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  • Integers are numbers that have no fractional component like the number 8.

  • Look at that beauty. Oh but what about a number like 0.8

  • Yikes! Problem here is we have a fractional piece. We have this .8

  • but don't worry! .8 may not be an integer but it is

  • rational which means it can be expressed as a ratio between two other integers. In

  • this case 0.8 is equal to 8 over 10. 8 divided by 10. 8 and 10 are clearly both

  • integers which means that since point 8 is equal to them it is at least rational

  • It can be expressed as a ratio between two integers. Now integers and rational

  • numbers are beautiful. Ancient mathematicians love them. There's a

  • problem. There were some things we could think of that didn't appear to be either

  • For instance think of a square with side length 1. Every side of this shape has a

  • length of 1. What is the length of this diagonal. I'll call this line

  • C. Well the Pythagorean theorem tells us that the length of C squared is equal to

  • the length of this side squared plus the length of this other side squared.

  • 1 squared plus 1 squared is just 2 so C squared equals 2 which means that C

  • equals the square root of 2. Perfect alright so the length of this diagonal is

  • square root of 2. What's the square root of 2? It could be an integer.

  • It could be rational. Well one thing you could do is you could measure the

  • diagonal of a perfectly drawn square of side 1 and measure it better and better

  • and better so that you've got more and more precision. As you did that you would

  • accumulate new digits for your answer to the square root of 2 and at each step of

  • the way you could find a ratio that equals that number but here's

  • the problem will you ever be done? Will you ever reach a point where you've

  • reached the last digit in the decimal expansion of the square root of two? If

  • you do then it's a rational number. The numerator and denominator might be

  • really big numbers but who cares at least it's not irrational or is it?

  • If it is how do you prove it? It is seems like the only way you could do it is by

  • calculating for some unknown possible infinite amount of time or making

  • completely infinitely precise measurements. Yikes.

  • But here's what is so fantastic about our universe. We have been able to prove

  • that the square root of 2 is not an integer and is not rational and today

  • we're gonna do just that but we need to cover four preliminaries so that this

  • proof is nice and complete. The first thing I want to do is define an even

  • number. What is an even number? We all are very familiar with even numbers 2 4 6 8

  • negative 2 negative 4 negative 68 these are all even numbers. What do they have

  • in common? Well they are divisible by two. That is a definition of evenness which

  • by the way means that 0 is even because 0 divided by 2 is just 0

  • there's no fractional piece left over so 0 is even but 1 and negative 1 on either

  • side are odd and that's how numbers go even odd even odd in that kind of a

  • pattern. Let's look at this definition: an even number like 8 is even because it

  • can be evenly divided by 2 8 divided by 2 equals 4 4 as a nice

  • whole number there's nothing left over that's perfect but what this also means

  • is that eight is equal to some number in this case four times two so here we have

  • a nice generalized definition of an even number. A number is even if it can be

  • expressed as two times some integer. I'll just call that integer C and because the

  • pattern of even odd even odd is what it is we can also define an odd number as

  • being equal to two times any integer plus one so negative 12 is even because

  • negative 12 can be expressed as two times negative six which is an integer

  • but negative 13 well negative 13 is odd because it can be expressed as two times

  • negative seven plus one. These are literally the definitions of even and

  • odd. The next thing we need to do is show that if you take an even number and

  • square it the result will also be even and if you take an odd number and square

  • it the result will always be odd. Now here's how we do that. Let's take an even

  • number which as we know is expressed by two times some integer and let's square

  • it. Now 2C squared equals two times C times two times C it's just two C times

  • itself. This is equal to four C squared. But what does 4c squared look like? We

  • can pull a 2 out of there and wind up with two times two C squared. Alright.

  • uh-oh look what we've got. This is 2 times some integer we know that two

  • times an integer is even so this is an even number. An even number squared is

  • even. Now let's square an odd number. An odd number looks like this. It's two

  • times any integer you like plus one. Now if we square this we wind up with my oh my

  • favorite thing, binomial multiplication. Let's take a look at this. We've got two

  • C plus one and it's squared so we're multiplying it by itself two

  • c plus 1 times 2c plus 1. Let's use foil to work this out. This means f we will take

  • we will find the product of the first two terms here. 2c times 2c is 4c

  • squared we actually already knew that. Then we're going to add to that the

  • product of the outer terms 2C times 1. Well that's just 2C. We add to that the

  • product of the inner terms 1 times 2 C which is just 2 C and then finally we

  • add the product of the last terms 1 times 1 which is 1. This simplifies into

  • 4 C squared 2 C plus 2 C is just 4 C and we've got this one on the end. Ohh alright.

  • Now how about this? Let's pull a 2 out of this thing

  • 2 times 2 C squared + 2 C and we've got this plus 1 at the end

  • Yowza! Look at this result. We have this thing right here in the parenthesis

  • which is some integer and we're multiplying it by 2 but then we're

  • adding 1. This is the form of an odd number so this is odd. An odd number

  • squared is odd and even numbers squared is even. How beautiful. Next I want to

  • talk about squaring rational numbers. Now this is something that we've all learned

  • before but I want to prove it. When we have some fractions like let's say a

  • over B and we want to multiply it by itself so we have A over B times a over

  • B. This is pretty easy to do. You literally just find the product of the

  • numerators, A squared, and divide them by the product of the denominators B

  • squared. Boom. Pretty simple but how can we be sure that is true because after

  • all fractions can be a little bit weird right I mean if I want to add A over B to

  • A over B I don't just add up the numerators and add up the denominators

  • instead I add up the numerators, 2A, and then I just keep the denominator the

  • same. This is very different than this.

  • what's going on? How can we be sure that we're doing this fraction multiplication

  • correctly? Well my favorite way to do this since we already kind of have an

  • idea that this is right unless our teachers have been lying to us our

  • entire lives is to just take advantage of the fact that multiplication and

  • division are inverse operations so let's take two fractions. I will call one A

  • over B and I will multiply it by another fraction C over D. What the heck is their

  • product going to look like? Well here is how we will make this easy. Let's go

  • ahead and multiply their product by B times D and divide by B times D because

  • multiplication and division are inverse operations this won't change anything. If

  • I multiply by some number and then divide by the same number I haven't

  • changed the thing I started with so all of this junk is equal to what we're

  • trying to study; the product of A over B times C over D. Now let's start

  • associating and commuting all of these little things. We can do that in

  • multiplication so I can take this B here for instance and multiply it by the

  • product of AB times CD or I can take this B and multiply it just by AB and

  • then bring in C over D. So let's do that because if I take A over B and I

  • multiply it by B and then I multiply C over D by that D. I still have to make

  • sure I don't forget that I'm also dividing by BD and would you look at

  • this. Multiplication and division are inverse so if you divide by B and then

  • multiply by B that's the same as just multiplying by one. So A stays the same.

  • Same over here. Dividing by D and multiplying by D gives us C so what

  • we're left with is A times C divided by B times D tada

  • A over B times A over B equals the product of the numerator and the product

  • of the denominator. Wonderful we are now ready to really

  • take a big bite out of rational numbers. Every single ratio of integers can be

  • reduced to lowest terms. In fact if you can imagine a ratio of integers that

  • cannot be reduced to lowest terms then it is not a ratio of integers and we do

  • this all the time when we're working with fractions. Take a look at a fraction

  • like 4/6. That's beautiful. That is a totally legitimate ratio of integers but

  • it's not in lowest terms because there are factors shared by 4 and 6.

  • By a factor I mean a number that evenly divides into them. What numbers evenly

  • divided into 4? Well 1 2 and 4. What numbers evenly divide into 6? Well 1 does so

  • does 2 so does 3 and so does 6. Yowzas. There is a common factor of 2. I can

  • divide both of them by 2. Now dividing by 2 over 2 is the same as dividing by 1 so

  • this ratio won't change, it'll just be in simpler terms. 4 divided by 2 is 2. 6

  • divided by 2 is 3 and boom 2/3. This is a very pretty looking fraction. It is equal

  • to 4/6 but the neat thing about it is that it is in a way complete because

  • it's a ratio between two integers that are co-prime. Co-prime means that two

  • numbers do not share any factors except for one. The factors of 2 are 1 and 2 and

  • the factors of 3 are 1 and 3. They share none in common but one so they are

  • co-prime. Every single ratio between two integers can be reduced to a ratio

  • between co-prime integers. There's another example

  • 14/15. This one doesn't feel as pretty but it's done these are lowest terms. The

  • factors of 14 are 1, 2, 7, and 14. The factors of

  • 15 are 1, 3, 5, and 15. The only factor they share in common is

  • once they are co-prime. 14/15 is in lowest terms. It is a reduced fraction. I

  • love it. The key here is that every single ratio

  • of integers can be reduced to a ratio between co-prime integers. If the square

  • root of 2 is indeed rational it should be 2. So here we begin our proof that the

  • square root of 2 is in fact irrational. We do this by contradiction. We just

  • start off by assuming that the square root of 2 is rational which means it

  • really does equal the ratio between two integers. We'll call them A and B. We

  • don't know what they are but we're just assuming they exist. If that's true then

  • A over B squared should equal 2. That's the definition of a square root. We've

  • also shown that a fraction squared means of course A over B times A over B and

  • when you multiply fractions you literally just find the product of the

  • numerators, A times A is A squared and the product of the denominators B times

  • B is B squared so A squared over B squared should equal 2 if the square

  • root of 2 is rational. Now we can rearrange this by multiplying both sides

  • by B squared. This gives us A squared equals 2B squared oh my goodness

  • gracious. Look at that. Look at that. Now B squared is some integer we don't know

  • what it is but it's being multiplied by 2 which means that this term is even.

  • It's divisible by 2. This is the definition of an even number. Where's my

  • even page look an even number as we said is 2 times some integer. This is 2 times

  • some integer so 2 B squared is clearly even but if it is even and it is equal

  • to A squared then A squared must also be even. Now we know that an even

  • number squared is even so if A squared is even then A must be even as well. Now

  • that's pretty interesting. It means that we can represent A as two times some

  • integer. Let's call it...let's call it C. I think that'll be clear enough and then

  • let's take this representation of A and plug it right back into this equation so

  • if A is 2 C then we have A squared so that means 2 C squared equals 2 B

  • squared. Now this 2 C squared is just 2 times C times 2 times C so 4 C squared

  • equals 2 B squared. Good. Oh we can divide both sides by 2 so

  • that 2 goes away and this 4 becomes a 2. Now oh my goodness gracious look what we

  • have. Now we have 2 times C squared which is some integer

  • well this is divisible by 2 so this is even but if this is even and it's equal

  • to B squared then B squared must also be even and since an even number squared

  • creates an even number B must be even and here we have our result. A must be

  • even and B must be even but if both of them are even they cannot be co-prime

  • because they both share 2 as a common factor. This could go on forever because

  • every ratio of integers must be reducible to the ratio between 2

  • co-prime integers and this one can't. The square root of 2 is not rational.

  • This result is beautiful because what we're able to do in this proof is learn

  • discover something about our universe using just mathematics and logic inside

  • our own minds without looking at the universe itself. Stay curious and as

  • always thanks for watching

Integers are numbers that have no fractional component like the number 8.

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證明二的平方根是無理的 (A Proof That The Square Root of Two Is Irrational)

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    林宜悉 發佈於 2021 年 01 月 14 日
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