Like many heroes of Greek myths, the philosopher Hippasus was rumored to

have been mortally punished by the gods. But what was his crime? Did he murder guests, or disrupt a sacred ritual? No, Hippasus’s transgression was

a mathematical proof: the discovery of irrational numbers. Hippasus belonged to a group

called the Pythagorean mathematicians who had a religious reverence for numbers. Their dictum of, “All is number,” suggested that numbers

were the building blocks of the Universe and part of this belief was that

everything from cosmology and metaphysics to music and morals followed eternal rules describable as ratios of numbers. Thus, any number could be written

as such a ratio. 5 as 5/1, 0.5 as 1/2 and so on. Even an infinitely extending decimal like

this could be expressed exactly as 34/45. All of these are what we now call

rational numbers. But Hippasus found one number

that violated this harmonious rule, one that was not supposed to exist. The problem began with a simple shape, a square with each side

measuring one unit. According to Pythagoras Theorem, the diagonal length

would be square root of two, but try as he might, Hippasus could not

express this as a ratio of two integers. And instead of giving up, he decided

to prove it couldn’t be done. Hippasus began by assuming that the

Pythagorean worldview was true, that root 2 could be expressed

as a ratio of two integers. He labeled these hypothetical integers

p and q. Assuming the ratio was reduced

to its simplest form, p and q could not have any common factors. To prove that root 2 was not rational, Hippasus just had to prove that

p/q cannot exist. So he multiplied both sides

of the equation by q and squared both sides. which gave him this equation. Multiplying any number by 2

results in an even number, so p^2 had to be even. That couldn’t be true if p was odd because an odd number times itself

is always odd, so p was even as well. Thus, p could be expressed as 2a,

where a is an integer. Substituting this into the equation

and simplifying gave q^2=2a^2 Once again, two times any number

produces an even number, so q^2 must have been even, and q must have been even as well, making both p and q even. But if that was true, then they had

a common factor of two, which contradicted the initial statement, and that’s how Hippasus concluded

that no such ratio exists. That’s called a proof by contradiction, and according to the legend, the gods did not appreciate

being contradicted. Interestingly, even though we can’t

express irrational numbers as ratios of integers, it is possible to precisely plot

some of them on the number line. Take root 2. All we need to do is form a right triangle

with two sides each measuring one unit. The hypotenuse has a length of root 2,

which can be extended along the line. We can then form another

right triangle with a base of that length

and a one unit height, and its hypotenuse would equal

root three, which can be extended

along the line, as well. The key here is that decimals and ratios

are only ways to express numbers. Root 2 simply is the hypotenuse

of a right triangle with sides of a length one. Similarly, the famous irrational number pi is always equal

to exactly what it represents, the ratio of a circle’s circumference

to its diameter. Approximations like 22/7, or 355/113 will never precisely equal pi. We’ll never know what really happened

to Hippasus, but what we do know is that his discovery

revolutionized mathematics. So whatever the myths may say,

don’t be afraid to explore the impossible.

People are smart.3:44

Golden Ratio!wow! 🙂If I saw this video before my descrete maths exam I would have understood methods of proof

What about transcendental numbers like e?

ben nasıl geldim buraya

All is number

Wasn't it Euclid who proved it?

Wow!!???? !!

Thank you for the knowledge beyond textbooks.

how irrational numbers are plotted on a line and graph or its not necessary to know.

but, is (q) is 1,5 is gonna be like 3, not even number, this bugged me, not every number multiplied by 2 is even

they didn't use algebra in those days

Sub to Pewdiepie

Thanks for the bits

I can disprove the fact that there are numbers that are irrational. If a number is rational, it means it can be written as a ration between two whole numbers. If that’s the case, then they can have common factor. I don’t understand why not. sqrt(2) = 1.414… so it can be written as (1.414… x 10^n)/10^n (where n is the number of digits of sqrt(2). The fraction would turn out to be 1414…/10… (1.414… is a whole number and 10… is a whole number too). QED!

Beautifully explained!!

my brain is hurted

Hippasus' colleagues punished him for discovering irrational numbers? Wow, that's truly irrational!

why you and many youtubers use word ,,dus,, instead of so or that

Now wait a minute. If pi is a ratio of circle's diameter and circumference, would it be then a RATIONAL number?

0:41 what’s the Kahoot code?

If squaring root(2) gives (2), doesn’t that mean that roots already exist? Then how is it his fault? Did they assume roots as rational number before?

If not then, how did he know squaring the root(n) gives the number(n)? Just curious.

Ganesh Pai always makes videos without any faults because sin (Pai) = 0

Cucumber.

Salutes to Hippasus…. The great

I bet you can’t find a fraction on 1.2345678910111213…

√4 ÷ √2 = √2 , 😁😁🤣🤣

Im German… And my last name is Pai. I have never ever seen anotherone have the name Pai. Thats awesome

My brain

So no one is talking about Ganesh Pai??

Roses are red

Violets are blue

I wonder what

Is square root 2

3:43 the Golden Ratio!

Who else watched this when they were hanging on the edge of their bed, trying to make their device to stand up, fell off once, and was in extreme pain the whole time?

I figured it out first

to me math is complicated and hard and i cant do it. but i find it really

extremlyinteresting is so interesting how everything relates to…EVERYTHING!Roses are red,

Violets are blue,

An irrational number

Is the square root of 2.

WAIT WAIT WAIT isn’t a square root squared a exponent-less number? So the square root of 2q squared would be 2q, not 2q squared. The square root and squared cancel out. Correct me if I’m wrong but that just seems off.

Some say he was killed to preserve the Pythagorean reputation.

i think schools smart classes should be more like this video and so should be teachers…

it seriously hurts my brain soooo hard when teachers goes like "IT IS HOW IT IS

just freaking get it already""Two times any number = an even number" what about 0.3 for an example

This whole lecture is in the initial chapters of NCERT class 9 and 10 😂😂… Like if u studied it!

Ganesh 3.14….hmm

why did the sqare root disappear after 2:12

3:42 oh, the fibonacci spiral. Well. Different math concepts are so related and connected to themselves. Like, here we accidentally connect geometry, the number line, and the fibonacci spiral.

so this number is irational 1.2837485758475846584775749856748576666666666666 in infinity

becouse it isnt a ratio of any prime number

Me trying to flirt based on someones how to flirt comment about roots and irrational numbers :” are you the square root of two?”

Her:”what?”

Me:”cus i feel irrational about you”

Her:”ok..

..”

So irrational numbers is not too irrational huh?

I know it's not the most strenuous proof ever but I do love the proof of root 2 being irrational by contradiction

You can assume that they have factors

Outstanding video

√2 = 1.414213562373295

Sorry, but try it out!

is it just me or she actually pronounced "belonged" as "belon-j-d"?

You could say the Gods thought finding such numbers was

irrationalYou can only theoretically represent an irrational number on a dot plot but not in real life. There is no such a thing as irrational or perfect shapes in the real world

Perfect proof

She doesn't sound like a Ganesh Pai

I did not understand anything

The Greek Gods are angry at mathematicians,so they said that all mathematicians will die at a point of time. Make a point

Nah,let's just use animation to make a diameter blanket the circle repeatedly until the circle is covered. Easier so that we don't near an Irrational number to multiply.

DAMN YOU GODDo you have an irrational problem, Well here’s the answer to it

I cant get it the time he squared both expression.Why the sqaure root lost and then the squared are still there? Plss someone help me

Hey! In India this been taught in Class 9 -students ageing 13-14

Discrete math student unite!

“Not quite” 😅

How simple and smart is that? Great video, as always.

👏🏽👏🏽🙌🏽

Just studied this in class 10 chapter 1 😔

Great explanation.

I have a degree in applied mathematics and have never heard this pontificated so well.

Thank you.

oh my goodness i just realized that "rational" means it can be expressed in ratios. I thought mathematicians just didn't think the numbers made sense.

I watch all these ted-ed videos but never really understand them

This was the question of first chapter in maths in 10th class in India

It was proved by the smae method

But what if you simplify p/q?

I 😍 Mathematics

Hippasus:

discovers irrational numbersGod: wait that’s illegal

Though we cannot EXPRESS IRRATIONAL INTO FRACTION but every irrational is a rational even log trig antilog etc

3:44 isn't that the Fibonacci Spiral?

His last name sounds like pi

1:30 just use a ruler.

5=5/1, sqrt(2)=sqrt(2)/1 so that's a very bad example

I did not understand

Irrational numbers…i don't understand them

If they like rational numbers so much they could've acted more… Rationally

FAKE NEWS

Michael from vsause made a video explaining the exact same thing, Hippasus just copied what he said and claimed it as his own. No wonder the gods hated him.

This is class 9th maths

for the diagonal just name it .5 because its .5 of a unit

Sadly, I don't think this was the original proof, since algebra had not been invented yet…ViHart has a better video on that.

We were taught this proof in school. So it has a history!

I like pretending i understand something from this

0:27 The man has blonde hair is doing the floss

This doesn’t help at all.

1.414215 squared? Lmao

Confused

I can't blame the gods for punishing him. I also want to punish him for the headache he's given me!

Isnt Ganesh a male name?

Mind blown

How to represent π on number line.

But you have to prove that root 2 exists first

The ratio of root 2 is root 2 divided by 1? …

Pythagorean theorem! My HS Algebra. 😭😭😭