## Ask Professor Puzzler

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Seventh grader Lucie from Australia asks, "How do you do scientific notation?"

That's a great question, Lucie. There are a couple reasons why we have scientific notation. The first reason is that many numbers are either so big or so tiny that they are hard to write out without scientific notation. For example, would you like to know the distance from the Milky Way to Andromeda galaxy? The distance is:

d = 23,600,000,000,000,000,000,000 meters

Would you like to know the size of a water molecule? The size is:

w = 0.0000000025 meters

Both of those numbers are a nuisance to write. The first one because it's so big, and needs a lot of zeroes, and the second one because it's so *small* and needs a lot of zeroes! If only there was a way to avoid writing all those zeroes! Well, that's the good news about scientific notation; it lets us avoid all that writing.

If you're writing a *big* number with scientific notation, you're going to ask yourself, "How many places would I need to move the decimal to the *left* in order to have only one digit in front of the decimal? In the case of the Andromeda distance, if you moved the decimal 22 places to the left, you would have the decimal right between the 2 and the 3. Once you've figured that out, you can write the number as:

d = 2.36 x 10^{22} meters

Notice that we didn't have to write all those zeroes! We put the decimal after the first digit, and then we wrote the number of places we had to move the decimal as an exponent of 10.

We do the opposite for very small numbers: we ask, "How many places would I need to move the decimal to the *right* in order to have one non-zero digit in front of the decimal?" In the case of the size of a water molecule, if we wanted the decimal to to be after the two (the first non-zero digit), we would need to move it 10 places to the right. Once you've figured that out, you can write the following:

w = 2.5 x 10^{-10} meters

Note that we used a *negative *exponent this time. That's because we were moving the decimal ten places *in the other direction*. And again, we get to skip all those zeroes, making it easier to write.

I said there were a couple reasons for writing numbers in scientific notation. One reason, as shown above, is to make them easier to write. The other reason is to make them easier to read and understand. Let me show you an example.

*Which number is bigger: 23,500,000,000,000,000,000 or 4,200,000,000,000,000,000?*

In order to answer my question, you had to count the number of groups in each number to find out which one was bigger. Suppose it was written this way instead:

*Which number is bigger: 2.35 x 10 ^{19} or 4.2 x 10^{18}?*

This is so much easier! If both numbers are properly written in scientific notation, the number with the larger exponent is bigger! And what if they have the same exponent? Then you compare the numbers in front of the multiplication. For example:

*Which number is bigger: 3.4 x 10 ^{32} or 4.5 x 10^{32}?*

In this case, both numbers are in scientific notation, and they both have an exponent of 32. So we just compare 3.4 and 4.5. Since 4.5 is bigger, we know that 4.5 x 10^{32} is the larger number.

I hope that helps, Lucie!

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