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Scientific Notation (Mathematics Glossary)

What Is Scientific Notation?

Scientific notation is a way of writing a number.

In scientific notation, a number is written as a number between 1 and 10 multiplied by a power of 10.

The number below is written in scientific notation. It is said as "1.2 times 10 to the 3".

1.2 times 10 to the 3

Dictionary Definition

The Oxford English Dictionary defines scientific notation as "a system of representing some given number as a product of a number with an absolute value between 1 and 10 and a power of 10 (as 2 × 102 for 200)"

Understanding Scientific Notation

A number represented in scientific notation is written as a number between 1 and 10 multiplied by a power of 10.

The power of 10 is written as a 10 with a raised number by its side. The raised number is called an exponent.

1.2 times 10 to the 3

It tells you how many times the 10 is multiplied by itself, or how many 0s there are after the 1.

1.2 × 103 = 1.2 × 10 × 10 × 10 = 1.2 × 1,000

So 1.2 × 103 represents the number given by 1.2 × 1000, which is 1,200.

1.2 × 103 = 1,200

Why Is Scientific Notation Useful?

Scientific notation is useful because it allows us to write very large or very small numbers in a shorter, standard way.

Some examples of numbers written in scientific notation are shown below, along with the numbers written in full. Notice that it can save you writing lots of 0s.

4 times 10 to the 2 equals 200. 1.2 times 10 to the 5 equals 120,000

When numbers are very small (less than 1), we can also save writing 0s by using scientific notation. In this case, the power of 10 can has a negative exponent:

5.1 times 10 to the minus 2 equals 0.051

Converting a Number into Scientific Notation

Scientific notation is a useful way to write long numbers.

123,000 equals 1.23 times 10 to the 5

To convert the number to scientific notation, we have to move the decimal point to the right of the first digit.

Even though it isn't written, the decimal point is at the end of the number. Count how many places it has to move left so it is to the right of the first digit (1):

The decimal point needs to move 5 places to the left

The decimal point has to be moved 5 places to the left to make 1.23. The number is written as 1.23 × 105.

Read more about converting numbers to scientific notation

Converting Scientific Notation to a Number

When you see a number written in scientific notation, you must know what number it represents.

1.23 times 10 to the 5 equals 123,000

To convert scientific notation to a number, we have to move the decimal point right. The exponent of the power of 10 is 5 so the decimal point needs to be moved 5 places.

You will need to add 0s so that the decimal point can be moved:

The decimal point needs to move 5 places to the right

The decimal point has to be moved 5 places to the right to make 123,000.

Read more about converting scientific notation to numbers

Interactive Test
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Here's a second test on scientific notation.
Here's a third test on scientific notation.

Note

Number Between 1 and 10

The number in scientific notation (that multiplies the power of 10) is between 1 and 10. It can include 1, but not 10.

Powers of 10

A power of 10 is 10 raised to a exponent.

For example, 102 is a power of 10.

The small 2 written beside the 10 means it is raised to an exponent of 2. This means 10 is multiplied by itself 2 times.

102 = 10 × 10

The answer will have 2 0s after the 1:

102 = 100

Other Ways of Writing Scientific Notation

To write scientific notation on a keyboard, the ^ symbol is used for the exponent of 10 (102 = 10^2):

5.2 × 103 = 5.2 × 10^3

On a calculator, the E symbol is used to represent a power of 10 (×102 = E2):

5.2 × 103 = 5.2E3

Engineering Notation

Engineering notation is similar to scientific notation, except the number is limited between 1 and 1,000, and the exponent of the power of 10 must be a multiple of 3.

1,200 = 1.2 × 103
12,000 = 12 × 103
120,000 = 120 × 103
1,200,000 = 1.2 × 106
1,200,000,000 = 1.2 × 109