Scientific Notation - Online Tutor, Practice Problems & Exam Prep

Scientific notation is a method used to express very large or very small numbers in a compact form, making them easier to work with in calculations. The notation consists of three parts: a coefficient that is greater than or equal to one but less than ten, a base of ten, and an exponent which is a whole number indicating how many places the decimal point has been moved. A positive exponent means the coefficient is made larger by moving the decimal point to the right, resulting in a larger number. Conversely, a negative exponent signifies that the coefficient is made smaller by moving the decimal point to the left, yielding a smaller number. To convert from standard notation to scientific notation, adjust the decimal point so that the coefficient falls between one and ten, and then set the exponent to reflect the number of places the decimal has been moved. Understanding scientific notation is crucial for handling complex calculations in chemistry and physics, where precision and efficiency are key.

Scientific Notation

Oftentimes when we deal with calculations in chemistry, we're going to run into the situation where we're dealing with extremely large numbers and extremely small numbers. Now, a convenient way to deal with this is scientific notation. Now, scientific notation is used to turn these small or large inconvenient numbers into manageable ones.

So here we have an example of something written in scientific notation. We say it's 6.88×10-12. Now what's in red is called our coefficient. Now the coefficient is just the beginning part of the value that is equal to or greater than one but less than 10. Next we have our base, which is here. This is the portion of the scientific notation value that is always 10. So that number is always going to be 10 if we're writing something in scientific notation.

And then finally the exponent, which we sometimes call the power. This is the number of places the decimal was moved to create the scientific notation value. Now we'll talk about in a couple of videos, what happens when it's positive versus negative. What effect does that have on my value overall? Now, another important thing when it comes to this exponent is that it must be expressed as a whole number integer. So -12 positive three -2 can't be decimals or fractions must be whole numbers.

Now that we know the basic layout of scientific notation, click on the next video and let's take a look at the example question.

Scientific Notation Example 1

So now that we know the major parts of scientific notation, let's put it to practice with this example question. Here it says which of the following scientific notation values is written correctly. So if we take a look at A here, we have 1.25×10-14. Well, if we look the coefficient, this part here is fine. It's a number between 1:00 and 10:00. Here our base is 10, but our error is that the exponent is not a whole number integer, it's -14. This is not written correctly.

Next, if we take a look at the very beginning, we see that our coefficient here is a number that is not between 1:00 and 10:00. This is not allowed for scientific notation. The base is correct and the exponent is a whole number. But again, the coefficient is not a number between 1:00 and 10:00, so this will not work.

First C we have our coefficient Wiz which is a number between 1:00 and 10:00, so this is allowed. Our base is 10 and our exponent is a whole number integer 3. So this is a correct way of writing something within scientific notation values. So this is our answer.

If we look at D though, for D our coefficient again is a number between 1:00 and 10:00. But our base here, our base is not ten, it's two. Remember, if you're writing something in scientific notation, the base is always 10. The exponent is correct, but it's the base that's making it incorrect overall.

So again, remember the three components of scientific notation and the limits that they each have. Following that ensures that your right scientific notation correctly every time.

Scientific Notation

Now that we know about scientific notation, it's going to become important on how to convert from scientific notation to standard notation. Now, standard notation itself is just the normal way of writing numbers. So here, in order to go from scientific notation to standard notation, it all deals with looking at the exponent.

Now we're going to say here that a positive exponent tells you to make the coefficient value larger. So if we take a look here we have 7.17×105. Here 5 is telling me that I need to make the coefficient larger by 5, so we're starting out with 7.1 77. To make it larger I'm going to move the decimal place over to the right. So we have to move it five spaces so 12345. So our new way of writing this would be 717, 000 so this becomes 717,000 as our new value.

Now what if the exponent is negative? Well a negative exponent tells you to make the coefficient value smaller. So if we take a look here this is -7 so we have 3.25. We need to make this number smaller, so we're going to have to move it to the left and we move it over 7 spaces, so 123-4567. So that's going to be, oops, .00000325. So these values here would represent the normal way of writing numbers.

Now, it's inconvenient because you can see how much writing is involved in writing both of these numbers, and if our exponents were even bigger positive or even more negative value, that'd be more movement of the decimal. That's the whole point of scientific notation. It changes these inconvenient, non manageable numbers into something that's easier for us to read.

Now that we've done this, click on the next video and let's take a look at a practice question where we put this into practice. Remember look at the exponent to determine if you want to make the coefficient larger or smaller.

Convert the following scientific notation values into standard notation.

a) 1.25 x 10^-4

b) 3.20 x 10^-9

c) 1.6100 x 10⁴

So now let's take a look at the following practice question. It says convert the following scientific notation values into standard notation. Remember, the goal or the key to this is to look at the exponent. That tells us how many places to move in order to make our coefficient larger or smaller. So if we look at the first one, it's negative 4. So we need to make our coefficient smaller. We have 1.25. In order to make it smaller, I have to move the decimal over to the left side, and I need to move it 4 spaces. So that would be 1, 2, 3, 4. So that means we're going to be 0.000125 as our standard notation value. For the next one, it's negative 9. So we're going to have to move it over to the left 9 spaces. So, yeah, that's a lot of movements. So you can see why scientific notation is so important. It helps make an inconvenient number into something that's more manageable. So here, we're going to have to move it over 9 spaces. I'm gonna put 8 zeros. So 1, 2, 3, 4, 5, 6, 7, 8, 9 spaces. So 0.000000320. So that is our standard notation value. And then finally, this one is a positive 4, so we need to make the coefficient larger by moving the decimal over 4 spaces. So 1, 2, 3, 4. So that'll be 16,100 as our standard notation value. So now that we know how to go from scientific notation to standard notation, what happens when we have to go in the reverse of that? How do we go from standard notation to scientific notation? To find out, click on the next video and let's look at the steps required.

Scientific Notation

Now we're going to take a look at how exactly do we change standard notation into scientific notation. Now the key is to look at the coefficient. Now we're going to say to convert a number into a scientific notation value. Make sure the coefficient is equal to or or greater than one but less than 10. So what we learned, the very beginning of scientific notation is the key to changing standard notation into scientific notation.

So if we take a look here, we're going to say increasing the coefficient makes the exponent value decrease. All right. So here we have a number of 0.000145. We need to make sure that that coefficient, which we're going to make, has to fall between 1:00 and 10:00. Here is our decimal place. We're going to have to move it 12345 spaces so that we can get 1.45, which is a value between 1:00 and 10:00. Now we made the coefficient larger. Since we made the coefficient larger, that means the exponent has to become smaller. So the coefficient increased by 5. So the exponent has to decrease by 5 because they're opposites of one another.

Now decreasing the coefficient value makes the exponent value increase. So here in this one there is. We could think of there being a decimal point there, and we're going to move it over enough spaces in order to make this coefficient a number between 1:00 and 10:00. So we're going to move 12345678 spaces, so that becomes 1.01325000 times 10. Now we moved over 8 spaces in order to make this smaller, so if the coefficient is decreasing by 8 spaces, that means our exponent has to be increasing by 8 spaces, so it'll be a +8 here. So that would be the scientific notation value for this number here.

Later on we'll talk about things such as significant figures, which will help us to even simplify this scientific notation even more because there's an awful lot of zeros here at the end. But for now, this would be the entire value that we place to show that we moved decimal place eight spaces over to make the coefficient smaller and thereby increasing the exponent by 8. Now that we've seen these examples, move on to the practice question left at the bottom of the page.

Convert the following standard notation values into scientific notation.

a) 377,000

b) 0.000101

c) 707.82

So now let's take a look at the following practice question. Here it says convert the following standard notation values into scientific notation. So for the first one, we have 377,000. We can assume that we have a decimal here at the end and we just have to move it enough spaces in order to make our coefficient a value between 1-10. So we move it over 1, 2, 3, 4, 5. We stop there because that gives me 3.77, which is a number between 1-10. Now I moved it over 5 spaces to make the coefficient smaller. If I make the coefficient smaller, that means that my exponent becomes larger, or it increases. So here we moved over 5 spaces to make coefficients smaller, so my coefficient increases by 5. So it's 3.77×105. For the next one, here is our decimal point here. Again, we need to move it enough spaces to make a coefficient of value between 1-10, so I move it 1, 2, 3, 4. Now I am making the coefficient larger, which means my exponent has to become smaller, has to decrease. And we moved it 4 spaces, so this becomes negative 4. And then finally, we have 707.82. We're going to move this in order to decrease my coefficient to make it a value between 1-7. So now it's 7.0782×102. Now I'm making the coefficient decrease, which means the exponent has to increase. So I moved it 2 spaces, so this is a positive 2. So just remember, scientific notation is just a way of making very large or very small numbers more manageable and easier to use. Sometimes, though, it's important to know how to do standard notation as well, because sometimes, scientific notation isn't necessary. Take to heart some of these questions that we've done, and you'll be able to convert between scientific notation and standard notation quite easily.

Here’s what students ask on this topic:

What is scientific notation?

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, mathematicians, and engineers to handle extremely big or tiny numbers more efficiently.

In scientific notation, a number is written as the product of two parts: a coefficient and a power of 10. The coefficient must be a number greater than or equal to 1 and less than 10, and it is followed by a multiplication sign and a power of 10. This power of 10 indicates how many times the coefficient should be multiplied or divided by 10.

For example, the number 6,500,000,000 in scientific notation is written as:

$6.5\times {10}^9$

The exponent 9 indicates that the decimal point in the coefficient 6.5 has been moved 9 places to the right. Conversely, a small number like 0.00000052 would be written as:

$5.2\times {10}^{-7}$

indicating the decimal has moved 7 places to the left.

Scientific notation simplifies complex calculations, as it allows for easier multiplication and division of large numbers by manipulating the exponents of 10.

How to do scientific notation?

Scientific notation is a way to express very large or very small numbers in a compact form. Here's how you can convert a number into scientific notation:

1. Move the decimal point in the number until you have a coefficient that is greater than or equal to 1 but less than 10. Count the number of places, n, that you moved the decimal point.
2. If you moved the decimal to the left (for large numbers), your exponent will be positive, and it will be the number of places you moved the decimal. If you moved it to the right (for small numbers), your exponent will be negative.
3. Write the number as the coefficient multiplied by 10 raised to the power of the exponent (n).

For example, the number 53,000 in scientific notation is:

$5.3\times {10}^4$

because the decimal was moved 4 places to the left to get the coefficient 5.3. Conversely, 0.0053 would be:

$5.3\times {10}^{-3}$

because the decimal was moved 3 places to the right.

How do you write a number in scientific notation?

Writing a number in scientific notation involves expressing it as the product of two factors: a decimal number and a power of ten. Here's how you do it step by step:

1. Move the decimal point in the number to the right of its first non-zero digit. Count the number of places you moved the decimal point; this will be the exponent of the power of ten.
2. The decimal number (now between 1 and 10) is the coefficient.
3. If you moved the decimal to the left to reach the coefficient, the exponent is positive. If you moved it to the right, the exponent is negative.
4. Write the scientific notation as the coefficient followed by "x 10" raised to the power of the exponent.

For example, to write 0.00052 in scientific notation:

• Move the decimal to the right of the first non-zero digit (5), which gives you 5.2.
• You moved the decimal 4 places to the right, so the exponent is -4.
• The scientific notation is $5.2x{10}^{-4}$.

For a large number, like 85,000:

• Move the decimal to the left to get 8.5.
• You moved the decimal 4 places to the left, so the exponent is +4.
• The scientific notation is $8.5x{10}^4$.