Scientific Notation Equation Calculator

Use our scientific notation equation calculator to learn how to calculate the algebraic sum, multiplication, and division when you write your number as a coefficient and an exponent. Keep reading our article to learn:

• What is scientific notation: a quick introduction.
• Adding and subtracting: scientific notation algebraic sum rules.
• The multiplication and division of scientific notation numbers.
• How to use our scientific notation equation calculator.

We'll give you quite a few examples of equations with scientific notation: multiplication and sum step by step, and much more!

Scientific notation is a handy way to express numbers of any magnitude without worrying about too large or too small values that, if left untouched, would make the representation quite clumsy. Scientific notation splits a number into two parts: a coefficient and an exponent, to which we need to add another quantity, the base:

• The coefficient is the "numerical" part of the number, its value, stripped of indication about the magnitude: the coefficient of $40$ is the same as that of $0.000004$ — both are 4.
• The exponent indicates the magnitude of the number — how big or small it is. It specifies the position of the coefficient in a positional numerical system.
• The base is the number raised to the power specified by the exponent. Of all three quantities, it's the one that concerns us the least, as it should always have a value of $10$: don't trust anyone who uses other values!

Here you can see a generic number written with scientific notation:

In this example:

• $1.3745$ is the coefficient: notice how it has a single "integer" digit: always keep only one number on the left of the decimal separator when writing scientific notation!
• $10$ is the base.
• $-5$ is the exponent.

The fact that the base is $10$ allows us to navigate a number written in scientific notation by only counting positions at which the decimal separator can be. Our scientific notation calculator can give you a deeper explanation of scientific notation, in case you need it!

Let's focus on how we can use operators written using scientific notation to calculate equations of any kind. We will deal with addition and subtraction first.

In calculating equations, the scientific notation can be either a curse or a blessing. As the positional notation kind of breaks down, since we express the position using the exponent, some operations (for example, adding scientific notation terms) require slightly more complex steps than when using the simple positional notation. On the other hand, division and multiplication with scientific notation are much easier than when dealing with all these zeros.

In the following two sections, we'll teach you how we execute our subtracting and adding of scientific notation numbers and how easy it is, on the other hand, to use scientific notation for multiplication and division calculations.

If our terms use scientific notation, adding and subtracting requires a few steps that we can't really avoid.
Notice that addition and subtraction are the same operations (algebraic sum), and we can treat them simultaneously!

Take two numbers written with scientific notation. As the sum is strictly based on the position of the digits in a number, we can't consider the coefficients alone in our operation. First, you must be sure that the digits of both numbers are in the same position. The easiest way to do so is to rewrite the numbers in positional notation (the one you can learn at the place value calculator). To do so, multiply them by the power of ten specified by the exponent.

However, if the two numbers are relatively close, you can keep them expressed with the scientific notation: in this case, make sure that both numbers have the same exponent. Here is an example:

Let's rewrite the second term so that it has exponent $4$ too:

Now that both terms use the same exponent of the scientific notation, adding them is much easier:

Thankfully, even when your numbers use scientific notation, adding and subtracting them won't require much thought: simply perform the proper operation digit by digit! What about significant digits? In the case of the algebraic sum, we keep the number of digits of the shorter term: we don't count the significant digits, but we consider the position of the largest significant digit and approximate our numbers to that position. If you need a refresh on sig-figs, visit our comprehensive significant figures calculator!

You'll understand this concept with an example: take the number $1.416\times10^{-3}$ and $0.46\times10^{-2}$. Let's sum them after converting them to the positional notation:

The red digits in the sum will eventually disappear, approximating them with the half-up method, giving us the result $0.0047$.

Calculating multiplication if scientific notation is used in our factors (or division as well) is super easy! Much easier than adding and subtracting, scientific notation will help you be even faster with multiplying and dividing numbers.

The trick here is that when multiplying powers, you can sum the exponents (but only if the base is the same). Every number written using scientific notation has the same base ($10$), so we can always sum the exponents. To compute the product or quotient of two numbers that use scientific notation:

1. Multiply or divide the coefficients; and
2. Add the exponents (if you are multiplying) or subtract them (if you are dividing).

Nothing easier. Let's check with an example: we multiply the numbers $5.16\times10^{-2}$ and $3.24\times10^{3}$. Forget converting them; let's jump to the math! Multiply their coefficients:

Let's now sum the exponents:

This gives us the result:

In the case of multiplication and division, we count the significant digits of the factors. We then approximate the result to the lowest number of significant digits among the factors.

Let's take two numbers written using scientific notation, and calculate the division and multiplication: we chose $6.3548\times10^{4}$ and $2.58\times10^{-2}$. The first number has 5 significant digits, while the second one only has three.

Let's calculate the multiplication with scientific notation first: start by multiplying the coefficients:

We can already approximate this result to the appropriate number of significant digits, in our case, three: $16.395384\rightarrow 16.4$.

It's time to sum the exponents:

The final result is then:

We increased the exponent by $1$ since we want the number on the left of the decimal separator to be at most one digit long.

Let's keep the same numbers written using scientific notation and calculate the division. Start with the coefficients:

Let's round it to three significant figures: $2.46310075...\rightarrow2.46$. We now subtract the exponents (in the order the respective numbers appear in the division):

Which gives us the result:

If you want to use our calculator for converting numbers into scientific notation and engineering notation, follow these simple steps:

1. Enter a number into the first field. If you want to insert them using scientific notation, use the $e$ notation (for example, $4.3e5$); however, you can also input them in decimal form!
2. Specify the desired number of significant digits for the results by ticking the Adjust significant figures checkbox. We'll calculate the scientific notation equations assuming ten significant digits if you don't specify your own number.