Case 1: Inexact quantity
Suppose we have some sort of salty food. On the label it says:
"Sodium: 300 mg"
Sig Figs: 1
Suppose we are counting ducks in a pond. We count them a few times to make sure we have an exact count:
"1, 2, 3, 4, ... 298, 299, 300"
"1, 2, 3, 4, ... 298, 299, 300"
"1, 2, 3, 4, ... 298, 299, 300"
Sig Figs: Infinite
Why: This is an exact count. You are absolutely sure that you do not have 299 or 301 or 302 or 300.5 ducks, so there is no uncertainty. It is not 3 sig figs, for we know that the tenths place is 0. it's not 4 sig figs for we know that the hundredths place is also 0. In fact, we know that there are 300.00000000000000... ducks on the pond.
Case 3: Defined Quantity
The conversion factor from inches to centimeters is defined as:
2.54 cm = 1 in.
Sig Figs: Infinite
Why: The conversion factor is a definition; there is no uncertainty. Even though the number goes to 2 decimal places, there is no possibility of 2.55 cm equaling 1 inch due to the way that the quantities are defined.
Case 4: Textbook Ambiguity
Suppose a textbook problem talks about a 300 L tank of water.
Sig Figs: Depends. 1 if interpreted as a measurement, infinite if interpreted as a definition.
Why: This really depends on the textbook.
Some textbooks might treat it as a measurement. Then it would be one sig fig for it might not be exactly 300L in real life.
Other textbooks might give an answer with more sig figs than should be needed. If so, it is either because they treat it as a theoretical example where numbers can be exact, or just provide extra sig figs so you can check your math before rounding.
All in all, check your textbook carefully. If the questions denote different numbers of sig figs differently, they probably treat numbers as measurements. If not, the textbook might treat it as a definition.
If you are not sure, talk to your teacher.
Avoiding Ambiguity
When writing numbers, there's a number of conventions to help determine how many sig figs for a measurement. Take these for example:
300 - 1 sig fig
300. - 3 sig figs. the decimal shows that we know up to the decimal place for certain
300 - 2 sig figs. an underline explicitly shows the last significant figure.
3x10^2 - 1 sig fig. In proper scientific notation, every number shown on the left of the x is significant.
3.0x10^2 - 2 sig figs. Same rule as the previous example.
I hope that helps to explain some of the more challenging aspects of sig figs, particularly zeros.
"Sodium: 300 mg"
Sig Figs: 1
Why: Due to rounding in the label and the fact that each serving might not have exactly the same proportions, you could easily have 290 mg, 320 mg, 310 mg, or anything that's between 250 and 340 mg. Otherwise, the number would be 200 or 400 mg. Since we can't figure out what the tens or ones place is, the only digit we know is exact is the 100s place, hence 1 sig fig.
Case 2: Exact quantity
Case 2: Exact quantity
Suppose we are counting ducks in a pond. We count them a few times to make sure we have an exact count:
"1, 2, 3, 4, ... 298, 299, 300"
"1, 2, 3, 4, ... 298, 299, 300"
"1, 2, 3, 4, ... 298, 299, 300"
Sig Figs: Infinite
Why: This is an exact count. You are absolutely sure that you do not have 299 or 301 or 302 or 300.5 ducks, so there is no uncertainty. It is not 3 sig figs, for we know that the tenths place is 0. it's not 4 sig figs for we know that the hundredths place is also 0. In fact, we know that there are 300.00000000000000... ducks on the pond.
Case 3: Defined Quantity
The conversion factor from inches to centimeters is defined as:
2.54 cm = 1 in.
Sig Figs: Infinite
Why: The conversion factor is a definition; there is no uncertainty. Even though the number goes to 2 decimal places, there is no possibility of 2.55 cm equaling 1 inch due to the way that the quantities are defined.
Case 4: Textbook Ambiguity
Suppose a textbook problem talks about a 300 L tank of water.
Sig Figs: Depends. 1 if interpreted as a measurement, infinite if interpreted as a definition.
Why: This really depends on the textbook.
Some textbooks might treat it as a measurement. Then it would be one sig fig for it might not be exactly 300L in real life.
Other textbooks might give an answer with more sig figs than should be needed. If so, it is either because they treat it as a theoretical example where numbers can be exact, or just provide extra sig figs so you can check your math before rounding.
All in all, check your textbook carefully. If the questions denote different numbers of sig figs differently, they probably treat numbers as measurements. If not, the textbook might treat it as a definition.
If you are not sure, talk to your teacher.
Avoiding Ambiguity
When writing numbers, there's a number of conventions to help determine how many sig figs for a measurement. Take these for example:
300 - 1 sig fig
300. - 3 sig figs. the decimal shows that we know up to the decimal place for certain
300 - 2 sig figs. an underline explicitly shows the last significant figure.
3x10^2 - 1 sig fig. In proper scientific notation, every number shown on the left of the x is significant.
3.0x10^2 - 2 sig figs. Same rule as the previous example.
I hope that helps to explain some of the more challenging aspects of sig figs, particularly zeros.
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