Course Notes 31 the Importance of Measurements Section Review
 | Math Skills Review Significant Figures |
There are two kinds of numbers in the world:
- exact:
- example: There are exactly 12 eggs in a dozen.
- example: Most people have exactly ten fingers and 10 toes.
- inexact numbers:
- instance: whatsoever measurement.
If I quickly mensurate the width of a piece of notebook newspaper, I might get 220 mm (ii significant figures). If I am more than precise, I might get 216 mm (3 pregnant figures). An even more than precise measurement would exist 215.6 mm (4 significant figures).
- instance: whatsoever measurement.
PRECISION VERSUS Accurateness
Accuracy refers to how closely a measured value agrees with the right value.
Precision refers to how closely individual measurements concur with each other.
In whatever measurement, the number of significant figures is critical. The number of significant figures is the number of digits believed to exist correct past the person doing the measuring. It includes one estimated digit. Then, does the concept of significant figures deal with precision or accuracy? I'll answer this question afterwards you peruse the next instance.
Let's look at an example where significant figures is of import: measuring volume in the laboratory. This can be washed in many ways: using
- a beaker with volumes marked on the side,
- a graduated cylinder, or
- a buret.
A rule of pollex: read the volume to 1/10 or 0.1 of the smallest segmentation. (This rule applies to whatsoever measurement.) This ways that the error in reading (chosen the reading error) is 
one/ten or 0.one of the smallest division on the glassware. If you are less sure of yourself, you can read to i/5 or 0.2 of the smallest segmentation.
| Chalice |  | The smallest division is x mL, so we tin read the volume to one/x of x mL or 1 mL. The volume we read from the beaker has a reading fault of one mL. The volume in this chalice is 47 So, How many pregnant figures does our volume of 47 |
| Graduated Cylinder |  Look in the textbook for a flick of a graduated cylinder. | First, notation that the surface of the liquid is curved. This is called the meniscus. This miracle is acquired past the fact that water molecules are more attracted to glass than to each other (adhesive forces are stronger than cohesive forces). When nosotros read the volume, nosotros read it at the BOTTOM of the meniscus. The smallest sectionalization of this graduated cylinder is 1 mL. Therefore, our reading fault will be How many meaning figures does our answer accept? 3! The "3" and the "vi" we know for sure and the "five" we had to guess a little. |
| Buret |  Look in the textbook for a picture of a buret. Note that the numbers get bigger as you lot go downwards the buret. This is different from the chalice or the graduated cylinder. This is because the liquid leaves the buret at the bottom. | The smallest division in this buret is 0.one mL. Therefore, our reading error is 0.01 mL. A expert book reading is 20.38 0.01 mL. An equally precise answer would exist xx.39 mL or 20.37 mL. How many meaning figures does our reply have? 4! The "2", "0", and "3" we definitely know and the "eight" we had to estimate. |
Conclusion: The number of significant figures is straight linked to a measurement. If a person needed just a rough gauge of volume, the beaker volume is satisfactory (two significant figures), otherwise one should use the graduated cylinder (3 significant figures) or better nonetheless, the buret (four significant figures).
Then, does the concept of meaning figures deal with precision or accuracy? Hopefully, you can run into that it really deals with precision only. Consider measuring the length of a metal rod several times with a ruler. Yous volition get essentially the same measurement over and over once more with a minor reading error equal to near 1/x of the smallest segmentation on the ruler. You have adamant the length with high precision. However, y'all don't know if the ruler was authentic to begin with. Perhaps it was a plastic ruler left in the hot Texas sun and was stretched. You don't know the accuracy of your measuring device unless you calibrate it, i.due east. compare it against a ruler you knew was accurate. Note: in the laboratory, a good analytical chemist always calibrates her volumetric glassware before using it by weighing a known volume of liquid dispensed from the glassware. Past dividing the mass of the liquid by its density, she tin decide the actual volume and hence the accurateness of the glassware.
Rules for Working with Significant Figures:
- Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant simply if the decimal point is specified.
Hint: Change the number to scientific annotation. It is easier to see. - Improver or Subtraction:
The last digit retained is set by the start hundred-to-one digit. - Multiplication or Partition:
The answer contains no more significant figures than the least accurately known number.
EXAMPLES:
| Example | Number of Pregnant Figures | Scientific Annotation | ||
| 0.00682 | 3 | 6.82 x x -3 | Leading zeros are non pregnant. | |
| one.072 | 4 | 1.072 (x 100) | Imbedded zeros are always significant. | |
| 300 | 1 | three 10 102 | Abaft zeros are significant only if the decimal point is specified. | |
| 300. | iii | iii.00 ten 10ii | ||
| 300.0 | iv | 3.000 10 ten2 |
EXAMPLES
| Addition |  | Even though your figurer gives yous the answer eight.0372, you must round off to 8.04. Your respond must only contain 1 doubtful number. Note that the doubtful digits are underlined. |
| Subtraction |  | Subtraction is interesting when concerned with significant figures. Even though both numbers involved in the subtraction have five significant figures, the answer only has 3 pregnant figures when rounded correctly. Recall, the respond must merely take 1 doubtful digit. |
| Multiplication |  | The answer must exist rounded off to ii significant figures, since i.half-dozen only has 2 significant figures. |
| Sectionalisation |  | The answer must exist rounded off to 3 significant figures, since 45.two has only 3 pregnant figures. |
Notes on Rounding
- When rounding off numbers to a certain number of significant figures, practice so to the nearest value.
- instance: Round to iii significant figures: 2.3467 x ten4 (Answer: 2.35 x 104)
- example: Round to 2 significant figures: ane.612 x 103 (Reply: 1.6 x 103)
- What happens if there is a 5? In that location is an arbitrary rule:
- If the number before the v is odd, round upward.
- If the number before the 5 is even, allow it be.
The justification for this is that in the grade of a series of many calculations, any rounding errors will exist averaged out. - case: Round to ii significant figures: two.35 x 102 (Answer: ii.four x 102)
- instance: Round to two significant figures: 2.45 x ten2 (Answer: ii.4 x 10two)
- Of course, if we round to two significant figures: ii.451 10 ten2, the respond is definitely ii.5 10 x2 since 2.451 10 tenii is closer to 2.5 x 102 than ii.4 x xtwo.
QUIZ:
| Question i | Give the right number of significant figures for 4500, 4500., 0.0032, 0.04050 |
| Question 2 | Give the reply to the correct number of pregnant figures: 4503 + 34.90 + 550 = ? |
| Question 3 | Requite the respond to the right number of significant figures: one.367 - 1.34 = ? |
| Question 4 | Requite the reply to the correct number of pregnant figures: (1.3 ten 103)(five.724 x x4) = ? |
| Question 5 | Give the answer to the right number of significant figures: (6305)/(0.010) = ? |
Answers: (one) 2, 4, 2, iv (2) 5090 (three pregnant figures - round to the tens place - prepare by 550) (3) 0.03 (1 significant effigy - circular to hundredths identify) (four) 7.4 x tenvii (2 significant figures - set by 1.iii 10 103) (5) half dozen.3 ten 10v (2 meaning figures - gear up by 0.010)
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Source: https://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
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