Scientific Notation: Definition and Examples

Scientific Notation: Definition and Examples

Scientific notation is a special way of writing numbers so they are easier to work with. This lesson will define scientific notations and show some examples of how to convert numbers from standard notation to scientific notation and back.

Just Like Baby Bear

Remember the story of Goldilocks and the Three Bears? Goldilocks goes into the house of the Three Bears and at one point in the story finds Papa Bear's chair is too big, Mama Bear's is too small, but Baby Bear's is just right. Now don't think I've gone off the deep end trying to relate Goldilocks with math; I do have a point.

In mathematics, especially as it relates to the sciences, there are often numbers that are very large or very small, and they can be difficult to work with. By writing these numbers in scientific notation, we can more easily solve problems with numbers that used to be too big or too small. Now the numbers are just right.

What is Scientific Notation?

So now you know that scientific notation is a way to write very large or small numbers in a way that makes them usable. But what does it look like? Here are some examples of numbers written in scientific notation:

34,000,000 = 3.4 x 10^7

0.0000613 = 6.13 x 10^-5

200 = 2 x 10^2

0.0099 = 9.9 x 10^-3

How Does Scientific Notation Work?

Here is how scientific notation works. Take the number 700, for example. We know that 700 is equal to 7 x 100. Well, 100 is the same thing as 10^2, so 7 x 100 = 7 x 10^2, which means that 700 is also equal to 7 x 10^2. Both 700 and 7 x 10^2 have the same value; they are just written in different ways.

Any number written in scientific notation has two parts.

The first part is the digits - written with a decimal point after the first number and excluding any leading or trailing zeros. Leading zeros are zeros between the decimal point and the first non-zero number in a number smaller than 1. For example, the number 0.0012 has 2 leading zeros. Trailing zeros are zeros after the last non-zero number in a number greater than 1. For example, the number 5,308,000 has 3 trailing zeros. The zero between the 3 and the 8 is not a trailing zero because there is a non-zero number (the 8) that comes after it.

The second part of a scientific notation number is the x 10 to a power. This part puts the decimal point where it should be. It shows how many places to move the decimal point.

What Moves Where?

Knowing how to move the decimal point is one of the more difficult parts of scientific notation. Do I shift it left or right? What is the difference between the negative exponent and the positive exponent? But it really just comes down to remembering two things:

1.) Large numbers will have a positive exponent.

2.) Small numbers will have a negative exponent.

So when you are converting numbers from standard notation to scientific notation, just remember these two things. Let's do some examples.

Convert 834,000 to scientific notation.

Even though there is no decimal point showing in this number, we know that it is at the end of the number. To convert the number to scientific notation, the first step is to move that decimal point from the end of the number to after the first non-zero number - in this case, the 8. Then we drop the trailing zeros, and the first part of our scientific notation is 8.34.

To find the second part of the scientific notation number, count the number of spaces that you moved the decimal point. For this example, the decimal point was moved 5 spaces. That number will be the exponent. And in this case it will be positive because the number is a large number. So the second part of the scientific notation for this example is x 10^5.

Putting it all together gives us 834,000 = 8.34 x 10^5.

Let's try another example: Write 0.002598 in scientific notation.

Again, the first step is move the decimal. It will stop right after the 2.

Next, count the number of spaces it moved, and that number will be the exponent. It will be negative in this case because the number is smaller than 1.

So, 0.002598 written in scientific notation is 2.598 x 10^-3.

Let's try an example going the other way:

Write 4.92 x 10^4 in standard notation.

We can learn a few things just by looking at the number. First, we know that the answer will be a large number because the exponent is positive. Next, we see that the decimal place will move 4 spaces.

When you move the decimal place, any spaces that are created where there is not a number will be filled with a zero.

So after moving the decimal place 4 spaces to the right, we see that 4.92 x 10^4 = 49,200.

Let's try one more example:

Convert 2.205 x 10^-3 to standard notation.

By looking at this problem, we notice that the exponent is negative, so the number will be small. And the exponent also tells us to move the decimal 3 spaces to the left. Again, any empty spaces will be filled with a zero.

So, 2.205 x 10^-3 = 0.002205.

Lesson Summary

Scientific notation is a method for writing numbers that makes very small and very large numbers easy to work with. It is especially useful for scientists who might be working with extremely large numbers like the distance between the Earth and the sun, or extremely small numbers, like the size of an atom.

To write a number in scientific notation, simply move the decimal place from its current location to the space directly behind the first non-zero digit in the number. This will be the first portion of the scientific notation number. The second part is always x 10 to some exponent. The exponent is the number of spaces that the decimal point was moved. It will be positive if the number is a large number and the decimal point was moved to the left and negative if the number was very small and the decimal point was moved to the right.