Algebra 1 8 4 Scientific Notation

All right, section 8.4 Sophia notation. Essential question for today is how can represent very big numbers or very small numbers. You will understand these memes more at the end. Let's get right into it. First of all, why do need scientific notation? As it turns out in science, there are some very big numbers. For example, the speed of light 300 million meters/s. That's extremely extremely fast. Mass of the Earth. I'm not even going to try to say this number. I'm not sure how would do it. The point is huge. Possible diameter of an atom. This is one particular atom. Very very very small. 000 0 etc. Now you could try writing these in your science papers or you could try writing these on test. However, these numbers are just very very big and we don't want to kill the rainforest. We want to write our numbers in more palatable format meaning easy to digest. So what scientists or somebody came up with, but probably scientists because it's called scientific notation, is they decided this. What if just moved the decimal and then said how many times moved the decimal. So what they're going to do is they're going to look after the first kind of regular number that's not zero. They're going to put decimal and then they're going to record how many times they moved the decimal. So from the starting decimal place of here, they move the decimal 1 2 3 4 5 6 7 8. So they're going to write three. They could write 3.0, but like it's just three, right? 3.0 is just three. But then they need to record how many times they moved the decimal. And they're going to do that with this. So I'm just going to do it and then I'll explain it. 10 eth. The reason it's times 10 to the eth is because think about what multiplying 10 means. Multiplying times 10 moves the decimals. So they're saying, hey, from this number from three, we're going to be moving the decimal 8 times. And that would get you back to here with this number. This next one, decimal starts here. Every comma is kind of three. So 3 6 9 12 15 18 21 24. We're going to put the decimal here. Look, there's one number and then 972. Remember, we're putting the decimal after the first number. Okay, but we have to record how many times we move the decimal place. In this case, it's 24. Also, we can use this for very small numbers except instead of using positive exponent because instead of making very big number, we're going to use negative exponent because as you know 10^ the -1 is kind of like dividing by 10. So for this number again, we're shooting to put the decimal after the first like nonzero number. So 1 2 3 4 5 6 7 8 9 10. So five, we can write 5.0 0 or just 5 in this case* 10 to the -10 and again it's negative because I'm thinking dividing by 10. Dividing by 10 is negative exponent. Dividing by 10 makes the number very very small. So this is why need scientific notation. Let's practice with it. So big reminders. First of all, numbers that are bigger than one, like this guy and this guy, they need positive exponents after the 10. So, 10 to positive. But decimal numbers between 0 and 1, we had to divide to get there. So, that's like this example, we're going to use 10 to the minus. Okay, so let's do some problems. We need to write the numbers. shouldn't have said standard form. They're already in standard form. want to write the numbers in scientific notation. Notation. Okay. So this one, this first one, need to move the decimal. Let's see. 1 2 3. Here it is. It's after the first nonzero number. And in this case, moved it three times. But is it plus or minus? Let's think about my reminders. Well, decimal numbers between 0 and one. Okay, that's this situation, right? need to use negative. And it's because to get from here to here, I'm thinking about dividing by 10 three times. This next number, the decimal, where is it? Is it here? Is it here? Is it here? No, silly. It's right after the number. And need to take that decimal and need to move it 1 2 3 4 5 6. Okay, so this is 1.065250 * 10 to the positive 6 because multiplied by 10 to get that big number. And you might be saying to yourself, "Isn't this just as long as if had just written the original number?" And would say, "Yes." This is silly example. Next problem. Looks like I'll be using negative exponent. Remember, want the exponent to show up right after the first nonzero number. So, how many times do need to move the decimal? 1 2 3 4 5 6. And that would be 5.26. Finally 3 6 7 7.2 * 10 to the This is positive 7 because I'm multiplying by 10 to make the number big. Let's go backwards and let's try to do it quickly here. Okay, so have 7.5, but they're telling me, hey, moved the number 10 times and did this in positive fashion. In other words, we're making the number big because we're multiplying by 10 seven times. So, let's move that decimal. 1 2 3 4 5 6 7. Everywhere where see Okay, so sorry. Now the decimal is here. Everywhere where see gap, just add zero because that's what multiplying by 10 is doing. It's multiplying by 10. That's adding zeros. And then we can go back in. guess don't need the decimal anymore. We can put in commas to make it look nice. 75 million. Next one. I'm moving the decimal six times, but since it is negative, that's kind of like dividing by 10. So, in other words, this number is going to become very small negative number less than zero. Remember, negative exponent means small decimal number between 0 and 1. So, 1 2 3 4 5 6. Put the decimal here. How many zeros? 1 2 3 4 5. Okay. So, let's write that little nicer. I'll also put zero before the decimal. would like to see that. Then it looks like now have 1 2 3 4 five zeros and then two three three 10 the 4th. It's multiplying by 10 four times. So move that decimal positively. 1 2 3 4. Looks like I'll have two extra gaps for zeros. So 3 0 3 0 0. This is an insanely boring video to do. apologize for this. All right, last one. Negative exponent means move the decimal to make this number very small number because remember times 10^ the 6 is like dividing by 10. So 1 2 3 4 5 6. Here's the decimal. This is 1 2 3 4 5 zeros. So 0 five zeros 1 2 0 3 4 I'm exhausted. Am done? No. okay. So okay, I'm just comparing two numbers. So let's see. This number is in standard form already. This number though is in scientific notation. So let's take this number and put it in regular notation. Time 10 5th means move that decimal five times to make the number bigger because I'm multiplying by 10 five times. So I'm moving the decimal to make the number bigger. 1 2 3 4 5. It's 4 04 0 0 1 This one has 0.1 at the end. This one has just straight up one at the end. So, I'm going to say that this number is bigger. This is the bigger number. Next, okay, I'm moving the decimal. Since it's like dividing by 10 because of the negative exponent, I'm going to move it three times to make the number smaller. So, 1 2 3. That throws an extra two zeros at the beginning. Two zeros after the decimal. And then have three three nine. I'm flying through this because feel like you got the point. look. These numbers are exactly equal. That's surprise. Yay. All right. Next. Finally, move it seven times to make the number big. 1 2 3 4 5 6 7. So, 6 0 4 5 8 9. I'm adding two extra zeros. Let's throw in some commas. this one definitely has more digits than this one. This is the bigger number because look, more digits. This guy is only in the 60 millions. This guy is in the 600 millions. Well, that was fun. Am done? no. I'm not done yet. Okay, now want to tell you how to do this in your calculator. Here, we're taking number in scientific notation. We're multiplying it by another number in scientific notation. You could enter this straight into your calculator and it should be able to do it for you. Same with this. But most of your calculators have button that looks like this. So try to find this in your calculator. it's comma button and then above it is Okay. So you need to obviously hit second in order to access the part. But this is what we can do. And if you've ever seen this in your calculator now you know what it means. If hit 4.4 So again hit second and then hit this button. That will put the there. minus 5 use the negative like negative 5. hear the bell. I'm recording this at school. How exciting. This means 10 to the So if do 4.4 - 5, that means 4.4 * 105. So would put this in parenthesis. Then you can multiply times 5.8 2. So let me just put this into my calculator right now. 4.4 4 e5 * 5.8 e2. It gives me this answer. 02552. If want to put this in scientific notation, let's do that. I'm just moving the decimal twice. It would be 2.552 * 10 -2. If your calculator happens to be in scientific notation form, it would tell you 2.552 minus 2. And you know, at this point, the video is just getting extremely boring. Let me just set up this for you and you can do it on your own. Like this would be 5.4 e5. This would be 1.8 -2 if you wanted to put it into your calculator and you could find out what that is. But think you get the point. And honestly, this video got way too long for pretty simple concept. But anyway, there you go. Summary. If you have big numbers, make sure you're doing 10 to the positive. If you have decimal numbers between 1 and zero, make sure you're doing time 10 to the negative. Because in order to get those decimals, we know that we're kind of more dividing by 10. And that's what negative exponent is doing. Thanks for watching. See you.