Scientific notation is very useful, especially when one is working with huge numbers or microscopic ones. This lesson will help make scientific notation less confusing and clarify when and how it is used.

## What Is Scientific Notation?

If you were planning a trip to the nearest multiple-planet solar system to ours (Gliese), maybe to visit our alien neighbors, you would have to travel about 145 trillion kilometers. Numbers like that can make your eyes cross just trying to keep track of the zeroes! Here’s another one. The latest guess for the radius of a proton (the small, positive particle in the nucleus of an atom) is 841 quintillionth meters, or 0.841 femtometers. That’s less than a quadrillionth of a meter for the radius of that positive particle that defines so much of our world. **Standard form** is the way numbers normally appear. Most numbers you run into will be in standard form. As you can see, however, standard form can get a little unwieldy when the numbers get too large or too small.

Huge numbers and incredibly tiny ones can be very difficult to write and keep track of. Imagine trying to do math with numbers that have 25 digits each. You’d get a hand cramp just trying to type the problem into your calculator! Fortunately, there’s an easier way to work with numbers that are impossibly huge or tiny. **Scientific notation** is a way to take the most significant digits and then use a power of 10 to express just how big or small they are. Let’s break that down a little bit. **Significant digits** are the parts of the number that make the most difference. The **power of ten** is the number of times that 10 is multiplied by itself, often represented by how many places the decimal is moved. So, scientific notation is a way to represent large numbers as smaller numbers that are multiplied by a power of 10.

Let’s look at how the distance to Gliese and the radius of a proton may be written in scientific notation.

Notice that the decimal places have been replaced by powers of 10. Instead of hundreds of trillions, you have 10 to the 14th power. Instead of quintillionths, you have 10 to the negative 16th power. That’s why they came up with scientific notation.

Okay, let’s work with changing numbers back and forth, between scientific notation and standard form.

## Changing Notation

You can use these steps to convert a number from standard form to scientific notation. First, find the most important non-zero digit in the number. It will always be the non-zero digit farthest to the left. In 123, the most important digit would be the 1. In .0000234, the most important number would be the 2. If you have fewer than three non-zero digits, you only use the ones you have, you don’t have to add 0s to the right.

Next, rewrite the number with that digit to the left of the decimal point and the next two or three digits to the right of the decimal point. For 123, you would end up with 1.23. For .0000234, you would get 2.34. Next, find out what power of 10 you need to make the scientific notation correct. Count the decimal places you had to slide to get that most important digit to its place on the left of the decimal point. For 123, you had to slide it to the left twice. For .0000234, you had to slide it five places to the right.

Finally, write a multiplication sign, the number 10, and then the number of decimal places you had to slide as an exponent for the 10. If the number was less than 1, the exponent will be negative.

What if you had more than three or four digits in your number? You’ll only take the most significant three or four, and will round up if the next digit is larger than 4. For example, if your number was .0056789, you would round up the 7 to an 8 (or the 8 to a 9, if you’re keeping three digits after the decimal point in your scientific notation).

Now let’s see if we can convert our new scientific notation numbers back to their standard form. Begin by grabbing the number (without the power of 10) to start the process. Then move the decimal point one place for every power of 10 you have. If the powers of 10 are positive, move the decimal point to the right. If they’re negative, move the decimal point one place to the left. Fill in with 0s when you run out of digits. Finally, if your number is huge, make sure you separate the decimal places to the left of the decimal point into groups of three, separated by commas.

## Lesson Summary

**Standard form** is the way you normally see numbers written. 15 and .000000178 are both written in standard form. **Scientific notation** is a way to separate out the most significant digits of a number, and then multiply by a power of 10 to keep their size accurate. The figures in the lesson show what scientific notation looks like. Converting from one to the other is a matter of taking specific steps, as described in the lesson, and can become quite automatic for you when you get used to it. Remember, scientific notation always has a power of 10 and only a single digit to the left of the decimal point.