Converting numbers to scientific notation and back to standard format

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Written by MM

This article explains how to convert numbers to scientific notation and back again to standard format. It also contains two calculators that will enable calculations to be checked and that provide feedback on common data entry input errors.

How do you convert numbers to scientific notation?

In mathematics, science and engineering students frequent have to work with very small, e.g. 0.000005, and very large, e.g. 3200000000 numbers. For example, students in microbiology are often required to write the number of colony forming units (CFU)/mL or gram in scientific notation. To avoid dealing with these large and small numbers mathematicians, scientists and engineers have developed a particular way of expressing numbers; this is called scientific notation.

In scientific notation each number is written in the form:

b × 10 ^{y}

where b is a number between 1 and 10 and y is a positive or negative whole number.

The following examples may help to illustrate how scientific notation works, 10 can be written as 1 x 10 ^{1}, 0.1 as 1 x 10 ^{-1 }, 100 as 1 x 10 ^{2 }, and 0.01 as 1 x 10 ^{-2 }.

Note that 10 ^{0 }is 1 and that the number 1 and numbers between 1 and 10 (but not 10!) can also be written as number x 10 ^{0} e.g. 3 can be written as 3 x 10 ^{0} . While this is mathematically valid this is not necessary since these numbers are already in scientific notation.

Note this form of scientific notation where a number is written in the form of a single number before a decimal point to a power of 10 is often called normalised scientific notation. Final answers to calculations should be expressed in this format.

For example 0.350 can be written in a number of ways including 3.5×10 ^{-1}, or 35 ×10 ^{-2}, or 350×10 ^{-3}. However only 3.5×10 ^{-1} is written in correct normalised scientific notation.

So 6000 CFU/mL can be written as 6 x 10 ^{3 }CFU/mL.

The calculator below can be used to convert numbers, to scientific notation this is sometimes referred to as converting regular notation into scientific notation.

How do you convert numbers in scientific notation back to decimals?

Above we have seen that numbers such as 6 x 10 ^{3 }or 3.5 x 10 ^{-1} are expressed in normalised scientific notation. How do you convert a number in scientific notation back to standard format again?

First we will define terms. Looking at 3.5 x 10 ^{-1}, you can see that we have a number, 3.5, multiplied by 10 (base) to the power of -1.

The number before 10 to the power of -1 is usually called the fractional coefficient and the -1 is referred to as the exponent.

In this case we must make the fractional coefficient smaller. So if you wish to convert 3.5 x 10 ^{-1} to a standard number simply move the decimal point to the left by one place; the exponent indicates the number of places to move the decimal point. So if the exponent was -4 you would move the decimal point 4 places.

Let’s look at what happens when the exponent is positive. In this case we are making the fractional coefficient larger and we will move the decimal point to the right. Again, the exponent indicates the number of places that the decimal point should be moved.

Converting numbers from one format to another can easily be done using spreadsheet programmes such as Numbers or Excel or the calculator below.

The figure below illustrates how you might set up an Excel spreadsheet to convert, say, 3.2×10 ^{-4 }which is in scientific notation^{ }to standard format. Prepare two columns, label one Fractional coefficient (A1) and the other Exponent of base 10 (B1). The fractional coefficient is then entered at A2 and the exponent, in this case -4, entered into cell B2. The result of the calculation 0.00032 is given in cell, B3.

To make the conversion, 3.2 is multiplied by 10^-4 as shown in the figure below. Hence =B2*10^B2 is entered into B3.