Scientific notation

Scientific notation is an essential component of both mathematics and science. You may already be familiar with its rules and uses, but just in case, let’s start with the basics.

Scientific notation is comprised of three parts:

1. Your simplified number

2. $\times$ (multiplied by)

3. $10$ to some power

Scientific notation is nothing if not practical. It is used to make huge numbers smaller and easier to read and write. That being said, scientific notation does not change the value of a number. It merely changes the way that it looks. Let’s see a short example:

Consider the number $15$. In this example, it is not very helpful to convert the number using scientific notation, but it will help visualize the process to a simpler degree.

$15$ written in scientific notation is $1.5\times 10^1$. Try working this out in your head: $10^1$ is just $10$. And $1.5\times 10=15$. Scientific notation is nothing fancy, it just gets a bit more complex as the numbers get bigger. In this chapter we will primarily focus on how to fundamentally convert to and from scientific notation so that you don’t need to worry about what the number is; you will follow the same process no matter what it may be.

Converting from standard to scientific notation

The goal of scientific notation is to edit your number so that there is one digit to the left of the decimal point. There can be varying numbers to the right of the decimal, but we will just work with one to the right as well.

This means that the way we convert to scientific notation will wholly revolve around how we reduce our big number into a smaller one, or how we grow our small number into a bigger one. To do that, we will count decimal places.

Take a look at this more complex example: $12,345,678$ is a very large number. If we were to change it to scientific notation, we would need to make the number smaller until it looks like $1.2$ multiplied by some power of $10$. Follow the steps below to get a more concise explanation of how to get there by counting decimal places.

1. View the whole number to the left of the decimal: $12,345,678.0$

2. Move the decimal over one place to the left (this signifies that you are dividing the number by $10$, because decimals work in groups of tens): $12,345,67.80$

3. Continue to repeat this step until you end up with $1$ as the only number to the left of the decimal: $1.23456780$

4. Count how many times you had to move the decimal (how many groups of ten were removed): you had to move the decimal points $7$ times.

5. Round off your big number so that there is one digit on each side of the decimal: $1.2$

6. Recognizing that you removed $7$ groups of $10$, you will now need to add that onto the end of your new, smaller number in the form of “$\times 10^7$”: this would be $1.2\times 10^7$

7. Nice work! You just converted to scientific notation!

Working with a very small number also may require scientific notation, so we will examine that, too. However, this will seem very similar to the process of converting a very large number to scientific notation. One important distinction is that when we make sure there is only one digit to the left of the decimal, it must be a non-zero digit. We will use the number $0.000437$.

1. We already see the decimal, so we can skip the usual first step.

2. Move the decimal over one place to the right. We are moving it to the right this time because that is how we will make sure there is only one non-zero digit to the left of the decimal: $0.00437$

3. Move the decimal over until we get just one non-zero digit to the left of the decimal point: $0004.37$

4. Count how many times you had to move the decimal (how many groups of ten were added): you had to move the decimal points $4$ times.

5. Round off the rest of the number to the right of the decimal so that you only have one digit on both sides: $4.4$.

6. Recognizing that you added $4$ groups of $10$ to enlarge your number to what you now currently have, you will now need to add that onto the end of your new, bigger number in the form of “$\times 10^{-4}$”: this would be $4.4\times 10^{-4}$

7. It is very important that you remember that you are multiplying by a negative power of $10$. This ensures that we move the decimal back to where we found it in case we need to backtrack.

8. If you are ever uncertain about whether you have the right sign on your power of $10$, try multiplying it out! Remember, you didn’t change the number. If you put $4.4\times 10^{-4}$ into the calculator, you will get $0.000437$ as your response. Since we got back out the number we started with, this means we did it correctly!

Count your decimal places, and remember that for larger numbers your power of $10$ is positive while for your smaller number (to the right of the decimal) the power of $10$ is negative.

Converting from scientific to standard notation

Converting backward from scientific notation to standard notation should be fairly straightforward if you are comfortable with what we did above. We actually did this conversion when we checked to see if our answer was correct. In order to convert in this way, we simply carry out the expression that is given to us by the scientific notation. Let’s figure out what that means:

$2.6\times 10^3$ is a simple example of scientific notation. If we are to convert from this notation to standard notation, we will simply do as the expression says:

1. $2.6$ multiplied by $10^3$

2. That is actually it! Doing this with a calculator, all we need to do is Step 1. The number we receive will be the conversion in standard notation.

3. If we do this without a calculator, we would need to add zeros equal to the power of $10$ included in the scientific notation.

4. In the calculator, $2.6\times 10^3=2600$

5. Doing this without a calculator, we will say that we must multiply the number ($2.6$) by ten to the power of three ($10^3=1000$). we know this because the exponent indicates the number of zeros to attach to $1$. So, $2.6\times 1000=2600$.

6. You can also think about this as moving decimal places, just like we have been doing. If the exponent on the $10$ is positive, you move the decimal that many places to the right. If the exponent is negative, you move that many decimal places to the left. Fill any gaps in between the decimal and the numbers with $0$s.

Addition and subtraction in scientific notation

The easiest way to do any sort of math with scientific notation is to use your calculator. However, we will explore the basics of what we are trusting our calculator to do as we first look at addition and subtraction in scientific notation.

Addition and subtraction are a bit tedious with scientific notation. The most reliable way to perform these operations is to convert the numbers to standard form, add them, then revert them back into scientific notation. Make sure you are comfortable with the conversions above so that this method is easier to follow.

When you are ready to look at an example, try to solve this question on your own:

What is the sum of the following expression?

$$(5.1\ast 10^8)+(6.5\ast 10^8)$$

You’ll notice that when we added the two numbers together in standard form we increased the number of digits by one, so we had to increase the exponent $10^8$ to $10^9$. Then, we rounded off the decimals to one decimal place.

Multiplication and division in scientific notation

For multiplication, we can similarly convert everything to normal numbers just like we did above. Or, we can perform the operation leaving the numbers as they are. To do this, we can view the exponent portion of scientific notation (the part that looks like $10$ to the power of some number) as the remainder of our primary number (the portion that does not include the ten raised to some power). Doing this allows us to ignore the exponent and perform multiplication and division on our simple numbers. Let’s do an example to make it easier to understand:

What is the product of the following expression?

$$(2.4\ast 10^8)\ast (8.9\ast 10^3)$$

We focus our attention on only the simple part of the numbers. Doing so gives us $2.4\ast 8.9$.

This simpler product is $21.4$ (rounding the decimal). What you should notice is that we have just increased the number of digits before the decimal to two digits ($21$). In order to revert the format to proper scientific notation, we have to move the decimal place one space to the left. Do you remember what will happen to our exponent term when we do this? It will increase by one in the exponent!

So, after our simple product we are left with $2.1$ (rounding to one decimal place after our modification), our increase of our exponent by $1$ (think of it as “carrying the one”), and our original exponents,

$$10^8\ast 10^3$$

We will simply multiply these numbers together too! Their product is $10^{11}$, since the rule for multiplying exponents is to add them together (as long as the base numbers are the same, in this case, $10$). This will be discussed more in the section on [Elementary Algebra].

Now we put it all together:

$2.1\ast 10^{11}$, carrying the one for our exponent. Let’s add that in as well!

This gets us our final answer of $2.1\ast 10^{12}$.

Division is very similar to multiplication. You will perform the same steps, doing division where you would normally do multiplication. The one big difference is that if your number changes by a lot you will be moving your decimal to the right, which will subtract one from your original exponent. So, you will be “carrying a negative one” and will subtract it from your total exponent instead of adding it.

If you were to perform the above example using division instead of multiplication the final answer would be $2.7\ast 10^4$. Can you get to that answer?

One last note!

You can also use scientific notation on your calculator. Look for an “$EE$” button. On the TI-83 and TI-84, you can use it by pressing “2nd” and then the comma “$,$.” The $EE$ stands for “$\times 10^x$.” For example, if I were to type $1.25\times 10^7$ in my calculator, it would look like this: $1.25EE7$ (though it will usually only show one capital E on the screen instead of both). If you have a different calculator, be sure to Google it and make sure you feel comfortable using it. Try some of the above exercises using your calculator until you’re confident.