# Making sense of irrational numbers – Ganesh Pai

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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.

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1. 3:44 Golden Ratio! wow! 🙂

2. If I saw this video before my descrete maths exam I would have understood methods of proof

3. What about transcendental numbers like e?

4. ben nasıl geldim buraya

5. Wasn't it Euclid who proved it?

6. Thank you for the knowledge beyond textbooks.

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

8. 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

9. they didn't use algebra in those days

10. Thanks for the bits

11. 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!

12. Beautifully explained!!

13. my brain is hurted

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

15. matija sevaljevic

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

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

17. 0:41 what’s the Kahoot code?

18. 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.

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

20. Cucumber.

21. Salutes to Hippasus…. The great

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

23. √4 ÷ √2 = √2 , 😁😁🤣🤣

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

25. So no one is talking about Ganesh Pai??

26. Roses are red
Violets are blue
I wonder what
Is square root 2

27. 3:43 the Golden Ratio!

28. 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?

29. ARandomGuy aka FE2 and More -Roblox

I figured it out first

30. to me math is complicated and hard and i cant do it. but i find it really extremly interesting is so interesting how everything relates to…EVERYTHING!

31. ARandomGuy aka FE2 and More -Roblox

Roses are red,
Violets are blue,
An irrational number
Is the square root of 2.

32. 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.

33. Some say he was killed to preserve the Pythagorean reputation.

34. 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 "

35. can we get 1000 subscribers without content

"Two times any number = an even number" what about 0.3 for an example

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

37. Ganesh 3.14….hmm

38. why did the sqare root disappear after 2:12

39. 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.

40. so this number is irational 1.2837485758475846584775749856748576666666666666 in infinity
becouse it isnt a ratio of any prime number

41. 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..

..”

42. Haesar Prauditra

So irrational numbers is not too irrational huh?

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

44. You can assume that they have factors

45. Anuradha Mukherjee

Outstanding video

46. The LEGO Cuber

√2 = 1.414213562373295

Sorry, but try it out!

47. Mattia Cucciaglioni

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

48. gerry obadiah

You could say the Gods thought finding such numbers was irrational

49. Do not subscribe to Me

You 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

50. Perfect proof

51. She doesn't sound like a Ganesh Pai

52. Wouter Van der hoeven

I did not understand anything

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

54. 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.

55. DAMN YOU GOD

56. Do you have an irrational problem, Well here’s the answer to it

57. 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

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

59. swordandgunandsteel

Discrete math student unite!

60. “Not quite” 😅

61. BronzeJourney

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

62. Jericca Young

👏🏽👏🏽🙌🏽

63. Just studied this in class 10 chapter 1 😔

64. Great explanation.

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

Thank you.

65. 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.

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

67. This was the question of first chapter in maths in 10th class in India
It was proved by the smae method

68. But what if you simplify p/q?

69. I 😍 Mathematics

70. Hippasus: discovers irrational numbers

God: wait that’s illegal

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

72. 3:44 isn't that the Fibonacci Spiral?

73. His last name sounds like pi

74. 1:30 just use a ruler.

75. Olivier Kokkedee

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

76. I did not understand

77. Irrational numbers…i don't understand them

78. Amit Mendelevitch

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

79. 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.

80. This is class 9th maths

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

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

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

84. I like pretending i understand something from this

85. Super Nicholas The MLG BACON HAIR

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

86. Colobrinus With a side of cube

This doesn’t help at all.

87. 1.414215 squared? Lmao

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

89. Isnt Ganesh a male name?

90. Mind blown

91. How to represent π on number line.

92. But you have to prove that root 2 exists first

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

94. Pythagorean theorem! My HS Algebra. 😭😭😭