What is the Constructibility of the Square Root of 2?

[MonkeyKid]

Sorry for any mispellings, English is not my first language.

So, I'm studying irrational numbers and I got curious about something and my teacher couldn't give me the answer. I understand Pi must exist because it's the simple result of a division (perimeter by diameter). But how can the mathematicians say there is the square root of 2? If there is no number that we can write down to a piece of paper and then multiply this number by itself and obtain 2 as a result, then there is no square root of 2, in my opinion.

Is the square root of 2 like something made up to make mathematics work? (like factorial of 0)?

Thank you.

 

[Mark44]

Sorry for any mispellings, English is not my first language.

So, I'm studying irrational numbers and I got curious about something and my teacher couldn't give me the answer. I understand Pi must exist because it's the simple result of a division (perimeter by diameter). But how can the mathematicians say there is the square root of 2? If there is no number that we can write down to a piece of paper and then multiply this number by itself and obtain 2 as a result, then there is no square root of 2, in my opinion.

You can write $\sqrt{2}$ on a piece of paper. Then, if you do this multiplication - $\sqrt{2} \sqrt{2}$ you get a result of 2.

Maybe you mean the decimal representation of $\sqrt{2}$, which has infinitely many digits. The fact that you can't write an infinite number of decimal digits is irrelevant to the existence of this number. Rational numbers also have an infinite number of digits in their decimal representations; for example, 2/3 = 0.6666... Would you say that 2/3 does not exist (since you can't write all of its decimal digits on a piece of paper)?

You can approximate $\sqrt{2}$ by truncating the decimal representation at some place. Each of these numbers has a finite number of decimal places, so conceivably could be written on a piece of paper.

Is the square root of 2 like something made up to make mathematics work? (like factorial of 0)?

No, it's one of many (an infinite number of) real numbers whose decimal representations aren't finite. Most of the numbers in any real interval are irrational

 

[pasmith]

Sorry for any mispellings, English is not my first language.

So, I'm studying irrational numbers and I got curious about something and my teacher couldn't give me the answer. I understand Pi must exist because it's the simple result of a division (perimeter by diameter). But how can the mathematicians say there is the square root of 2? If there is no number that we can write down to a piece of paper and then multiply this number by itself and obtain 2 as a result, then there is no square root of 2, in my opinion.

Is the square root of 2 like something made up to make mathematics work? (like factorial of 0)?

Thank you.

If the side of a square is $a$, then the length of its diagonal is $a\sqrt 2$.

 

[MonkeyKid]

You can write $\sqrt{2}$ on a piece of paper. Then, if you do this multiplication - $\sqrt{2} \sqrt{2}$ you get a result of 2.

Maybe you mean the decimal representation of $\sqrt{2}$, which has infinitely many digits. The fact that you can't write an infinite number of decimal digits is irrelevant to the existence of this number. Rational numbers also have an infinite number of digits in their decimal representations; for example, 2/3 = 0.6666... Would you say that 2/3 does not exist (since you can't write all of its decimal digits on a piece of paper)?

You can approximate $\sqrt{2}$ by truncating the decimal representation at some place. Each of these numbers has a finite number of decimal places, so conceivably could be written on a piece of paper.



No, it's one of many (an infinite number of) real numbers whose decimal representations aren't finite. Most of the numbers in any real interval are irrational

But those other numbers we can write as rational numbers as you did it yourself so there is a simple division that results in 0.666666... But I think there is no actual number that, when multiplied by itself, yields 2.

If the side of a square is $a$, then the length of its diagonal is $a\sqrt 2$.

Well, yes. But unless I didn't understand your proof, that prooves only that we need $\sqrt{2}$ but that doesn't prove it is a number that actually exists. Well, maybe it does... my head is a bit confused right now.

 

[KCC2]

Real numbers are constructed in such a way that $\sqrt 2$ exists.

http://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

One way to state the completeness is that every nonempty set that has an upper bound has a least upper bound which is called supremum. Now consider a set of all rational numbers x that satisfy the rule: $x^2 < 2$. This set is nonempty and it has an upper bound, so it has a supremum and it's easy to proof that the supremum is $\sqrt 2$

So the existence of irrational squareroots and all the other irrational numbers come from the axioms of real numbers. Maybe it's "made up", but real numbers make math fun.

 

[SteamKing]

You can't write pi down on a piece of paper, either.

You've got to overcome this notion that if you can't see something or make something, ipso facto, it doesn't exist. Otherwise, you are not going to be successful studying mathematics or science.

(ipso facto is Latin for "by this fact itself")

 

[MonkeyKid]

Real numbers are constructed in such a way that $\sqrt 2$ exists.

http://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

One way to state the completeness is that every nonempty set that has an upper bound has a least upper bound which is called supremum. Now consider a set of all rational numbers x that satisfy the rule: $x^2 < 2$. This set is nonempty and it has an upper bound, so it has a supremum and it's easy to proof that the supremum is $\sqrt 2$

So the existence of irrational squareroots and many other irrational numbers come from the axioms of real numbers. Maybe it's "made up", but real numbers make math fun.

Reading the article made me realize I've stumbled on something far beyond that which my current scholar knowledge can grasp.

You can't write pi down on a piece of paper, either.

You've got to overcome this notion that if you can't see something or make something, ipso facto, it doesn't exist. Otherwise, you are not going to be successful studying mathematics or science.

(ipso facto is Latin for "by this fact itself")

I'll do that. From now on, I'll take a little dose of faith on the mathematicians when I study math. If they say something is true it's because it probably is, and it's better to believe than to ask them to prove why, because the proof may give you a headache lol. I can't understand the demonstrations and concepts of the link above and I shouldn't be so skeptical about something the whole world agrees upon. Thank you all for your patience.

 

[DrClaude]

I understand Pi must exist because it's the simple result of a division (perimeter by diameter).

Well, yes. But unless I didn't understand your proof, that prooves only that we need $\sqrt{2}$ but that doesn't prove it is a number that actually exists. Well, maybe it does... my head is a bit confused right now.

Even by your own criterion it exists: $\sqrt{2}$ is the simple result of the division of the diagonal of a square by its side.

 

[MonkeyKid]

Even by your own criterion it exists: $\sqrt{2}$ is the simple result of the division of the diagonal of a square by its side.

You are right! Finally I can put my mind to rest. But what a weird number it is the square root of 2. The fact that I can now envision it geometrically makes it a lot more real and conceivable to me. Thank you!

I'm looking foward to demonstrate this to my teacher who couldn't come up with an answer! hehehe! I hope she doesn't get mad at me though

 

[ClamShell]

The Pythagoreans had the same logical problem with sqrt(2).
They threw Hippasus into the Mediterranean(so rumor has it),
for even suggesting that sqrt(2) is irrational.

 

[SteamKing]

I'll do that. From now on, I'll take a little dose of faith on the mathematicians when I study math. If they say something is true it's because it probably is, and it's better to believe than to ask them to prove why, because the proof may give you a headache lol. I can't understand the demonstrations and concepts of the link above and I shouldn't be so skeptical about something the whole world agrees upon. Thank you all for your patience.

The difference between science and faith is, science gives you the tools to prove that something you can't see or can't write down on a piece of paper exists and has meaning. Sqrt(2) doesn't exist because of consensus; it can be proven to exist.

 

[MonkeyKid]

The difference between science and faith is, science gives you the tools to prove that something you can't see or can't write down on a piece of paper exists and has meaning. Sqrt(2) doesn't exist because of consensus; it can be proven to exist.

I know, but the proof that can be found here
http://en.wikipedia.org/wiki/Completeness_of_the_real_numbers is beyond my comprehension, so to me if that was the only proof, then it was a matter of faith in the mathematicians because they know what they are doing and they know how to prove it, even though i can't follow the proof.

But saying that it exists because it is the result of the division of the diagional of a square by it's side is something I can understand. because the result of that division is a segment that has a 2 dimensional length, and that length is the square root of 2, therefore, the square root of 2 exists.

And if you measure the square root of 2 segment with a rule you'll get an aproximation of it. The more precise the rule, the more precise your aproximation. But there's no rule in the universe that can measure the segment without leaving a little bit of it unmeasured. What a fascinating thing this is to me.

Edit: Unless the measurement unit of your rule is based on the square root of 2 (like square root of 2 divided by 100, square root of 2 divided by 99 and so on). But that would be cheating...

 

[ClamShell]

I know, but the proof that can be found here
http://en.wikipedia.org/wiki/Complet...e_real_numbers is beyond my comprehension, so to me if that was the only proof, then it was a matter of faith in the mathematicians because they know what they are doing and they know how to prove it, even though i can't follow the proof.

But saying that it exists because it is the result of the division of the diagional of a square by it's side is something I can understand. because the result of that division is a segment that has a 2 dimensional length, and that length is the square root of 2, therefore, the square root of 2 exists.

And if you measure the square root of 2 segment with a rule you'll get an aproximation of it. The more precise the rule, the more precise your aproximation. But there's no rule in the universe that can measure the segment without leaving a little bit of it unmeasured. What a fascinating thing this is to me.

Edit: Unless the measurement unit of your rule is based on the square root of 2 (like square root of 2 divided by 100, square root of 2 divided by 99 and so on). But that would be cheating...

Links don't work on my metro IE11, is this it?

en.wikipedia.org/wiki/Completeness_of_the_real_numbers

 

[MonkeyKid]

Links don't work on my metro IE11, is this it?

en.wikipedia.org/wiki/Completeness_of_the_real_numbers

Yes, sorry I don't know what went wrong with the link... I'll try to paste the correct one
Edit: It's fine now. I think I must have copy-pasted it from some post above.

 

[gopher_p]

I think the constructibility (https://en.wikipedia.org/wiki/Constructible_number) of $\sqrt{2}$ is the most compelling and understandable argument for why it should exist, especially given the OP's apparent beliefs regarding the existence of circles and lines.

 

[MonkeyKid]

I think the constructibility (https://en.wikipedia.org/wiki/Constructible_number) of $\sqrt{2}$ is the most compelling and understandable argument for why it should exist, especially given the OP's apparent beliefs regarding the existence of circles and lines.

this was really fun to follow although I wish my compass could change colors like the one from the image because the final result was a mess. The concept of conctructible numbers is much easier to understand and very interesting.

But after following the demonstration, I think I can simplify it in my mind, because I believe in the existence of squares. So after the demonstration I ended up with a square in which the size has a length of 1. Then the diagional of this square (the line segment BD in the figure) has a length of the square root of 2.

What I don't understand is why the constructible numbers deal only in points lines and circles, because to know that BD has a length equal to the square root of 2 you must at least assume the existence of the right triangle and it's properties, I think... Or maybe I failed to see something.

 

[Mark44]

But saying that it exists because it is the result of the division of the diagional of a square by it's side is something I can understand. because the result of that division is a segment that has a 2 dimensional length

There's no such thing as a two dimensional length. Length is one dimensional measure. On the other hand, area is a two dimensional measure, and volume is a three dimensional measure.

, and that length is the square root of 2, therefore, the square root of 2 exists.

And if you measure the square root of 2 segment with a rule you'll get an aproximation of it. The more precise the rule, the more precise your aproximation. But there's no rule in the universe that can measure the segment without leaving a little bit of it unmeasured. What a fascinating thing this is to me.

Edit: Unless the measurement unit of your rule is based on the square root of 2 (like square root of 2 divided by 100, square root of 2 divided by 99 and so on). But that would be cheating...

Even if you had a ruler marked in units of $\sqrt{2}$, you wouldn't be able to tell if what you were measuring was actually $\sqrt{2}$ in length. Measuring things is inherently inaccurate.

 

[gopher_p]

What I don't understand is why the constructible numbers deal only in points lines and circles, because to know that BD has a length equal to the square root of 2 you must at least assume the existence of the right triangle and it's properties, I think... Or maybe I failed to see something.

Perpendiculars are constructible with a compass and straight edge, so you don't need[\i] to assume the existence of right triangles in this setting. Note that you can't construct all right triangles since not all lengths or angles are constructible.

There are other kinds of constructions that are more "lenient" than compass and straightedge in terms of what is assumed/allowed (https://en.wikipedia.org/wiki/Compass_and_straightedge_constructions#Extended_constructions).

 

[MonkeyKid]

There's no such thing as a two dimensional length. Length is one dimensional measure. On the other hand, area is a two dimensional measure, and volume is a three dimensional measure.
Even if you had a ruler marked in units of $\sqrt{2}$, you wouldn't be able to tell if what you were measuring was actually $\sqrt{2}$ in length. Measuring things is inherently inaccurate.

you are right about the dimensions of length. Sorry, my mistake. About measuring, I was talking more of a hypothetical measuring with arbitrary precision. I thought that was implied. And what I meant was that even if you could measure things with perfect precision, unless you used a unit based on $\sqrt{2}$ you could never measure the whole segment

 

[PeroK]

You can look at this a different way. We know that 1.4 < √2 and 1.5 > √2. And we can keep going with every greater mathematical precision:

$$1.41421356^2 < 2 \ and \ 1.41421357^2 > 2$$

And so on. And, we can make these two numbers either side of √2 as close as we want. So, the question is simple: is there an "irrational" number in there that is equal to √2 or not?

Suppose not, then the number line has gaps in it. And, it would get quite difficult to do maths if there are gaps where you'd like numbers to be. For example, if you draw the graph of

$$y = x^2-2$$
This clearly crosses the x-axis twice (at -√2 and +√2). But, if there are gaps where these numbers should be, then the graph crosses the x-axis between numbers, so there is no solution to the equation. The graph sort of ghosts thru the x-axis missing all the numbers.

So, an axiom of mathematics is that there are no gaps. This is called the "completeness" axiom.

It might be quite interesting to see how far you could go without the completeness axiom. But, intuitively, I don't like the idea of a curve crossing the x-axis where there is no number! Surely, wherever the curve crosses the x-axis, there must be a number there?

 

[MonkeyKid]

You can look at this a different way. We know that 1.4 < √2 and 1.5 > √2. And we can keep going with every greater mathematical precision:

$$1.41421356^2 < 2 \ and \ 1.41421357^2 > 2$$

And so on. And, we can make these two numbers either side of √2 as close as we want. So, the question is simple: is there an "irrational" number in there that is equal to √2 or not?

Suppose not, then the number line has gaps in it. And, it would get quite difficult to do maths if there are gaps where you'd like numbers to be. For example, if you draw the graph of

$$y = x^2-2$$
This clearly crosses the x-axis twice (at -√2 and +√2). But, if there are gaps where these numbers should be, then the graph crosses the x-axis between numbers, so there is no solution to the equation. The graph sort of ghosts thru the x-axis missing all the numbers.

So, an axiom of mathematics is that there are no gaps. This is called the "completeness" axiom.

It might be quite interesting to see how far you could go without the completeness axiom. But, intuitively, I don't like the idea of a curve crossing the x-axis where there is no number! Surely, wherever the curve crosses the x-axis, there must be a number there?

Because of these numbers being necessary, I thought they were made up, so math could work, even though there would be no exact point whose distance from zero in the real number's line would be the $\sqrt{2}$, since I had no proof of it's existence. So, when I first asked the post, I thought you could get arbitrarily close to $\sqrt{2}$ but there was no actual $\sqrt{2}$.

After the geometrical demonstrations I was shown, I was convinced $\sqrt{2}$ indeed exists, because it is possible to draw a line segment with a length of $\sqrt{2}$. Then if you take that line segment and put it's beginning at 0 in the real numbers' line, it's end would be the exact point of the line that has a distance equal to $\sqrt{2}$ from 0, in other words, that point is where $\sqrt{2}$ lies in the real number's line.

Geometrically, it's easy to picture. Using just numbers, not so much, at least for me. I suppose my mind is limited to things more concrete like lines and points and circles, and is bad at conceiving purely abstract stuff.

 

[MonkeyKid]

Perpendiculars are constructible with a compass and straight edge, so you don't need[\i] to assume the existence of right triangles in this setting. Note that you can't construct all right triangles since not all lengths or angles are constructible.

There are other kinds of constructions that are more "lenient" than compass and straightedge in terms of what is assumed/allowed (https://en.wikipedia.org/wiki/Compass_and_straightedge_constructions#Extended_constructions).

I know this kind of strays from the original subject, but how can I, without the notion of the right triangle and it's properties, and armed only with the notion of perpendicular angles, know that the line that connects two line segments that are perpendicular to each other has the square of it's length equal to the sum of the square of the lengths of the two line segments? (I don't promisse to understand the explanation, but I will read it diligently, and try my best, hopefully learning new things along the way)

 

[micromass]

I know this kind of strays from the original subject, but how can I, without the notion of the right triangle and it's properties, and armed only with the notion of perpendicular angles, know that the line that connects two line segments that are perpendicular to each other has the square of it's length equal to the sum of the square of the lengths of the two line segments? (I don't promisse to understand the explanation, but I will read it diligently, and try my best, hopefully learning new things along the way)

Are you looking for a proof of the Pythagorean theorem?

 

[MonkeyKid]

Are you looking for a proof of the Pythagorean theorem?

Yes, one that doesn't assume the existence of triangles or any other geometrical entities other than lines, points and circles. The only one I was taught which I can't quite recall uses the area of four triangles and the area of the square to prove it.

 

[micromass]

Yes, one that doesn't assume the existence of triangles or any other geometrical entities other than lines, points and circles. The only one I was taught which I can't quite recall uses the area of four triangles and the area of the square to prove it.

Are you ok with an axiom system mentioning only lines and points? The axiom system will then develop notions of triangles and squares to help prove the Pythagorean theorem. Or do you want a text that only talks about lines, points and circles and nothing else (in which case I doubt you'll find anything useful).

Anyway, a very good book is https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20
In the second chapter he starts with constructing an axiom system using the undefined notions of "line", "point", "between" and "congruent". It might just be what you're looking for. The book is however a bit to the difficult side.

 

[MonkeyKid]

Are you ok with an axiom system mentioning only lines and points? The axiom system will then develop notions of triangles and squares to help prove the Pythagorean theorem. Or do you want a text that only talks about lines, points and circles and nothing else (in which case I doubt you'll find anything useful).

Anyway, a very good book is https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20
In the second chapter he starts with constructing an axiom system using the undefined notions of "line", "point", "between" and "congruent". It might just be what you're looking for. The book is however a bit to the difficult side.

I think the constructibility (https://en.wikipedia.org/wiki/Constructible_number) of $\sqrt{2}$ is the most compelling and understandable argument for why it should exist, especially given the OP's apparent beliefs regarding the existence of circles and lines.

The reason why I mentioned the possible need for a proof that does not involves triangles is because this article above uses something called constructible numbers to prove that $\sqrt{2}$ is a number. Using compass and a straightedge, it draws a square with side 1 and then concludes that the diagonal of the square is equal to $\sqrt{2}$. Well, unless this conclusion comes from something other than the knowledge of the pythagoras' theorem, I don't see the need for all the compass and straightedge operations. All you have to do is declare something like this: "say we have a right triangle where both catheti have a length of 1; according to the pythagoras' theorem the hypotenuse will have a length of $\sqrt{2}$". So, if the proof uses the pythagoras' theorem I can't see the point of all that hard work in using compass and straightedge, unless there's a way to prove the pythagora's theorem without taking right triangles into consideration.