Before you can determine the number of significant figures within a given value, you first need to understand the difference between exact and inexact numbers. Now, we're going to say the numbers you will encounter can either be exact or inexact. An exact number is a value or integer obtained from counting objects or as part of a definition. For example, there are 125 students in your lecture. This is something that you can determine by actually counting the number of students within your lecture, or there are 13 objects in a baker's dozen. So this is an actual thing. A baker's dozen is actually 13 and not 12. Now, an inexact number, this is a value obtained from calculations or measurements that contains some uncertainty. We're going to say your textbook is measured at a length of 12.53 inches. So you've determined this by taking out a ruler and measuring it. Now you might be a little bit off because maybe you didn't adjust to the exact edge of the book. So there is a little bit of uncertainty associated with this number. That would not be the case with an exact number. I count 125 students in your lecture. I can't say there is 124.8 because a student doesn't count as 0.8. A student is a student. A baker's dozen is 13, not 12.8, not 12.5, not 13.1. It's exactly 13. So just remember the difference between an exact and an inexact number.

- 1. The Chemical World9m
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# Significant Figures (Simplified) - Online Tutor, Practice Problems & Exam Prep

Understanding significant figures is crucial for precision in measurements. Exact numbers, like counting students, have infinite significant figures, while inexact numbers, derived from measurements, have uncertainty. To determine significant figures, follow three rules: count from the first non-zero digit left to right for numbers with a decimal, and right to left for those without. Exact numbers, such as 12 eggs, are known with certainty and also possess infinite significant figures. Mastering these concepts enhances accuracy in scientific calculations and data interpretation.

The number of **Significant Figures** for a value affects its precision.

## Exact Numbers, Inexact Numbers, & Sig Figs

### Significant Figures (Simplified) Concept 1

#### Video transcript

### Significant Figures (Simplified) Example 1

#### Video transcript

Here are example states, determine if the following statement deals with an exact or inexact number. The combined mass of all doses of bronchodilator administered to a patient measure 10.0 milligrams. Alright. So remember, when we're dealing with an exact number, we get an exact number either from a definition or by literally counting the number of objects. Remember, an inexact number happens when we do measurements or calculations. Because we're measuring it, that means that this is an inexact number. So there's a little bit of uncertainty associated with it. We may think we're administering exactly 10.0 milligrams, but maybe we're administering 10.0001 milligrams. Okay? So there's a little bit of uncertainty associated with this value. In this case, remember, this is an inexact number.

### Significant Figures (Simplified) Concept 2

#### Video transcript

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 are the numbers that contribute to the precision associated with any value. 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 rely on 3 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 non-zero 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 left to right. We're going to start counting once we get to our first non-zero number. So 0, 0, 0, 0. Here's our first non-zero number, this 2. We're going to start counting there and count all the way until the end. So 1, 2, 3. This number has 3 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, the exponent, don't matter. It has a decimal point, so we're moving from left to right. Our first non-zero number is this 8. We start counting there and count all the way until the end. So, we have 3 significant figures 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. The same rules apply. Start counting once you get to your first non-zero number and keep counting until you get to the end. Our first non-zero number is this 5. So that's 1, 2, 3, 4. So we have 4 significant figures here.

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 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. Twelve eggs. We can count each one of those individual eggs. So, 125 students within a lecture hall has an infinite number of significant figures. 12 eggs equal 1 dozen. That can also have an infinite number of significant figures. That's because, for example, if we're looking at the 125 students, it could be 125, which would have 3 significant figures, or it could be 125.0, that's still saying 125, that has 4 significant figures, or it could be 125.00, which has 5 significant figures, and it can go on and on because technically that is still saying 125. So just remember, the first two rules are pretty simple. They deal with the presence or absence of a decimal place. The third rule is a little trickier. You have to recall that 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 (Simplified) Example 2

#### Video transcript

So let's take a look at the following example question. Here it says determine the number of significant figures in the following value. Well, 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 non-zero number. Our first non-zero 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 4 significant figures. So remember, just rely on the 3 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-zero 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 was an 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 10^{3} mL

c) 10 apples

#### Problem Transcript

Indicate the number of significant figures in the following:

A liter is equivalent to 1.059 qt.

### Here’s what students ask on this topic:

What are significant figures and why are they important in scientific measurements?

Significant figures are the digits in a number that contribute to its precision. They include all non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number. They are crucial in scientific measurements because they indicate the precision of a measurement and help in maintaining consistency and accuracy in calculations. For example, if a length is measured as 12.53 inches, the significant figures (1, 2, 5, 3) show the precision of the measurement. Understanding significant figures ensures that the uncertainty in measurements is properly communicated and that calculations based on these measurements are accurate.

How do you determine the number of significant figures in a number with a decimal point?

To determine the number of significant figures in a number with a decimal point, follow these steps: Start counting from the left, beginning with the first non-zero digit, and continue counting all digits to the right, including zeros. For example, in the number 0.00456, you start counting from the first non-zero digit (4), so the significant figures are 4, 5, and 6, giving a total of three significant figures. This method ensures that all meaningful digits contributing to the precision of the measurement are included.

What is the difference between exact and inexact numbers in the context of significant figures?

Exact numbers are values obtained from counting objects or defined quantities, and they have an infinite number of significant figures. For example, there are exactly 12 eggs in a dozen. In contrast, inexact numbers are derived from measurements and contain some uncertainty. For instance, the length of a textbook measured as 12.53 inches is inexact because it involves measurement uncertainty. Understanding this distinction is important for correctly applying significant figures in scientific calculations.

How do you count significant figures in a number without a decimal point?

To count significant figures in a number without a decimal point, start from the right and move left, beginning with the first non-zero digit. Continue counting all digits to the left. For example, in the number 4500, you start counting from the first non-zero digit (5), so the significant figures are 4 and 5, giving a total of two significant figures. This method ensures that all meaningful digits contributing to the precision of the measurement are included.

Why do exact numbers have an infinite number of significant figures?

Exact numbers have an infinite number of significant figures because they are known with complete certainty and do not involve any measurement uncertainty. For example, if there are 125 students in a lecture hall, this count is exact and can be represented as 125, 125.0, 125.00, and so on, without changing its value. This infinite precision is crucial for ensuring that exact numbers do not limit the precision of calculations in which they are used.

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