Significant Digits

Summary

In this educational podcast, Mr. Andersen explains the concept of significant digits, crucial for scientific accuracy and precision. He illustrates the difference between accuracy and precision using a sniper analogy, then demonstrates how to determine significant figures in various numerical formats. The video covers rules for multiplication, division, addition, and subtraction, emphasizing that results should not exceed the precision of initial measurements. The goal is to practice until mastering the skill of applying significant digits correctly.

Takeaways

• 📚 Significant digits, also known as significant figures or sig figs, are crucial in scientific measurements to ensure accuracy and precision.
• 🎯 Accuracy in measurements refers to how close the result is to the true value, while precision refers to the repeatability of the measurements.
• 🐝 The example of measuring a wasp illustrates the concept of precision, where the ruler's capability determines the precision of the measurement.
• 📏 Significant digits include non-zero numbers, final zeroes after the decimal point, and 'sandwiched' zeroes between non-zero digits.
• 🔢 In scientific notation, all numbers are considered significant, and the number of significant digits is determined by the digits to the left of the decimal point.
• ❌ Place-holding zeroes (those between non-significant digits and the decimal point) are not considered significant and do not contribute to the count of significant digits.
• ✂️ The law of multiplication and division states that the answer should have the same number of significant digits as the number with the least significant digits in the calculation.
• 🔄 For addition and subtraction, the answer should have the same number of decimal places as the number with the fewest decimal places in the calculation.
• 📉 Rounding is necessary when the calculated result has more significant digits than allowed by the least significant digit rule.
• 📝 Practicing significant digit problems is essential for mastering the concept and ensuring accurate scientific reporting.
• 👨‍🏫 Seeking help when confused or lost in the concept of significant digits is encouraged to gain a better understanding.

Q & A

What are significant digits also known as?

Significant digits are also known as significant figures or sometimes just sig figs.

What does the term 'accuracy' refer to in the context of the podcast?

Accuracy refers to how close a measurement is to the right accepted answer or the truth.

How is 'precision' different from 'accuracy' as explained in the podcast?

Precision refers to the repeatability of measurements, indicating how closely grouped the results are, even if they are not necessarily close to the true value.

Why can't the measurement of the wasp be more precise than 2.5 centimeters initially?

The initial measurement of the wasp couldn't be more precise than 2.5 centimeters because the ruler used was not precise enough to provide more accurate readings.

What are the four types of digits that are considered significant?

The four types of digits that are considered significant are non-zero digits, final zeroes after the decimal place, 'sandwiched' zeroes between non-zero digits, and all numbers in scientific notation.

Why are the zeroes in 209 considered significant, according to the podcast?

The zero in 209 is considered significant because it is a 'sandwiched' zero, positioned between two non-zero digits, making it significant.

What is the rule for determining significant digits in multiplication and division?

The rule for multiplication and division is that the number of significant digits in the answer should equal the least number of significant digits in any of the numbers being multiplied or divided.

How does the rule for significant digits apply to addition and subtraction?

In addition and subtraction, the number of decimal places in the answer should be equal to the least number of decimal places in any of the numbers being added or subtracted.

Why is it important to round the answer to .759 instead of .7593 when calculating 10.6 meters divided by 13.960 seconds?

The answer should be rounded to .759 because the calculation can only have 3 significant digits, which is the least number of significant digits between the two measurements (10.6 has 3 significant digits and 13.960 has 5).

What advice does Mr. Andersen give for improving one's ability to handle significant digit problems?

Mr. Andersen advises practicing significant digit problems until one gets it right, and seeking help if needed.

Outlines

00:00

📏 Understanding Significant Figures in Measurement

In this paragraph, Mr. Andersen introduces the concept of significant figures, also known as sig figs, and their importance in scientific measurements. He explains the difference between accuracy and precision using the analogy of a sniper's aim. Accuracy is how close one is to the correct answer, while precision is the consistency of measurements. He then illustrates the concept with an example of measuring a wasp's length using different rulers, demonstrating how a more precise instrument can yield a more accurate measurement with more significant figures. The paragraph concludes with an explanation of the four types of significant digits: non-zero digits, final zeroes after the decimal point, 'sandwiched' zeroes between non-zero digits, and all numbers in scientific notation.

05:02

🔢 Identifying Significant Figures in Numbers

This paragraph delves deeper into the identification of significant figures in various numerical formats. Mr. Andersen clarifies which digits are considered significant, such as non-zero digits, final zeroes after a decimal point, and 'sandwiched' zeroes. He also addresses the significance of all numbers in scientific notation. Conversely, he explains that place-holding zeroes, which serve only to position other digits correctly relative to the decimal point, are not significant. The paragraph provides examples to illustrate the rules for determining significant figures, emphasizing the importance of adhering to these rules in calculations to maintain the integrity of the measurements.

10:05

⚖️ Applying Significant Figures in Calculations

In this final paragraph, Mr. Andersen applies the concept of significant figures to mathematical calculations, focusing on multiplication, division, addition, and subtraction. He outlines the law of multiplication and division, which states that the number of significant figures in the result should match the least number of significant figures among the numbers involved in the operation. For addition and subtraction, he explains that the number of decimal places in the result should be the same as the least number of decimal places among the numbers being operated on. The paragraph includes examples to demonstrate how to round numbers to the appropriate number of significant figures, ensuring that the final answer does not claim a false level of precision beyond the accuracy of the original measurements.

Keywords

💡Significant Digits

Significant digits, also known as significant figures, are the digits in a number that carry meaningful information about its precision. In the video, Mr. Andersen explains that significant digits are crucial for maintaining the accuracy of scientific measurements and calculations. For instance, when dividing 10.6 meters by 13.960 seconds, the result should only have as many significant digits as the least precise number in the operation, which in this case is three.

💡Sig Figs

Sig figs is a colloquial abbreviation for 'significant figures'. The term is used in the script to refer to the count of digits in a number that are known to be accurate. It's a fundamental concept in scientific measurement and calculation, ensuring that the precision of the result matches the precision of the least accurate measurement involved.

💡Accuracy

Accuracy in the context of the video refers to how close a measurement or calculation is to the true or accepted value. Mr. Andersen uses the analogy of a sniper hitting the bull's eye to illustrate the concept of accuracy, emphasizing that in both shooting and scientific measurements, the goal is to be as accurate as possible.

💡Precision

Precision is the degree to which repeated measurements or calculations produce the same result. In the video, Mr. Andersen contrasts precision with accuracy, using the example of a sniper's shots that are closely grouped but not necessarily hitting the target. Precision is important in scientific work to ensure repeatability of results.

💡Bull's Eye

The bull's eye is used metaphorically in the script to represent the target or the correct answer in measurements or calculations. Mr. Andersen discusses how a sniper's accuracy is determined by how close their shots are to the bull's eye, and similarly, a scientist's measurements are considered accurate if they are close to the accepted value.

💡Wasp Measurement

The wasp measurement is an example used in the script to demonstrate the concept of significant figures in action. Mr. Andersen describes measuring a wasp's length with different levels of precision, highlighting how the number of significant figures in a measurement can vary depending on the measuring tool's accuracy.

💡Ruler

A ruler is a measuring tool mentioned in the script to illustrate the concept of precision in measurement. Mr. Andersen explains how the precision of a ruler affects the number of significant figures one can accurately measure, such as moving from an approximate measurement of 2.5 cm to a more precise 2.55 cm with a better ruler.

💡Scientific Notation

Scientific notation is a way of expressing very large or very small numbers and is mentioned in the script when discussing significant figures. Mr. Andersen explains that all numbers in scientific notation are considered significant, regardless of the number of digits, which is important for maintaining the precision in calculations.

💡Place Holding Zeroes

Place holding zeroes are zeroes that do not contribute to the significant figures of a number. In the script, Mr. Andersen clarifies that these zeroes are not counted when determining the number of significant figures because they serve only to position the decimal point and do not reflect the precision of the measurement.

💡Law of Multiplication and Division

The law of multiplication and division, as explained in the script, dictates that the number of significant figures in the result of a multiplication or division operation should be the same as the number of significant figures in the least precise number used in the calculation. This law is crucial for ensuring that the final result is no more precise than the least accurate measurement.

💡Addition and Subtraction

In the context of significant figures, addition and subtraction have specific rules that differ from those of multiplication and division. Mr. Andersen explains that the number of decimal places in the result should match the least number of decimal places in the numbers being added or subtracted, ensuring that the result is not artificially precise.

Highlights

Introduction to significant digits, also known as significant figures or sig figs, and their importance in scientific measurements.

Explanation of the difference between accuracy and precision in the context of measurements.

Illustration of accuracy and precision using the analogy of a sniper's aim.

The concept of significant digits in measuring objects, such as a wasp's length.

Demonstration of how the precision of a ruler affects the number of significant digits in a measurement.

Identification of four types of significant digits in numerical values.

Clarification that non-zero numbers contribute to the count of significant digits.

Explanation of the significance of final zeroes after the decimal point.

Discussion on 'sandwiched' zeroes and their contribution to significant digits.

Importance of all numbers in scientific notation being considered significant.

Identification of place-holding zeroes as non-significant in a number.

Application of the law of multiplication and division to significant digit calculations.

Example calculation demonstrating how to round to the correct number of significant digits after multiplication.

Explanation of how to handle division when determining significant digits in an answer.

Difference in handling significant digits between multiplication/division and addition/subtraction.

Practical example of rounding in addition to match the least number of decimal places.

Final problem solving example applying the concept of significant digits to a division problem.

Encouragement to practice significant digit problems for better understanding and mastery.

Transcripts

Hi. This is Mr. Andersen and today I'm going to give you a podcast on significant

digits, also known as significant figures or sometimes we call them just sig figs. And

so if I do my job right, you should be able to take a problem like this, 10.6 meters divided

by 13.960 seconds and come up with an answer that not only has the right number of units

or the right units, but also has the correct number of significant digits. So let's get

started. We've got some snipers here. And what snipers try to be is they try to be both

accurate and precise. What does that mean? Well accuracy refers to truth. In other words

how close you are to the right accepted answer. Precision however reports to the repeatability.

And so let's look at the bull's eyes down here. This bull's eye down here, this sniper

has been fairly accurate. In other words all the shots are pretty close to the bull's eye

which is going to be right in the middle. So we would call this accurate shooting. But

not precise. If we look over to here, this time all the shots are way off to the side.

And so it's not true anymore. In other word's it's not accurate, but it's really precise.

In other words they have a really tight grouping right here. And so what do we hope to be as

a sniper? We hope to be both accurate and precise. And what do we hope to be as a scientists?

We hope to be accurate and precise as well. So let's say you have a wasp that you want

to measure. And so if we measure this wasp from its head down to the need of its body,

we find that it is 1, 2 and somewhere between 2 and 3. And so I might say that the wasp

has a length of 2 point, let me approximate, 5 centimeters in length. Why can't I get more

precise than that? Well, my ruler is no better than that. And so if I get a better ruler,

now I see we've got a 1 here. We've got a 2 here. We've got a 3 here. But I also have

these delineations as well. And so this is a 2.5. And this right here is a 2.6. And so

I can be more precise in my measurement. And so what is the length of the wasp right now?

Well it is 2.55 centimeters. And so this right here is a more precise measurement because

I have a more precise measuring device. Or a more precise ruler. These number, 1, 2,

3, are called significant digits or significant figures. And so this measurement would have

3. And this measurement would only have 2. So let's play around with some of these things.

What kind of digits are significant? And there are 4 types of digits that are going to be

significant. And so if you are working through a problem and you see a non-zero number, so

let's say you see 32.6, how many significant digits are there in that number? Well the

3 is. The 2 is. The 6 is. And so there would be 3 significant digits. Or let's say we had

this measurement. 12.48. That would have 4 significant digits. Because there are no zeroes

in it. So that's pretty easy. Let's go to the next one. Final zeroes after the decimal

place are always going to be significant as well. So what does that mean? Let's say we've

got 2.0. How many significant digits are there? Well this 2 is. And this 0 is also significant

because it's a final 0, in other words at the end. And it's also after the decimal place.

And so this would have 2. Or if we did something like this. 28.40 Well, 1, 2, 3, and now this

one, according to that second rule is also going to be significant. So we would have

4 significant digits right there. What else is significant? I like to refer to these next

ones as "sandwiched" zeroes. And so let's say that we have 209. Well this is significant,

that is significant, because they're not zeroes. But this one is sandwiched between the two,

and so it's also significant. And so you could have for example 12.090. Let's apply all of

our rules. How many do we have now? Well these guys are all significant. This 0 is sandwiched

between the 2 and the 9. So it's significant. And this one is a final 0 after the decimal

place. And so this one right here would have 5 significant digits. So it seems like everything

is significant. Let's go to the next one. All numbers in scientific notation are significant

as well. What does that mean? Let's say I have a number like this. 3800000. In science

we use what is called scientific notation to write this out. And so if the decimal place

is here, remember I can count back 1, 2, 3, 4, 5, 6. And so we would write this as 3.8

times 10 to the 6th. That's significant. That's significant. And so this would have 2 significant

digits. Alright. So then let's go to the next page. What actually is not a significant?

So what numbers aren't going to be significant? Well there is only one group of numbers that

aren't. And those are place holding zeroes. And so an example of that. Let's say you had

230. Well this is significant. So is this. But this 0 right here is just spacing the

numbers 2 and 3 from the decimal place. So it's a place holder. And so we would now say

that's not significant. This only has 2. Or if we take a number like this. 0.00069. How

many significant digits are there? Well all of these zeroes are simply place holders.

So they're not significant. And so we'd only have two significant digits there. Okay. So

what do we do? Well in calculations you have to make sure that your answer is no more precise

than the measurements that you actually make. And so we're going to try some calculations

or try some practice. And if this doesn't make sense, slow it down, go back again and

take a look. So let's start with the law of multiplication and division. Law of multiplication

and division says, the number of significant digits in the answer should equal the least

number of significant digits in any of the numbers being multiplied or divided. What

does that mean? Let's try one. So for example let's say we take, I have one down here, 26.4

and we multiply that times 120. Okay. If we multiply those numbers in a calculator we

get a really large number. It is 3 1 6 8 point 0 0 0. So it keeps going like that. So what do we get

for an answer? We'll this has 1, 2, 3 significant digits. This one has 1, 2, that is not significant

because it's just a place holder. So that has 2 significant digits. And so since this

one has three and this one has two, my answer can only have 2 significant digits. So what

does that mean? I'm going to have to round. And so there's one significant digit. The

next one, the 1 is the second significant digit. And since this number right to the

right of it is larger than 5, or equal to 5, I'm going to round this up. And so what

is the right answer? The right answer is 3200. How many significant digits does this have?

Well these two zeroes here are just place holders. And so this is going to have two

significant digits. Which is equal to the least number is my two calculations. And why

do we do that? Well we want to make sure that the measurements we make are no more precise

than the answer that we get at the end. Or the answer we get is no more precise than

those measurements. Let's try another one of those. So let's say we're doing division

for a second. We'll make an easier one. Let's say we take the 19 and we divide that by the

number 3. What do we get for an answer? Well in our calculator we get 6.333333. It just

keeps repeating like that. But you would never turn in an answer like this in science class

or in math class because it's not, it's way more precise than the measurements we actually

made. And so let's go through and use our rules. How many significant digits does this

have? Two. How many significant digits does this measurement have? One. And so how many

significant digits can my answer have? Well it can only have one significant digit. And

so what is my answer? Well this is a 6. This is a point 3. And so my answer would be 6.

In other words I'm going to use this number to round so I can get to one significant digit.

And so the answer wouldn't be 6.333333. The answer would simply be 6. And so significant

digits actually make your job a little bit easier. Now addition and subtraction are a

little bit different. In addition and subtraction it's the number of decimal places in the answer

that should be equal to the least number of decimal points, or decimal places in any of

the numbers being added or subtracted. What does that mean? Let's say we have a measurement

like this. 13 plus 1.6 equals blank. Okay. Now in this one we have to look at the number

of decimal places. In other words this one is measured to the ones place. And this one

is measured to the tenth places. And so even though the answer if we add these up, you

can see is going to be 14.6, my answer can't go and give me another decimal place right

here. And so the right answer would be 15. In other words, I have to round that 4 up

to a 5. Because I can't get an answer that has more decimal places than my least decimal

place answer to the right. And so addition and subtraction work that way. Sometimes when

I'm solving these ones what I'll do is I'll line them up. So all the decimal places are

on top of each other. And then I can see which one has the least number of decimal places.

Okay. So if I go to the end I said after you watch this you should be able to answer a

question like this. So let's take a stab at it. So this 10.6 meters. How many significant

digits would that have? It's going to have 3. Now we've got 13.960. How many significant

digits does that have? 5. And so my answer can only have 3 significant digits. So even

though my calculator might say the answer is .759312321. I don't want to turn this answer in. I want

to get an answer that has 3 significant digits. And so the right answer would be .759. That's

it. Because this is 3, I'm not going to round this nine. And so the right answer would be

.759. So that's how you use significant digits. The best way to get better at doing significant

digit problems is to just practice them until you eventually get it right. And so I hope

that's helpful. And always come ask for help if you get lost.