02 iii) 1.70 x 103 + 1.34 x 105 iv) (33.4 + 112.7 + 0.032)/(6.487) E) Convert these measurements: i) 1.0 cm = ______ m, ii) 0.0390 kg = ______ g, iii) 1.7 m = _____ mm
Worksheet: Scientific Notation, Part 2
Name____________________
Express these numbers in scientific notation. 1. 49,100
_____________
4. 0.02045 _____________
2. 588.39
_____________
5. 0.111
_____________
3. 7.27
_____________
6. 300.6
_____________
Express these numbers in decimal notation. 1. 8.32 x 10 -2_____________
4. 1.457 x 102_____________
2. 5.4 x 104
_____________
5. 3.00 x 10-1 _____________
3. 9.67 x 103 _____________
6. 2.22 x 10-6 _____________
Use your calculator to solve the following. Be sure to express your answer in scientific notation even if your calculator does not. 1. (3.0 x 105) + (5.0 x 104)
_____________
2. (8.5 x 104)(6.0 x 10-4)
_____________
(1.0 109 )(9.1 10 5 ) 3. (1.4 102 )
_____________
(4.84 106 ) 4. (1.936 10 5 )
_____________
PHYSICSFundamentals © 2004, GPB
1-08
Name___________________________________ Date___________________ Pd _____________
Scientific Notation Practice 1. When the number 7312 is expressed in scientific notation, it is written as __________ x 103 2. When 3314 is written in scientific notation, the exponent indicating the power of ten will be _______. 3. When the number 4.512 x 103 is written in standard notation, it is expressed _____________. 4. When 0.0021 is written in scientific notation, the exponent is (positive/negative), whereas when 4540 is written in scientific notation, the exponent is (positive/negative). 5. For each of the following numbers expressed in scientific notation, write the number as standard notation a. 7.214 x 10-4 b. 6.015 x 105 c. 4.299 x 103 d. 9.089 x 10-6 6. For each of the following numbers, if the number is rewritten in standard scientific notation, what will be the value of the exponent for the power of ten? a. 0.000067 b. 9,331,442 c. 1/10,000 d. 163.1 x 102 7. Express each of the following numbers in scientific notation. a. 12,500 b. 37,400,000 c. 602,300,000,000,000,000,000,000 d. 375 e. 0.0202 f. 0.1550 g. 0.000 0104 h. 0.000 000 000 000 000 000 129 8. Mixed practice … If the number is in scientific notation, write it in standard notation … If the number is in standard notation, write it in scientific notation. a. 6.9193901 x 107 b. 8.114 x 10-3 c. 52.3 d. 54,331,000 e. 1.801 x 104
Name: _____________
Scientific Notation and Metric Prefixes 1. Write the following numbers in scientific notation: a. 31,210.5
________________________
b. 0.0032105 ________________________ c. 61,000,000 ________________________ d. 0.061
________________________
2. Write the following numbers in standard notation: a. 6.789 x 10-6 ________________________ b. 6.789 x 106
________________________
c. 4.29 x 101
________________________
d. 9.42 x 10-1
________________________
3. Name the following quantities using SI prefixes and also write the symbol for each. a. .1 m =
__________________ & ___________
b. 1,000,000,000 m
__________________ & ___________
c. 103 m
__________________ & ___________
d. 10-6 m
__________________ & ___________
4. For each of the following, rank the units from bigger (highest value) to smallest (least value). a. m, Mm, km = ____________________________________ b. pm, cm. mm = ____________________________________ c. mm, Mm, μm = ____________________________________ d. dm, mm, nm = ____________________________________
MATH IN PHYSICS Name
Date Metric System
DIRECTIONS: Convert each of the given measurements to the unit indicated. 1. 100 cm to m 2. 500 g to kg 3. 0.01 m to cm 4. 250 ml to L 5. 35 kg to g 6. 0.89 L to mL 7. 3.484 cm to mm 8. 15.93 mg to dg 9. 435.8 ms to s 10. 89.05 ml to dL 11. 0.00467 kg cg 12. 84.59 A to cm 13. 9.45 µm to mm 14. 10.4 cm to mm 15. 34. 2 mm to cm 16. 192.3 cg to mg 17. 4.006 L to mL 18. 70.5 cm3 to mL 19. 2.58 µg to cg 20. 1.5 x 104 g to cg 21. 3.2 x 10‐3 cm to mm 22. 0.5 x 10‐9 A to mm 23. 8.22 x 10‐5 µm to A 24. 14.8 x 10‐8 cm to A 25. 5.552 x 1014 µm to km 26. 3.680 x 10‐9 cm to µm 27. 9.92 x 107 mm to m 28. 14.82 x 10‐3 g to cg 29. 4.82 x 105 mg to kg
1. Convert a measurement of 50 cm to its equivalent in mm 2. Convert a measurement of 25 mg to its equivalent incg 3. Convert a measurement of 36 mL to its equivalent in dL 4. Convert a measurement of 89 cg to its equivalent in mg 5. Convert a measurement of 22.9 mLto its equivalent in µL 6. Convert a measurement of 60.3 to its equivalent in cg 7. Convert a measurement of 82.5 to its equivalent in mL 8. Convert a measurement of 0.8209 cg to its equivalent in mg 9. Convert a measurement of 264.77 nm to its equivalent in cm 10. Convert a measurement of 0.0489 µm to its equivalent in mL 11. Convert a measurement of 1.1194 dm to its equivalent in mm 12. Convert a measurement of 0.00804 cm to its equivalent in mm 13. Convert a measurement of 37.85 mg to its equivalent in cg 14. Convert a measurement of 1.0 x 10‐4 cg to its equivalent in mg 15. Convert a measurement of 5.00 x 103 mm to its equivalent in cm 16. Convert a measurement of 0.588 x 103 mg to its equivalent in dg 17. Convert a measurement of 7.755 x 103 µg to its equivalent in mg
