I wouldn't say it is a pity, but - it is s bit mysterious. It really freaked out the ancient Greeks when they proved that root two is irrational.
That's a simple proof, far easier than to see that pi is irrational, do you know it?
BTW irrational is different from having an endless decimal.
For instance
1/3 = 0.3333333... goes on endlessly in base 10
(in base 3 then 1/3 is 0.1).
Any decimal expansion that repeats itself is a rational as well as any that stops.
Anyway here goes, it's quite a beautiful proof in its way, many mathematicians when asked give this as an example of a beautiful elegant proof.
It is a bit tricky though to follow, if not a mathematician. Goes back to around the time of Pythagoras.
To show that root two is irrational, need to show that there is no rational solution to
(m/n)² = 2.
And the way you show that there is no solution in maths is to suppose that it did have a solution and derive a contradiction.
After all - how else could you prove mathematically that it doesn't have a solution, except to assume it has one and show that that is impossible?
Then the next thing is to notice that if you have a rational as a solution, you can always simplify it until it has no common factors at top and bottom. For instance if m and n are both divisible by 2, then you can just divide them both by 2 to get a simpler fraction.
So - if it has a solution, it must have a solution in its simplest form as well, one with no common factors top and bottom. We will derive a contradiction by showing that such a solution to the equation is impossible.
So then, if it does have a solution, then we must have some m and n such that
m²/n² = 2
with no common factors for m and n.
Then multiplying both sides by n²,
m² = 2 n²
So m² is even. So m must be even (because the square of an odd number is odd).
So call it 2k
Then you have (2k)² = 2 n², so
4k² = 2 n²
so. 2k² = n²
This means that n² is even also, and so n is.
So we've found a common divisor. That's a contradiction.
Or to put it another way, if
(m/n)² = 2.
had a solution
then by our argument just given, any such solution has top and bottom both divisible by 2.
So, divide both by 2 and result is still a solution (obviously) and applying our argument again, both must be divisible by 2, so we can divide again, and then by same argument again, both must be divisible by 2 once again, and that has to go on endlessly because there is no way for it to end.
That's obviously impossible if m and n are both finite.
And - that actually gives a bit of insight into the proof - it's possible to get a "kind of" rational solution if we permit numbers that look finite from one point of view and infinite from another point of view - in "non standard arithmetic".
You can get a sort of geometry though with PI rational.
Easiest one is taxicab geometry. Suppose you live in a city with streets that go EW and NS in square blocks and you measure the distance along taxicab routes. Then that's taxicab geometry.
In that geometry it turns out that the taxicab equivalent of PI = 4.
This is what a circle looks like in taxicab geometry
Taxicab Geometry - Circles
Other geometries have other values for circumference of a circle.
PI in maths always refers to the Euclidean geometry (flat plane, normal way of measuring distances). So - not really differing values for PI, but other constants that play the same role as PI.
I still think, myself, that it is somehow a bit mysterious that such a simple thing as the circumference of the circle, so easy to describe, turns out to be so complicated.
It's not just irrational. It's not the solution to any equation involving powers at all, no matter how complex. It's not an "algebraic number". Or, as they say, it's "transcendental" but that just means "not algebraic".
Mathematically it all hangs together and makes sense once you know. But philosophically - why does it have to be so complicated?
Why isn't ordinary geometry simple like taxicab geometry with PI = 4?
For that matter, why do we need irrationals, why can't all distances and measurements be rational as the Ancient Greeks thought?
(that includes any finite decimals like 0.13143532 as those also are rationals, that one is 13143532/100000000)
Though those questions just can't be phrased in maths as far as I know.
I think it sort of makes sense to ask that question, somehow on a philosophical / ordinary language way of thinking level, but not sure there is any answer to it.
That's a simple proof, far easier than to see that pi is irrational, do you know it?
BTW irrational is different from having an endless decimal.
For instance
1/3 = 0.3333333... goes on endlessly in base 10
(in base 3 then 1/3 is 0.1).
Any decimal expansion that repeats itself is a rational as well as any that stops.
Anyway here goes, it's quite a beautiful proof in its way, many mathematicians when asked give this as an example of a beautiful elegant proof.
PROOF THAT ROOT TWO IS IRRATINAL
It is a bit tricky though to follow, if not a mathematician. Goes back to around the time of Pythagoras.
To show that root two is irrational, need to show that there is no rational solution to
(m/n)² = 2.
And the way you show that there is no solution in maths is to suppose that it did have a solution and derive a contradiction.
After all - how else could you prove mathematically that it doesn't have a solution, except to assume it has one and show that that is impossible?
Then the next thing is to notice that if you have a rational as a solution, you can always simplify it until it has no common factors at top and bottom. For instance if m and n are both divisible by 2, then you can just divide them both by 2 to get a simpler fraction.
So - if it has a solution, it must have a solution in its simplest form as well, one with no common factors top and bottom. We will derive a contradiction by showing that such a solution to the equation is impossible.
So then, if it does have a solution, then we must have some m and n such that
m²/n² = 2
with no common factors for m and n.
Then multiplying both sides by n²,
m² = 2 n²
So m² is even. So m must be even (because the square of an odd number is odd).
So call it 2k
Then you have (2k)² = 2 n², so
4k² = 2 n²
so. 2k² = n²
This means that n² is even also, and so n is.
So we've found a common divisor. That's a contradiction.
ANOTHER WAY OF LOOKING AT THE SAME PROOF - ENDLESS DESCENT
Or to put it another way, if
(m/n)² = 2.
had a solution
then by our argument just given, any such solution has top and bottom both divisible by 2.
So, divide both by 2 and result is still a solution (obviously) and applying our argument again, both must be divisible by 2, so we can divide again, and then by same argument again, both must be divisible by 2 once again, and that has to go on endlessly because there is no way for it to end.
That's obviously impossible if m and n are both finite.
And - that actually gives a bit of insight into the proof - it's possible to get a "kind of" rational solution if we permit numbers that look finite from one point of view and infinite from another point of view - in "non standard arithmetic".
GEOMETRY WITH THE EQUIVALENT OF PI WITH VALUE 4
You can get a sort of geometry though with PI rational.
Easiest one is taxicab geometry. Suppose you live in a city with streets that go EW and NS in square blocks and you measure the distance along taxicab routes. Then that's taxicab geometry.
In that geometry it turns out that the taxicab equivalent of PI = 4.
This is what a circle looks like in taxicab geometry
Taxicab Geometry - Circles
Other geometries have other values for circumference of a circle.
PI in maths always refers to the Euclidean geometry (flat plane, normal way of measuring distances). So - not really differing values for PI, but other constants that play the same role as PI.
MYSTERY THAT SOMETHING SO EASY TO DESCRIBE IS SO COMPLICATED
I still think, myself, that it is somehow a bit mysterious that such a simple thing as the circumference of the circle, so easy to describe, turns out to be so complicated.
It's not just irrational. It's not the solution to any equation involving powers at all, no matter how complex. It's not an "algebraic number". Or, as they say, it's "transcendental" but that just means "not algebraic".
Mathematically it all hangs together and makes sense once you know. But philosophically - why does it have to be so complicated?
Why isn't ordinary geometry simple like taxicab geometry with PI = 4?
For that matter, why do we need irrationals, why can't all distances and measurements be rational as the Ancient Greeks thought?
(that includes any finite decimals like 0.13143532 as those also are rationals, that one is 13143532/100000000)
Though those questions just can't be phrased in maths as far as I know.
I think it sort of makes sense to ask that question, somehow on a philosophical / ordinary language way of thinking level, but not sure there is any answer to it.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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