Significant Figures - Online Tutor, Practice Problems & Exam Prep

Understanding significant figures is crucial for accurately reporting measurements and calculations in science. Significant figures represent the precision of a value, reflecting the meaningful digits in a measurement. To determine the number of significant figures:

1. For numbers with a decimal point, start counting from the left with the first non-zero digit until the end. For example, 0.0025 has two significant figures (2 and 5), and 8.366 written in scientific notation also has three significant figures (8, 3, and 6).
2. For numbers without a decimal point, count from the right with the first non-zero digit until the end. The number 45000 has two significant figures (4 and 5).
3. Exact numbers, such as counted values (e.g., 125 students in a class or a dozen eggs), have an infinite number of significant figures because they are known with complete certainty.

These rules help maintain precision and accuracy in scientific data, ensuring that reported figures are as accurate as the measurements allow. Remember that significant figures are a fundamental aspect of data integrity in scientific inquiry.

Significant Figures

So when we're dealing with any type of question or trying to write down the answer to a question, we need to take into account significant figures. Now, significant figures. These are the numbers that contribute to the precision associated with any value. Now, we're going to say here that there is an easy way, and of course a hard way to approach significant figures. Luckily for us, we're going to focus on the easy way.

Now, the hard way has a lot of rules, and it has terms that sometimes might be confusing, such as leading zeros and trailing zeros. We're going to avoid all of that, and we're going to rely on three simple rules to help us determine the number of significant figures associated with any value. Now the first rule, if your number has a decimal point. So if it has a decimal point, you're going to move from left to right. Start counting once you get to your first nonzero number, and keep counting until the end.

So here we have our first two examples. One is written in standard notation, one is written in scientific notation, but that doesn't matter. If we look at the first one, we're moving from right on from left to right. We're going to start counting once we get to our first non 0 number. So 0000. Here's our first nonzero number. This two we're going to start counting there and we count all the way until the end, so 123. This number has three sig figs or three significant figures.

For the next one. It's written in scientific notation, but that doesn't matter when it's written in scientific notation. Focus on the coefficient. So this part here the base which is the 10 and the power of the exponent don't matter. It has a decimal point so we're moving from left to right. Our first non 0 number is this eight. We start counting there, and we count all the way until the end, so 123. So we have three sig figs in this one as well.

Next, if your number has no decimal point, then we're going to move from right to left, so we're going to go this way. Same rules apply. Start counting once you get to your first non 0 number, and keep counting until you get to the end. Our first nonzero number is this 5, so that's 1234. So we have 4 sig figs here as our number significant figures.

Now this third rule, This third rule is a little bit different. So this third rule deals with exact numbers. Now, an exact number is a value or integer, so that means it has to be a whole number that is known with complete certainty. We're going to say here for an exact number, there are an infinite number of sig figs or significant figures. So for example, your lecture class has 125 students. That's something we can know with certainty because we can literally count the number of students that we see within the room. Or a dozen eggs equals 12 eggs. This is something that is known with complete certainty, 12 eggs. We can count each one of those individual eggs.

So 125 students within a lecture hall is an is has an infinite number of sig figs. 12 eggs equal 1 dozen that can also have an infinite number of SIGVIX. That's because, for example, if we're looking at the 125 students, it could be 125, which would have three sig figs, or it could be 125.0 that's still saying 125 that has four sig figs. Or it could be 125.00, which has five sig figs and it can go on and on and on and on because technically that is still saying 125.

So just remember the 1st 2 rules are pretty simple. That deals with decimal place or no decimal place. The third rule is a little bit trickier. You have to recall if this is an exact number, something that can be counted, that you can know for certain 100% that it's that number. Those have an infinite number of significant figures. Now that we've taken a look at these three rules, let's move on to the example question in the following video and see if we can determine the number of correct significant figures.

Significant Figures Example 1

So let's take a look at the following example question here. It says determine the number of significant figures in the following value. While our value has a decimal point, when it has a decimal point you move from left to right and we start counting once we get to our first nonzero number.

Our first nonzero number is this 3, so that's where we start counting. So that's going to be 1, 2, 3, 4. You count all the way to the end once you start counting. So here this would just simply have four significant figures. So remember just to rely on the three rules that we know in terms of determining the number of significant figures.

If it has a decimal point such as this one, we move from left to right. Start counting once you get to your first non 0 number and keep counting until the end. If it had no decimal point then we'd move from right to left and follow the same exact rules. If it wasn't exact number then it would have an infinite number of significant figures.

Now that we've done this example question, move on to the practice question.

How many sig figs does each number contain?

a) 100. min                             

b) 17.3 x 10³ mL

c) 10 apples

So here, let's take a look at this practice question. It says, "How many significant figures does each number contain?" Alright. So, if we take a look at the first one, it has a decimal point right there. That means we have to count from left to right. We start counting once we get to our first non-zero number, which is this one right here, and we count all the way to the end. So 1, 2, 3. So we have 3 significant figures for this first one. For the next one, we have it in scientific notation. So again, with scientific notation, just pay attention to the coefficient portion. So we're going to say here, our first non-zero number is this 1. So 1, 2, 3. This also has 3 significant figures. And then finally, we have 10 apples. Now, this is something we can count and know with exact certainty. Okay? Because it's something we can count with exact certainty, that means it's an exact number, and because it's an exact number, it would have an infinite number of significant figures. So, again, if you're able to count it — 125 students in a lecture room, a dozen eggs is 12 eggs, ten apples — these are exact numbers that we can know with certainty. They all, because they're all exact numbers, have an infinite number of significant figures. Alright. Now that we've done this practice question, let's continue on with our talks on significant figures.

Indicate the number of significant figures in the following: 

A liter is equivalent to 1.059 qt. 

• A. 0
• B. 1
• C. 3
• D. 4
• E. infinite

How many significant figures are contained in the following measurement? 

3,482,005 mg 

• A. 0
• B. 5
• C. 6
• D. 7
• E. infinite

Here’s what students ask on this topic:

What are significant figures?

Significant figures, often referred to as "sig figs," are the digits in a number that carry meaning contributing to its precision. They include all the non-zero digits, any zeros between non-zero digits, and any trailing zeros in a decimal number. The purpose of significant figures is to ensure that the numbers used in calculations reflect the precision of the measurements from which they are derived.

For example, in the number 0.00540, there are three significant figures: 5, 4, and the trailing zero, because it indicates precision. However, the leading zeros are not significant because they serve only as placeholders.

When you perform calculations, the number of significant figures in the final result should reflect the least precise measurement used in the calculation. This practice avoids implying a greater degree of precision than the measurements justify. Understanding how to use significant figures is crucial in scientific and engineering calculations to accurately report results.

How do you calculate significant figures?

Calculating significant figures involves identifying all the digits in a number that contribute to its precision. Here's a quick guide:

1. Non-Zero Digits: All non-zero digits are significant. For example, in 123.45, all five digits are significant.
2. Leading Zeros: Zeros that come before all non-zero digits are not significant. For instance, in 0.056, the zeros are not significant, and there are only two significant figures.
3. Captive Zeros: Zeros between non-zero digits are significant. So, in 1002, all four digits are significant.
4. Trailing Zeros in a Decimal Number: Zeros at the end of a number and to the right of a decimal are significant. For example, in 78.00, there are four significant figures.
5. Trailing Zeros in a Whole Number: Zeros at the end of a whole number may or may not be significant, depending on whether a decimal point is specified. In 1300 without a decimal point, it's ambiguous; with a decimal point (1300.), all four are significant.

When performing calculations, the number of significant figures in the result should be based on the original numbers with the least number of significant figures. For addition and subtraction, the result should have the same number of decimal places as the number with the least decimal places.

How do you count significant figures?

Counting significant figures involves identifying all the digits in a number that contribute to its precision. Here's how you do it:

1. Non-Zero Digits: All non-zero digits are significant. For example, in 123.45, all five digits are significant.
2. Leading Zeros: Zeros that come before all non-zero digits are not significant. They merely indicate the position of the decimal point. For instance, in 0.00123, the leading zeros are not significant.
3. Captive Zeros: Zeros that fall between non-zero digits are significant. In 1002, all four digits, including the zeros, are significant.
4. Trailing Zeros with Decimals: Trailing zeros in a number containing a decimal point are significant. So, in 12.300, all five digits are significant.
5. Trailing Zeros without Decimals: Trailing zeros in a whole number without a decimal point are ambiguous and may or may not be significant, depending on the context or if a decimal point is explicitly stated. For example, in 1500, it is unclear if the zeros are significant unless additional formatting or notation is used (like scientific notation).

To ensure clarity, especially with trailing zeros, numbers are often written in scientific notation, where all digits explicitly written in the coefficient are significant.