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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.

Like many heroes of Greek myths,

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B2 TED-Ed root ratio hypotenuse irrational length

【TED-Ed】Making sense of irrational numbers - Ganesh Pai

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    黃于珍 posted on 2016/06/10
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