Representing numbers using powers of 10

0:00- [Sal] What I want to do in this video is introduce you to the idea that there's many different ways of writing a number, and depending on the scenario, sometimes it's easier or maybe it uses less writing to write a number in one way or another. For example, if I had the number 4,200,000, I could write it as 420,000 times 10.

0:27Now, you might say, "Why is this at all useful?" This seems like more work than just writing 4,200,000." And you'd be right in that scenario, but let's keep running with it. I could also write this as 42,000, 42,000 times 100. You're saying, "Hey, you know what?

0:46This is still a lot more writing than what we had up there, but I could also write it as 4,200, 4,200 times 1,000. I think you could see where this is going. I could also write this as 420 times 10,000. I could also write this as 42 times 100,000. And I could also write this as, if I really want to, 4.2 times one million.

1:21These are all equivalent ways, and I know what you're thinking, "Why is Sal going through this trouble?" Well, there's ways that I could simplify how I write a million, how I write 100,000. I could rewrite them as powers of 10. One million is what? It is 10 to the sixth power.

1:38If you take 10 and you multiply, or if you take six tens and multiply them all together, you'll have one followed by six zeros. So I could also write that this is the same thing as 4.2 times 10 to the sixth power. Similarly, this one over here would be 42 times 10 to the fifth power.

1:56And now, you might see why something like this or even something like this might be cleaner ways of writing this. For the number of 4,200,000, you might say, "Hey, yeah, it takes up less space, but this is a little bit complex. You have to complicate it, you have to think about it a little bit."

2:12What if it was something that had way more zeros? What if it had 15 zeros or 20 zeros? Then you could imagine writing in some of these other formats, where using powers of 10 could be a lot more convenient. And we could take that idea and also apply it to really, really small numbers.

2:28For example, if I were to take the number 0.00078, how else could I write that? Well, I could write that as 78. Actually, let me think about it this way. I could think of this as 78. Let me get out my place values. This is tenths. This is hundredths. This is thousandths.

2:53This is ten thousandths. This is hundred thousandths. So one way I could view that this is the same thing as 78 over 100,000. And you might say, "Okay, well, why is that useful?" Well, that's the same thing as 78, that's the same thing as 78 divided by 10 to the sixth, or 10 to the fifth, I should say, let me be careful.

3:24I have five zeros here, 10 to the fifth power. And that's the same thing as 78 times 10 to the -5 power. So that's kind of interesting. Well, another way I could do this is I could divide this value by 10 and I could multiply that value by 10. If I divide by 10 and multiply by 10, I'm still at the same value, so if I divide the first thing by 10, this would be 7.8.

3:50And if I multiply this one by 10, that would be times 10 to the -4 power. So once again, you might be a little bit on the fence of whether writing it like this or writing it like this is really one's better than the other, but again, imagine if instead of just three leading zeros here, you had 30 leading zeros.

4:10Then it definitely becomes more interesting to write in one of these other formats.