Infinity is bigger than you think – Numberphile

Video Transcript

JAMES GRIME: We’re going to break a rule. We’re break one of the rules of Numberphile. We’re talking about something that isn’t a number. We’re going to talk about infinity. So infinity. Now like I said, infinity is not a number. It’s a idea. It’s a concept. It’s the idea of being endless, of going on forever. I think everyone’s familiar with the idea of infinity, even kids. You start counting 1, 2, 3, 4, 5– you might be five years old, but already you’re thinking, what’s the biggest number I can think of. And you go, oooh, it’s 20. You get a bit older, and you go, maybe it’s a million. It never ends, does it? ‘Cause you can keep adding 1. So that’s the idea of infinity. The numbers go on forever.

But I’m going to tell you one of the more surprising facts about infinity. There are different kinds of infinity. Some infinities are bigger than others. Let’s have a look. The first type of infinity is called countable. And I don’t like the name countable. And Brady gave me a little bit of a hmm, just then. Because if you’re talking about infinity, you can’t count infinity, can you? Because it goes on forever. I think it’s a terrible name. I prefer to call it listable. Can we list these numbers? All right.

Let’s do these simple numbers, 1, 2, 3–

BRADY HARAN: You’re not gonna do all of them, are you James?

JAMES GRIME: 4. How long have we got?

BRADY HARAN: (LAUGHING) 10 minutes.

JAMES GRIME: Right. 5, 6– so you can list the whole numbers. So this is called countable. Listable, I prefer. What about the integers? All the integers. That’s all the negative numbers as well. So there’s 0. Let’s have that. But there’s 1 and minus 1, there’s 2 and minus 2, there’s 3, and minus 3. Now, that is an infinity as well. And in some sense, it’s twice as big, because there seems to be twice as many numbers. But it is infinity as well. They’re both infinity, and they’re both the same type of infinity. They both can be listed.

Perhaps more surprisingly, the fractions can be listed as well. But you have to be a bit clever about this. Let’s try and list the fractions. I’m going to write out a rectangle. 1 divided by 1. That’s a fraction. [INAUDIBLE]. Let’s have 1 divided by 2, 1/3, 1/4, 1/7– OK, that goes on. Let’s do the next row and have two at the top. 2/1, 2/2, 2/3, 2/4. Let’s do the next one. 3/1, 3/2. 4/6, 4/7. That goes on and we can keep going. So here, I’ve made some sort of an infinite rectangle array of fractions. Now if I want to make it a list like this, though, If I went row by row, you’re going to have a problem. If you go row by row, I’ll go– there’s 1, 1/2, 1/3, 1/5, 1/6, 1/7– and I’ll keep going forever. And I’m never going to reach the second row. I can’t list them. Not that way. You can’t list them that way. You’ll never reach the second row.

This is how you list them. Slightly more clever than that. You take the diagonal lines. Now, I can guarantee that every fraction will appear on one of those diagonal lines. And you list them diagonal by diagonal. So that’s the first diagonal. Then you list the second diagonal– there it is. Then you list the third diagonal, then you take the fourth diagonal, and the fifth. So eventually, you are going to do this every fraction. Every faction appears on a diagonal, and you’re going to list them. Now, if you take all the numbers, right? That’s the whole number line.

Let’s try that. Look, I’m going to draw it. It’s a continuous line of numbers. These are all your decimals. You’ve got 0 there in the middle, and you’ll go 1 and 2 and 3. But it has a 1/3. It will contain pi, and e, and all the irrational numbers as well. Can you list them? How do you list them? 0 to start with, and then 1? But hang on. We’ve missed a half. So we put in the half. Hang on, we’ve missed the quarter. We put in the quarter. But we’ve missed 0.237– so how do you list the real numbers? It turns out you can’t. In fact, rather remarkably, I can show you that we can’t list them, even though were talking about something so complicated as infinity.

BRADY HARAN: Do it, man!

JAMES GRIME: We need paper.

BRADY HARAN: We need an infinite amount of paper here, I think.

JAMES GRIME: (LAUGHING) It’s a big topic. Imagine we could list all the decimals, right? We can’t, actually. But pretend we can. What sort of– what would it look like? We’ll start with all the 0-point decimals. Let’s pick some decimals. 0.121– dot dot dot dot dot. Let’s pick the next one. Let’s say the next one is 0.221–. Next one, let’s do 0.31111129–. And let’s take another one, here. 0.00176–. Now I’m going to make a number. This is the number I’m going to make. I’m going to take the diagonals here.

I’m going to take this number and this number and this number and this number and this number. And I am going to write that down. So what’s that number I’ve made? It’s 0.12101– something, something, something. Now this is my rule. I’m going to make a whole new number from that one. This is the number I’m going to make. If it has a 1, I’m going to change it to a 2. And if it has a 2 or anything else, I will change it to a 1. So let’s try that. So I’m going to turn this into– 0-point. So if it has a 1, I’m going to turn it into a 2. If it’s anything else, I’m going to turn it into a 1. So that will be a 1. I’m going to change 1 here into a 2. I’m going to change that one into a 1. I’m going to change that one into a 2– that was my rule.

And I’ll make something new. That does not appear on the list. That number is completely different from anything else on the list, because it’s not the first number, because it’s different in the first place. It’s not the second number, because it’s different in second place. It’s not the third number, because it’s different in the third place. It’s not the fourth number because it’s different in the fourth place. It’s not the fifth number, because it’s different in the fifth place. You’ve made a number that’s not on that list. And so you can’t list all the decimals, in which case it is uncountable. It is unlistable. And that means it’s a whole new type of infinity. A bigger type of infinity.

BRADY HARAN: Surely we could, James, because all we’ve got to do is keep doing your game and making them and adding them to the list. And if we keep doing that, won’t we get there eventually?

JAMES GRIME: But you could then create another number that won’t be on that list. And so the guy who came up with is a German mathematician called Cantor. Cantor lived ’round about the turn of the 20th century. He was ridiculed for this. For this idea that there were different types of infinity, he was called a charlatan. And he was called– it was nonsense, it was called. And poor old Cantor was treated really badly by his contemporaries, and he spent a lot of his later life in and out of mental institutions, where he died, in the end. Near the end of his life, it was recognized. It was true. It was recognized. And he had all the recognition that he deserved.

BRADY HARAN: And now he’s on Numberphile.

JAMES GRIME: And now he’s on Numberphile, the greatest accolade of all. Georg Cantor.

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Math Antics – What Are Percentages?


Video Transcript

Hi! Welcome to Math Antics. Now that you know all about fractions, from watching all of our fractions videos, it’s time to learn about something called “percentages”. Percentages are super important. Have you ever been in a math class and heard another student ask the teacher: Um.. excuse me… teacher… Ah… when are we ever gonna use this stuff? Ya know… like in real life? Well when it come6s to percentages, the answer is one-hundred percent of the time. Well alright… maybe not a hundred percent of the time… but a lot!

Percentages are used every day to calculate things like: …how much sales tax you pay when you buy something. …how much something costs when it’s on sale. …how much fiber is in your granola bar. …or how much money you can make if you invest it in the stock market. That’s all real life stuff for sure. So, you can see that it’s really important to understand percentages and how we use them in math. Alright then… are you ready to learn the key to understanding percentages, or percents as they’re called for short? Drum roll please… A percent is a fraction!

Whaaaat? That’s right… a percent IS a fraction!

And since you already know all about fractions, learning about percents is gonna be easy. But a percent isn’t just any old fraction. A percent is a special fraction that always has 100 as the bottom number. If it’s a percent, then no matter what the top number is, the bottom number will be 100. In fact, because the bottom number of a percent is always 100, we don’t even write it. Instead, we use this handy little symbol (%) called a percent sign. Whenever you see this symbol after a number, it means the number is a percent. It’s really a fraction with 100 on the bottom, but it’s just being written in this more compact form. …like this number 15 here. It’s got the percent sign after it, so we read it as “15 percent”, and because a percent is really a fraction that always has 100 as the bottom number, we know that it means the same thing as 15 over 100. Percents make even more sense if you know what the word percent means. The prefix of the word (per) means “for each” or “for every”. Ya know like if someone said, “only one cookie per person”. And the root word (cent) is Latin for 100.

That’s why there’s 100 cents in a dollar. So, percent literally means “per 100” and that’s why they’re shortcuts for writing fractions that have 100 as the bottom number. Alright then, so whenever you see a percent like this, you know it can be replaced with (or converted) to a fraction. Let’s look at a few examples so you see the pattern. 3% means 3 over 100 10% means 10 over 100 25% means 25 over 100 and 75% means 75 over 100 These are percents… and these are the fractions that they stand for. There’s a few other interesting percents that we should take a look at. …like this one: 0% …can you have 0% ? Yes! 0% would just mean 0 over 100. It’s what we like to call a “zero fraction” cuz its value is just zero.

Remember, it’s okay to have zero on the top of a fraction, but not the bottom! Alright then, what about 100%. Well 100% just means 100 over 100. That’s what we like to call a “whole fraction”. The top number is the same as the bottom, so its value is just one whole, or 1. Okay then, 0% is just zero, and 100% is just 1. But what about numbers bigger than 100? Can you have 126% ? Yep, it works exactly the same way. 126% just means 126 over 100. And you know from the fractions videos, that’s what we call an “improper fraction”. The top number is bigger than the bottom number, so the fraction’s value will be greater than 1. Alright team, I want you to go out there and give me a-hundred and TEN percent effort in today’s game! But coach… it would be “improper” for us to give a-hundred and ten percent effort in today’s game.

Okay, so now you know the key to percentages. …that they’re just special fractions that always have 100 as the bottom number. But there’s one more thing that I need to tell you about in this video, and that’s decimals. Do you remember in the video about fractions and decimals that you can convert any fraction into its decimal value? Sometimes it was kind of tricky converting to a decimal if we had to divide the top number by the bottom number. But other times, like when we had “base-10” fractions, it was easy because decimal number places are made for counting base-10 fractions, (like tenths, hundredths and thousandths). Well guess what… Percents ARE base-10 fractions! They are hundredths because their bottom number is always 100. That means it’s really easy to re-write a percentage as a decimal number. You can do it the same way as we did in the base-10 fractions video.

For example, we know that 15% is just 15 over 100, right? That’s its fraction form. But it also has the decimal form 0.15 because THIS is the hundredths place and 0.15 means 15 hundredths. So, we can re-write 15% as a fraction (15 over 100) OR as a decimal (0.15) And now that you know WHY we can easily convert a percentage to a decimal, let me show you a really simple trick for doing it. First, you start with the number in percent form like this: 35% Next, you imagine where the decimal point should be in the number 35. It’s not shown, but if it was, it would be right here next to the ones place. (Now remember, 35 and 35.0 are the same value.) Now that you know where the decimal point is, just move it two number places to the left (away from the percent symbol) and draw it in right there. Last of all, once you have moved the decimal point, you erase the percent sign because you don’t have a percent anymore. Moving the decimal point two places to the left converted it into the decimal value of that percent. Let’s try converting a few more percents into their decimal values so you can get the hang of it. For 62 percent, we move the decimal point two places to the left and get 0.62 (Remember, we can put an extra zero in front of the decimal point to be a place holder and to make the decimal point easier to notice.) For 75 percent, we move the decimal point and get 0.75 For 99 percent, we move the decimal point to get 0.99

Pretty Cool, huh?

Okay, but what about 4% ? You might wonder how we can move the decimal point two places over when our number only has one digit. But all we need to do is use a zero as a place holder in the number place that’s missing. Then, when we move the decimal point two places over, we end up with the decimal value of 0.04. Now that makes sense because 4 is in the hundredths place and 4% is 4 over 100. And in the same way, 1% would just be 0.01. Again, we need that extra zero placeholder. Here’s a few more interesting examples: 0% would be just 0.00 And if we have 100% and we move the decimal point two places to the left, we end up with 1.00 But 1.00 is the same value as 1. That’s why 100% represent one whole. And if we have 142%, we move the decimal point to get 1.42 That’s a value greater than one which is what we’d expect because 142% is really an improper fraction (142 over 100) Its value should be greater than 1. Alright, so now you know that a percent is a special fraction that always has 100 as the bottom number. And you know that you can re-write percents in either their fraction form OR their decimal form. 25% is 25 over 100 or 0.25 But keep in mind that you could go the other way too. If someone gives you a fraction with 100 as the bottom number, you can re-write it in percent form. If you get 12 over 100, you can say that’s 12% And if you get 80 over 100, you can say that’s 80% OR… If you get the decimal 0.10, you can say that’s 10% and if you get the decimal 0.38, you can say that’s 38% So, that’s the key to percentages. They’re another way to write fractions and decimals. But there’s a lot more to learn about how they‘re used in math, and we’ll learn more about that in the next few videos. But for now, you should be sure that you really understand the basics of percentages by doing the exercises for this section. Thanks for watching Math Antics, and I’ll see ya next time!

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