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How to Convert Scientific Notation to Standard Form

Scientific notation is very useful, especially when you're working with huge numbers or microscopic ones. The notation can be confusing, but once you get it down, it's both simple and easy to use.

How to Convert Scientific Notation to Standard Form

What is Scientific Notation and Standard Form?

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. That's 145,000,000,000,000! 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 (distance from the center to the edge) of a proton (the small, positive particle in the nucleus) is 0.841 femtometers. That's .000000000000000841 meters for the radius of that positive particle that defines so much of our world. Standard form is the way numbers normally appear, like the examples above. 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 write out the problem! 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 (the parts of the number that make the most difference), and then use a power of 10 (number of times that 10 is multiplied by itself) to express just how big or small they are. For example, the distance to Gliese and the radius of a proton may be written as shown in 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 Notice that the decimal places have been replaced by powers of 10.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 tenths of quadrillionths, you have 10 to the negative 16th power.

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

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.

Converting Between Scientific Notation and Standard Form

You can use these steps to convert a number from standard form to scientific notation (see Figure 2):

Find the most important non-zero digit in the number. It will always be the non-zero digit farthest to the left. In 123, it would be the 1. In .0000234, it 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).
Rewrite the number with that digit to the left of the decimal point and the next two (or three, if required) digits to the right of the decimal point. For 123, you would end up with 1.23. For .0000234, you would get 2.34.
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.
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).

Figure 2. Steps to convert standard numbers to scientific notation

Now let's see if we can convert our new scientific notation numbers back to their standard form (Figure 3).

Grab the number (not the power of 10 part), and start the process.
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.
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.

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Figure 3. Converting scientific notation to standard numbers