So scientific notation. We're going to talk about what it is, why it's important, and then we're going to do some example problems so that you'll get really good at this stuff. So scientific notation is like an abbreviation.
It's a shorthand way to write numbers that are really big and really small. Here's why that's important. In science, we spend a lot of time dealing with really huge numbers. For example, this is the number of miles that it is from Earth to the closest star outside of our solar system.
Really big number. We also look at a lot of really small numbers. This tiny number is the amount of time that it takes for light to go from one side of an average sized bedroom to the other side.
That's about 30 feet. And these numbers aren't even the biggest and smallest that we use a lot. This is the number of atoms in an average sized glass of water. And this number is the weight in pounds.
Notice there's a decimal place here, so it's a really tiny number. This is the weight in pounds of a proton in an atom. Look at all these zeros.
These zeros are such a pain, right? Imagine having to multiply this number by this number. Or do division. Do any of this by hand. It would be such a pain.
Even if you're using a calculator or a computer, just touch it. typing in all these zeros and making sure that you haven't forgotten one or added in a couple more, that's really a big pain. So scientific notation comes in handy because it lets us take numbers like these really big ones and these really small ones, get rid of all of the zeros and write them in a very compact, very simple way that's much easier to use.
Let me show you how we do it. Let's start with this really big number in miles. Here is what we do to compress it down and write it in scientific notation.
The first thing is we figure out where would the decimal place be. A lot of these numbers won't actually have a decimal place written but you know that it should be right here. So I'll write in where the decimal place would be. There it is. Now I move it and I count the number of spots.
Watch what I do. Here we go, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, 6, 7, 8, 9, 10, 11, 12, 13. And here is the new spot. I move it until the decimal place is sitting here and there is only one number, one digit I should say, one digit to the left of the decimal place. The decimal place is here and then one digit there.
So I moved it 13 spots in this direction. to move it to a place where there was only one digit to the left of it. After I do that I write the number with the decimal place in the new position. So it's going to be 2.53 and then I get rid of all of the zeros after this three here. So this is the first part of the number.
Now I need to account for the fact that I moved the decimal place sixty 16 spots, I mean 13 spots in this direction. So I say that I moved 13 spots in this direction by writing 10 raised to the 13 power. You see this number here, the exponent on 10 is the same number of spots that I moved the decimal place to its new position. So 2.53 times 10 to the 13. I move the decimal place in this direction and I get a number.
number 13 on 10. Okay, now here's a very small number. I do the same but instead of getting bigger the exponent on 10 is going to get smaller. Let me show you what I mean. Decimal place is here, it's already written in the number. Watch what I do.
I'm going to move it 1, 2, 3, 4, 5, 6, 7, 8 spots. Okay? So now the decimal place is here.
and there's one digit that's not a zero to the left of it. So 1, 2, 3, 4, 5, 6, 7, 8, I had to move it 8 in this direction. Now I rewrite my number with a decimal place in its new position.
So it's going to be 2.03. And now to show that I've moved the decimal place eight spots in this position, in this direction, I'm going to do 10 to the... Negative 8. So, when I move the decimal place in this direction, I get a positive number.
The exponent on 10, it goes up. On the other hand, when I move the decimal place in this direction, the exponent on 10 goes down. So, I have positive 13 in this direction and I have negative 8 in this direction.
So, I've summed up these rules here. Here's a decimal place. You move it left, the exponent goes up. You move it right, the exponent goes down.
component goes down. It's negative in this direction, positive in this direction. This is the kind of thing that is best learned by just getting a lot of practice.
So now we're going to do some example problems. First we're going to take numbers like this and we're going to write them in scientific notation and then we're going to take numbers that have already been written in scientific notation and show how to turn them back into regular numbers. Okay.
Let's start with the first one. So we're going to take the number of Let's start with this, we'll write it in scientific notation. The first thing that I want to do is find out where the decimal place is.
Decimal place is here. I'm going to keep in mind that when I move my decimal left the exponent goes up. Now I think it's easiest to think about starting at 10 to the 0, okay? When the decimal place is right here the exponent part of the number is 10 to the 0. I'm going to start with the decimal place. I move it one spot, okay, that's 10 to the first.
Move it again, 10 to the second, 10 to the third, 10 to the fourth, 10 to the fifth, 10 to the sixth, the seventh, eighth, and ninth. So I move it nine spots here and it's going to be 10 to the ninth. Now I'll rewrite my number with the decimal place in its new location.
It's going to be 3.749 times 10. 10 to the ninth. So what I did here, it's like I put the decimal place here and I imagine that I was starting at 10 to the zero and then that number on 10 went up each time I moved the decimal place to the left. Let's try it with this but instead of going up, the exponent is going to be going down.
Okay, so we start at 10 to the zero and now we go down. 10 to the negative first, 10 to the negative second, 10 to the negative third, 10 to the negative fourth, 10 to the negative fourth, 10 to the negative fifth, 10 to the negative to the negative fifth. So five in this direction times ten to the negative fifth because I just counted down from ten to the zero. And my new number is going to be five point two six nine times ten to the negative fifth. Okay, this one right here.
Put in the decimal place and start at ten to the zero. Ten to the zero, ten to the first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, 9th, 10th, 11th, 12th, 13th, 14th, 14 in this direction so 10 to the 14th as I was counting up and I stop right here so there's one digit to the left of the decimal place and I write it as 8.13 times 10 to the 14th. This one here I'm moving it in this direction so we're starting at 10 to the 0 and then the exponent is going to be going down. So 10 to the 0, 10 to the negative 1, negative 2, negative 3. negative 4, negative 5, negative 6, negative 7, negative 8. 8 in this direction times 10 to the negative 8 and my number, I keep the decimal place here so there is one non-zero digit to the left of it. It's going to be 2.14 times 10 to the negative 8. So that's how you take a number that's written kind of like regularly.
We call it in decimal notation. how you take a decimal notation number and transfer it into scientific notation. I'm going to do a couple more practice problems at the end of this video if you want a little more practice on this.
But now I want to move on to numbers that are written in scientific notation and I want to undo it and turn them back into regular numbers, numbers that are written in decimal notation. Okay? Here's how I do this.
I like to think that we need... We need to take the number and get the exponent back to 10 to the 0. Here's what I mean by that. Let's start up here. 7.921.
Right now, we're at 10 to the 8. We want to get this 10 to the 8 back down to 10 to the 0. So when we move the decimal place right, the exponent goes down. So watch this, okay? Here it is at 10 to the 8. 10 to the 7. the right one spot, okay? 10 to the sixth, move it down another spot.
10 to the fifth, move it down another spot. Now I'm going to have to move it a few more times and I'll add in the zeros after I'm done. Okay, so I went 10 to the eighth, seventh, sixth, five, four, three, two, one, zero.
So, I had to move the decimal place a few more spots and I ran out of numbers. So, for each of these places in between where I moved the decimal place, I can write in a new zero here. You see how I did that?
I just put a zero in between every place where I had to move the decimal. And so, now my new number is this. I just took the exponent and moved the decimal to the right until I got it back to 10 to the 0. Okay, now for this one here, 8.2 times 10 to the negative 5th. I want to get this exponent back to 0. 0. So I want the number to go up because it's a negative number. So I want to move the decimal place to the left.
So right here it's 10 to the negative 5th. Move it one spot it's 10 to the negative 4th. It went up one. Okay, 10 to the negative third, 10 to the negative second, 10 to the negative first, and 10 to the zero. So I'm going to put in a new zero between each one of these decimal places.
And then to be neat, I'll rewrite this number. 0.000082 is how you write this in decimal notation. Okay, two more.
4.13 times 10 to the third. So that means that the decimal place is 10 to the 3rd now. I want to get this 3 back down to 0. 10 to the 2nd, 10 to the 1st, 10 to the 0. That's my new spot and I add in a 0 there. So my answer is 4,130. Here's the last one.
5.2216. When the decimal place is here, we're at 10 to the negative 9th. So I want to go up to get it back to 0. So I'm going to be moving the decimal place to the left to get this exponent up. So negative 9th here.
negative 8th negative 7 negative 6 negative 5 negative 4 negative 3 negative 2 negative 1 0 there's my new decimal place location and I fill in all of the space in between the points with zero and then I can rewrite my number just like this and here it is written in decimal notation. That is how you take a number that's written in scientific notation and get it back into kind of a regular, a decimal notation number. Now, I'm just showing you how to do this here.
I'm not talking about the math behind it, what's actually going on with this exponent, with this... number with this 10 with the exponent. So, it's really good if you can do this now.
But I hope that you'll find the time to watch the video called Really Understand Scientific Notation, alright? Because it's one thing to just be able to work through this converting one type of number into another. It's a more important thing to be able to understand how the math is actually working.
So please watch that video so you can actually understand the math. So many people hate scientific notation and I think that's because they don't really understand it. So please watch that video if you have time about really understanding scientific notation.