Making sense of irrational numbers – Ganesh Pai


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.


  1. 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!

  2. 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.

  3. 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?

  4. 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!

  5. 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.

  6. 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 "

  7. 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.

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

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


  10. 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

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

  12. 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.

  13. 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

  14. Great explanation.

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

    Thank you.

  15. 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.

    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.

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


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