How to Convert Scientific Notation to Standard Form

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  • 0:03 What Is Scientific Notation?
  • 2:11 Changing Notation
  • 4:23 Lesson Summary
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Lesson Transcript
Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

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.


Figure 1. Distance to Gliese and the radius of a proton in standard and scientific notation
Sci Notation Figure 1


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.

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