Scientific Notation - Explained!

In science, we often have to deal with some very large numbers. For example, the mass of the sun. That is the mass of the sun. Two followed by thirty zeros in units of kilograms. That is two thousand billion billion billion kilograms. There has got to be a better way to write that. Can you imagine if there wasn't?

And so we find that the mass of the sun is two zero zero zero zero zero zero zero zero zero. Yes? Sorry, could you repeat that? I was with you up to two zero zero zero zero zero? Ahh? It was ahh... two zero zero.

So, to represent large numbers easily, we use something called scientific notation. Scientific notation takes advantage of powers of ten. For example, ten to the power of two means take two tens and multiply them together. And you get 100. Ten to the power of three means multiply three tens together. And you get 1000. Ten to the power of four means take four tens and multiply them together. And you get 10,000.

So you should notice a pattern developing. When it's ten to the power of two, our final answer has a one followed by two zeros. Ten to the power of three, our final answer has a one followed by three zeros. Ten to the power of four, again, four zeros. So if we wanted to represent the mass of the sun in scientific notation, we would need thirty zeros. So we represent that using ten to the power of thirty.

This means that we're multiplying ten by itself thirty times, which gives us a one followed by thirty zeros. But the mass of the sun is actually twice that. It's two followed by thirty zeros so we can multiply two by ten to the thirty in units of kilograms, and that is the mass of the sun. It's much easier to write and it takes up much less space on the page.

The other problem is we have to deal with some tiny numbers. For example, the mass of a proton is... That is the mass of a proton. Zero point, and there are 26 zeros, and then 1673 kilograms. A truly tiny number, so how do we deal with this? Well again, it's using a similar trick. If we raise ten to the power of minus one, it means divide by ten, not multiply by ten.

So this means divide by ten, which gives you zero point one. Ten to the power of minus two means divide by ten twice, which gives us zero point zero one, a hundredth. Ten to the power of minus three means divide by ten three times, or zero point zero zero one. So again, you see a pattern developing. The exponent here tells you how many places to the right of the decimal that the one is.

So in this case, ten to the minus two, the one is two places to the right of the decimal. Here, ten to the minus three, the one is three places to the right of the decimal. So in this case, I actually have a one that is 27 places to the right of the decimal.

So I can write the mass of a proton as 1.673 times ten to the minus 27 kilograms. And this ten to the minus 27 has the function of putting this number, 1.673, 27 decimal places to the right of the decimal point.

So as a challenge question to see if you've understood it: if the sun were made entirely of protons, how many protons would there be in the sun?