By Mark Zegarelli

When you understand how scientific notation works, you’re in a better position to understand why it works. Suppose you’re working with the number 4,500. First of all, you can multiply any number by 1 without changing it, so here’s a valid equation:

4,500 = 4,500 x 1

Because 4,500 ends in a 0, it’s divisible by 10. So you can factor out a 10 as follows:

4,500 = 450 x 10

Also, because 4,500 ends in two 0s, it’s divisible by 100, so you can factor out 100:

4,500 = 45 x 100

In each case, you drop another 0 after the 45 and place it after the 1. At this point, you have no more 0s to drop, but you can continue the pattern by moving the decimal point one place to the left:

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What you’ve been doing from the beginning is moving the decimal point one place to the left and multiplying by 10. But you can just as easily move the decimal point one place to the right and multiply by 0.1, two places right by multiplying by 0.01, and three places right by multiplying by 0.001:

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As you can see, you have total flexibility to express 4,500 as a decimal multiplied by a power of ten. As it happens, in scientific notation, the decimal must be between 1 and 10, so the following form is the equation of choice:

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The final step is to change 1,000 to exponential form. Just count the 0s in 1,000 and write that number as the exponent on the 10:

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The net effect is that you moved the decimal point three places to the left and raised 10 to an exponent of 3. You can see how this idea can work for any number, no matter how large or small.