In this explainer, we will learn how to perform arithmetic operations with numbers expressed in scientific notation.
When ordinary numbers (which we describe as being in “normal form”) have a very large or a very small absolute value, writing them implies writing a lot of digits; for instance, 156 000 000 or 0.00000000324. Recall that we have already learned a compact way of writing numbers, called scientific notation (also referred to as standard form). It proves particularly useful for numbers with a very large or a very small absolute value.
Definition: Scientific Notation
A number written in scientific notation (also called standard form) is of the form where .
The exponent is positive for numbers with a large absolute value. It is negative for numbers with a very small absolute value.
Before looking at how to perform calculations with numbers in scientific notation, let us quickly recap how to convert numbers between the normal form and scientific notation.
Considering the number 156 000 000, we need to establish how many times we would need to divide 156 000 000 by 10 to reduce it in magnitude to a number that is between 1 and 10. This would be 8 times, and the number would be 1.56. Hence, if we multiplied 1.56 by 10 eight times, we would get back to our original number. So, 156 000 000 in scientific notation is
Similarly, consider the number 0.00000000324. In order to convert this number into scientific notation, we first need to establish how many times we would need to multiply it by 10 to increase its magnitude to a number that is between 1 and 10. This would be 9 times, and we would get the number 3.24. If we then divided 3.24 by 10 nine times, which is the same as multiplying by nine times, we would get back to our original number. Hence, in scientific notation, we would write 0.00000000324 as
Notice here that we have used negative indices to avoid writing a division sign.
Now that we have recapped how to convert numbers into scientific notation, let us look at how to perform calculations with them in this form. In order to do this, we need to recall some key mathematical concepts.
Firstly, in the real number system, numbers are commutative under multiplication. In other words,
This means that we can reorder all the terms in a multiplication as we please.
Secondly, if we multiply two numbers with the same base, we can just add the exponents; this is known as the product rule:
Thirdly, if we divide two numbers with the same base, we subtract the exponents; this is known as the quotient rule:
In order to see the relevance of these rules, let us look at an example. Consider the multiplication
To evaluate this, we can first apply the commutative property to reorder the multiplication as
We can then evaluate the multiplication and simplify the powers of 10 to get which is an expression in scientific notation. If we wanted to write this expression in normal form, we would write 6 multiplied by 10 fifteen times: 6 000 000 000 000 000.
This was a fairly introductory-level example where the answer was given straight away in scientific notation. Let us now look at an example where this is not the case.
Example 1: Multiplying Numbers in Scientific Notation
Find the value of , giving your answer in standard form.
Answer
Recall that a number written in scientific notation (also called standard form) is of the form , where .
Here, we have two numbers written in scientific notation that we need to multiply together. First, we use the commutative property that to rewrite the expression as
Then, by evaluating the multiplication and also simplifying the powers of 10 by using the exponent product rule , we get
Note, however, that this result is not yet written in scientific notation (standard form) because 36.9 is not a number between 1 and 10. The trick we use is to rewrite 36.9 itself in scientific notation, as . We then get which simplifies to
Reviewing the answer options, we conclude that C is correct.
Now let us look at division. Here, we take a very similar approach but have to be a little careful when we rearrange the expressions. We will demonstrate the method with another example.
Example 2: Dividing Two Numbers in Scientific Notation
Calculate , giving your answer in scientific notation.
Answer
Recall that a number written in scientific notation is of the form , where .
First, we rewrite the given expression as
This can then be rewritten as a multiplication of two fractions:
Next, we simplify each fraction separately. For the first one, we evaluate it by doing the division . For the second one, we simplify the powers of 10 by using the exponent quotient rule . Therefore, we get
Since 2 lies between 1 and 10, this answer is written in scientific notation.
From the solution above, we see that, to evaluate a division of two numbers written in scientific notation as we transform it into the multiplication of two fractions, which then simplifies as
If the front number lies between 1 and 10, our answer will be in scientific notation. If it does not, we will transform our answer into the required form by rewriting the front number in scientific notation and then combining its powers of 10 with by using the exponent product rule.
Next, we consider addition and subtraction for numbers written in scientific notation. Although these operations are slightly more complicated than multiplication and division, they follow a similar pattern to each other. Let us start with a simple example of addition, such as
As the two numbers have the same power of 10, we can just add the front numbers while keeping the power of 10 the same as follows:
This answer is already in standard form, but if it was not, we would just rewrite the front number in standard form and then combine its power of 10 with the existing one by using the exponent product rule.
Similarly, the subtraction would simplify as
Often, however, the addition or subtraction calculations we meet involve two numbers with different powers of 10. The general method of solution is to rewrite one number so that both numbers have the same power of 10, then carry out the addition or subtraction, and lastly make sure that the answer is written in scientific notation.
We shall illustrate this method through the next example, which involves subtraction.
Example 3: Subtracting Numbers in Scientific Notation
Find the value of , giving your answer in standard form.
Answer
Recall that a number written in scientific notation (also called standard form) is of the form , where .
Here, observe that we have two numbers in scientific notation with different (negative) powers of 10. We start by rewriting one to make the powers of 10 the same. For negative powers, it is easiest to rewrite the one with the higher power (i.e., the least negative power) in terms of the one with the lower power. Therefore, we rewrite as and then work backward from the exponent product rule to get
Then, since both numbers now have the same power of 10, we can do the subtraction
Note that , so our answer will be negative. When we meet situations like this, it is simplest to swap the order of the numbers to make it positive, do the subtraction, and then change the sign of the answer. Hence, we have , so , giving the result
Lastly, as does not lie between 1 and 10, we must convert our answer to standard form (scientific notation). Thus, we rewrite in standard form and then simplify by using the exponent product rule as follows:
In some questions, we are asked to evaluate more complicated expressions involving numbers written in scientific notation. In such cases, it is important to remember the correct order of operations so that we carry out the different stages of the calculation in the right sequence. Let us look an example that tests this skill.
Example 4: Simplifying Numerical Expressions Involving the Scientific Notation
Calculate the value of , giving your answer in standard form.
Answer
Recall that a number written in scientific notation (also called standard form) is of the form , where .
In this question, we have a fraction made up of numbers written in scientific notation. Our strategy will be to add the two numbers in the numerator and then divide the result by the denominator.
First, to do the addition , we must rewrite the expression so that both numbers have the same power of 10. For positive powers, it is usually easiest to rewrite the one with the higher power in terms of the one with the lower power. Therefore, we rewrite as and then do the addition
Note that since 66.36 does not lie between 1 and 10, this number is not in scientific notation. However, this does not matter as it is not our final answer, so we can proceed with the rest of the calculation. We now have
Recalling our method for division, we can rewrite this expression as a multiplication of two fractions:
Next, we simplify each fraction separately. For the first one, we evaluate it by doing the division . For the second one, we simplify the powers of 10 by using the exponent quotient rule . Therefore, we get
Since 8.4 lies between 1 and 10, this number is in standard form.
We conclude that the value of the given expression in standard form is .
Since many quantities from real-world situations can be either very large (for example, the area of a country in square kilometres) or very small (for example, the mass of an atom in grams), then scientific notation is extremely useful in helping us to describe the numbers involved. Let us finish with an example of this type, which features a large number.
Example 5: Solving Word Problems Involving Multiplication of Numbers in Scientific Notation
Light from the Sun takes 13 minutes to reach one of the planets. Given that light travels at a speed of m/s, determine the distance between the Sun and that planet.
Answer
In this question, we need to work out the distance from the Sun to a planet when given the speed of light and the time it takes to travel that distance. Recall that the formula for distance is given by
Before we can substitute the values for the speed and the time into this formula, we need to make sure that their units are compatible.
We are told the speed is m/s. Note first that the number is written in scientific notation, which means it is of the form where .
Since the speed is given in metres per second, it makes sense to convert the time of 13 minutes into seconds. To do this, we need to multiply 13 by the number of seconds in 1 minute, which is 60. Hence, the time taken for light to travel from the Sun to the planet is .
Substituting into the distance formula, we get
Notice that this number is not written in scientific notation, because 2 340 does not lie between 1 and 10. Therefore, rewriting 2 340 as , we then get
By applying the exponent product rule to the powers of 10, this simplifies to
Finally, since the speed was in metres per second and the time was in seconds, this distance is in metres. We conclude that the distance between the Sun and the planet is metres.
Let us finish by recapping some key concepts from this explainer.
Key Points
- To rewrite and simplify numbers in scientific notation, we can use the commutative property , the exponent product rule , and the exponent quotient rule .
- To do the multiplication , we rewrite it as If the front number does not lie between 1 and 10, we rewrite it in scientific notation and then combine its powers of 10 with by using the exponent product rule.
- To do the division , we rewrite it as If the front number does not lie between 1 and 10, we rewrite it in scientific notation and then combine its powers of 10 with by using the exponent product rule.
- To do additions or subtractions, first we make sure that both numbers have the same power of 10, which may involve rewriting one of the numbers. Then, we carry out the addition or the subtraction. Lastly, we make sure that the answer is in scientific notation, rewriting it if necessary.
