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General discussion (archive)A math question(54 posts)(54 posts)
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This is my favourite topic
Posted October 03, 2012
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Nirth: Since this is a math thread I might as well jump in. I always enjoyed math in school but never really excelled at it because we hardly learned any tips, tricks or decent memory solution to easen the burden but nevertheless; there's a question that always eluded me.

What's the opposite of zero?

When you explain, you can keep in mind that I've read courses on basics when it comes to algebra, calculus and imaginary numbers but that's where it stops.

I'm sure this is the kind of question you can't really answer or requires more mathemathical knowledge than I've but humour me, I take a bad or weird answer before nothing. ;)
Oddly enough, my math teacher went over this with us today. 0 is neither positive nor negative, therefore it has no opposite.
Posted October 03, 2012
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Nirth: Since this is a math thread I might as well jump in. I always enjoyed math in school but never really excelled at it because we hardly learned any tips, tricks or decent memory solution to easen the burden but nevertheless; there's a question that always eluded me.

What's the opposite of zero?

When you explain, you can keep in mind that I've read courses on basics when it comes to algebra, calculus and imaginary numbers but that's where it stops.

I'm sure this is the kind of question you can't really answer or requires more mathemathical knowledge than I've but humour me, I take a bad or weird answer before nothing. ;)
I'd say (but that's not necessariky a math answer) that you'd have to define precisely what you mean with "opposite". It's a contextual notion. "Soft" is the opposite of "hard", or "rugged" or "harsh"... Zero can be the opposite of infinite but so can minus infinite if you use the word differently. Or the opposite of orez, or the opposite of one (in electronics, or in ontology). Or zero itself (as in : minus zero). Did you have a specific mathematical univocal sense of opposite in mind, such as "1/x" ?

(In french, we have a distinction between "inverse" et "opposé", for maths. I'm not sure of the terms in english.)
Post edited October 03, 2012 by Telika
Posted October 03, 2012
I'm not sure which univocal sense but I think 1/0 is the closest one, because it's not -0 or just 0 as it's defined in some situations. Speaking of which, x/0, is that defined? Is it the same as Tan 90°?

I gather that what I'm after is sort of a mix answer from mathematical and philosophical perspectives but I'm not sure.
Posted October 03, 2012
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ThomasPierson: What is the actual purpose of a square root? I understand what a square root is and how it is used, but my question is why does it exist at all when it is basically a question that has to be answered. Why not simply write the answer?
As a mathematician I can tell you that it's a tool for finding the length of the side of a given square area. So if we are given the information that a square area is 100m^2, we have something to represent the conversion of that information into our unknown length, which is 10m in this case. Just as the inverse (square) serves as a tool to convert a 10m length into the area of a square with a side of said length. This is the original primary purpose of these operations and since advanced mathematics developed, there are now several other applications e.g. in linear algebra for matrices and complex calculus.

This demonstrates why every mathematical operation has an opposite. They are all tools we made up to help calculate practical things in life.
Post edited October 03, 2012 by MrAlphaNumeric
Posted October 03, 2012
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Nirth: I'm not sure which univocal sense but I think 1/0 is the closest one, because it's not -0 or just 0 as it's defined in some situations. Speaking of which, x/0, is that defined? Is it the same as Tan 90°?

I gather that what I'm after is sort of a mix answer from mathematical and philosophical perspectives but I'm not sure.
I'm not sure this would be of much help, but anyway here is some reasoning. ^_^

From the strict algebraic point of view, for any x the expression x / 0 is an abbreviation of x * 0^-1, where 0^-1 is a value that would be "opposite"(or inverse) of 0 with respect to multiplication. So depending on the context in which you evaluate it (e.g. what kind of numbers you work with), x / 0 might or might not make sense.

For example, for some applications, there are good reasons to have real numbers that form a field, in which case having an inverse for 0 would contradict the field axioms. But in other cases such approach may be too restricting, so people introduce a value for 0^-1 called infinity, by sacrificing the field structure. The IEEE standard for CPU floating point representation allows for infinity, and it makes some things a lot simpler.
Posted October 03, 2012
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ThomasPierson: snip
In mathematics equations are used to represent relationships between abstract quantities. Algebra, on its most basic level, is the study of equations, the different operations one can use to compose an equation (addition, multiplication), the solvability of equations, the uniqueness of solutions and so on. Square roots first arise as solutions to a certain class of equations, the quadratic polynomials. A quadratic polynomial is an equation that can be written in the form

ax^2 + bx + c = 0

where x is a variable, and a, b and c are constants. In algebra you study equations of this and similar types to find out whether they can be solved. Solutions of the kind of equation I wrote above can be expressed in a general way if you introduce the concept of the square root. As it turns out, you can write down a general solution to the above equation which looks like this:

x = -b/2 +- sqrt(b^2 - 4ac) / (2a)

You would not be able to express this general relationship without the square root concept. Generally, you can think of the root as the inverse (opposite) operation to exponentiation. Raising a quantity to the n-th power and then taking the n-th root is equivalent to doing nothing, which is what defines an inverse operation.

The study of roots lead people to the discovery that certain equations do not possess solutions within the number systems that they were using at the time, which motivated them to introduce more general concepts of numbers than they had previously considered (rational numbers -> irrational numbers -> complex numbers -> ...).
Posted October 03, 2012
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xiongmao: I'm not sure this would be of much help, but anyway here is some reasoning. ^_^

From the strict algebraic point of view, for any x the expression x / 0 is an abbreviation of x * 0^-1, where 0^-1 is a value that would be "opposite"(or inverse) of 0 with respect to multiplication. So depending on the context in which you evaluate it (e.g. what kind of numbers you work with), x / 0 might or might not make sense.
That actually makes sense but in the end that makes it infinity. I've had this idea that the opposite of zero could be the end of infinity, e.g where infinity is going but never reaches. Sort of like half-life of radioactive subjects. They will never reach zero, it takes an unlimited amount of time for that to happen. However, in theory (or pre-theory?) when it does that situation would be what I would call opposite zero or end-infinity. I know I'm rambling nonsense but still, that's what I was going for.

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xiongmao: But in other cases such approach may be too restricting, so people introduce a value for 0^-1 called infinity, by sacrificing the field structure.
I don't understand that part about sacrificing the field structure. Could you elaborate?
Posted October 03, 2012
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Nirth: That actually makes sense but in the end that makes it infinity. I've had this idea that the opposite of zero could be the end of infinity, e.g where infinity is going but never reaches. Sort of like half-life of radioactive subjects. They will never reach zero, it takes an unlimited amount of time for that to happen. However, in theory (or pre-theory?) when it does that situation would be what I would call opposite zero or end-infinity. I know I'm rambling nonsense but still, that's what I was going for.
If I understand you correctly, then I think the infinity is exactly what you're looking for.
It's surely a weird concept to grasp, but in this case infinity represents just a single value rather than some arbitrarily large region of real line. I guess the name "infinity" somewhat contributes to the confusion, so it would be simpler if that value is called something abstract like "omega".

So starting from real numbers you just define the following:

omega = 0^-1
for any real x, x < omega

this way, you have a value that just sits there at the end of the real line, and can't be reached by any common operations on real numbers. For example if you start with 1 and keep multiplying it by 2, after arbitrarily large count of steps, the result would still be less than omega.

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xiongmao: But in other cases such approach may be too restricting, so people introduce a value for 0^-1 called infinity, by sacrificing the field structure.
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Nirth: I don't understand that part about sacrificing the field structure. Could you elaborate?
Well, it just arises from the application of field axioms. In particular, to form a field the real numbers need to have at least these properties:

(a) 1 != 0
(b) for any x, 0 * x = 0
(c) for any x for which x^-1 is defined, x * x^-1 = 1

Now, let's suppose that real numbers with omega added is still a field. Then the following should hold:

1 = (app. of (c)) = 0 * omega = (app. of (b)) = 0

But this clearly contradicts the property (a), so the real numbers with omega cannot be a field.

Hopefully it all makes some sense ^_^;
Posted October 04, 2012
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ThomasPierson: If a square root must be used to represent an approximation then the concept of accuracy ceases to be absolute and becomes relative.
Actually it's exactly the opposite :-) Square root of a given number is an exact value. Only if you try to calculate it and represent as a decimal number you get an approximation.

As for why it exists in the first place - if you like a more philosophical answer, it's a link between stuff in two dimensions (surface of a square) and stuff in one dimension (side of a square). If you want to go from three dimensions to one (cube volume to cube height) you use a cube root, and so on :-)
Posted October 04, 2012
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Nirth: I'm not sure which univocal sense but I think 1/0 is the closest one, because it's not -0 or just 0 as it's defined in some situations. Speaking of which, x/0, is that defined? Is it the same as Tan 90°?

I gather that what I'm after is sort of a mix answer from mathematical and philosophical perspectives but I'm not sure.
x/0 would work out to be infinity, which is basically the point at which mathematics turns into philosophy.

http://en.wikipedia.org/wiki/Infinity_%28mathematics%29#Mathematics

The value of tan(90) would be a slightly different scenario. The graph of y = tan(x) is not defined for x = 90. So, the value y = tan(90) is not infinity. In that respect there is a clear difference between tan(90) and 1/0. The first is not defined, the second is defined (as infinity).

Have fun with maths! :-)
Posted October 04, 2012
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ThomasPierson: If a square root is not a replacement for a specific number, and irrational numbers cannot be resolved, there is a flaw in the accuracy of every mathematical system that uses the square root. If a square root must be used to represent an approximation then the concept of accuracy ceases to be absolute and becomes relative.
The square root is not an approximation at all, this is an important point. The approximation part comes in when humans need a decimal format number and go "screw it, I'm fine with five decimals of accuracy here, I don't really need the full 167840 decimals that would represent the exact value of this square root".

Which is a completely necessary and practical thing to do when using math for actual real-world applications. For which you need to calculate the square roots of stuff rather a lot.

I'm not sure what you mean by "irrational numbers cannot be resolved". They can not be written as a/b, where a and b are integers (b != 0), no. But they're still there. Almost all of the real numbers are in fact irrational.
Posted October 04, 2012
Real numbers are pretty cool. You have stuff like pi and e, which are both really handy and have many uses that you couldn't do with rational approximations, unless maybe if you made some crazy new math rules, but those might not be as handy. The difference between rational numbers and real ones is that one axiom that went something like "a non-void upper(lower) bound subset of R (<-real numbers) has a supremum(infimum)". If we took, for example, the group {x in R | x>0 & x*x <2} (the positive real numbers the square of which is less than 2), this axiom suggests that there exists an actual number the square of which is the smallest, uhh (I study in Finnish, I'm improvising these English terms), number that's greater or equal to any member of the group. If we pick any rational number that's in the group, we can go a bit closer to 2 with another rational number, and if we pick a rational number the square of which is over 2, we can again go closer with another rational number, but there exists a number that is the supremum (a single one). There's a proof or 7 that there is no rational number the square of which is 2, but I can't remember any right now. Anyway, since there is a number like that, we probably should have a way to write it. Notice how cumbersome my text is with the "a number the square of which is 2" notification. Thus we say that the number with that supremum feature for that group is the square root of 2. It's as exact as it comes. It's as well defined as saying that "the number that has the feature 1x=x=x1 for all x that aren't 0 in the group" could be written as "1".
Posted October 04, 2012
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Adzeth: Real numbers are pretty cool. You have stuff like pi and e, which are both really handy and have many uses that you couldn't do with rational approximations, unless maybe if you made some crazy new math rules, but those might not be as handy. The difference between rational numbers and real ones is that one axiom that went something like "a non-void upper(lower) bound subset of R (<-real numbers) has a supremum(infimum)". If we took, for example, the group {x in R | x>0 & x*x <2} (the positive real numbers the square of which is less than 2), this axiom suggests that there exists an actual number the square of which is the smallest, uhh (I study in Finnish, I'm improvising these English terms), number that's greater or equal to any member of the group. If we pick any rational number that's in the group, we can go a bit closer to 2 with another rational number, and if we pick a rational number the square of which is over 2, we can again go closer with another rational number, but there exists a number that is the supremum (a single one). There's a proof or 7 that there is no rational number the square of which is 2, but I can't remember any right now. Anyway, since there is a number like that, we probably should have a way to write it. Notice how cumbersome my text is with the "a number the square of which is 2" notification. Thus we say that the number with that supremum feature for that group is the square root of 2. It's as exact as it comes. It's as well defined as saying that "the number that has the feature 1x=x=x1 for all x that aren't 0 in the group" could be written as "1".
You could also take a maybe more direct, but probably less comprehensible definition of the real numbers as the completion of the space of rational numbers: basically, you take the equivalence classes of Cauchy sequences in the rational numbers. While this may appear convoluted, it is quite natural and the decimal representation of some number is such a sequence: it may not be finite or periodic, but the partial sums of the first n digits form a Cauchy sequence.
One of the standard proofs of irrationality of sqrt(2) is by contradiction: you write such a quotient in coprime form, write down the equation defining sqrt(2) and get a contradiction.

Anyway, a square root is quite natural in its own right. It's like "halving" something, but the composition is multiplication in this case. And it will pop up wherever squares play a role. Like in quadratic equations, when throwing a ball or something and calculating the spot where it will land (hooray, a real-world application!).
Posted October 04, 2012
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ThomasPierson: If a square root is not a replacement for a specific number, and irrational numbers cannot be resolved, there is a flaw in the accuracy of every mathematical system that uses the square root. If a square root must be used to represent an approximation then the concept of accuracy ceases to be absolute and becomes relative.
I think stonebro's post covers this the best. The square root is a replacement for a specific number.

Thing is, at the most fundamental level we know the sqrt is x=y*y. It may sound trivial at this stage, but later on it will be used to explain certain functions. For example, in regressions with discrete variables, you may want to find the variance (or the numerical 'spread of the data'). To do that you sum up the squares of all the data values minus the average value.

The standard deviation is the square root of the variance, and again this may seem trivial. You may think why the hell do we need variance and std dev? Well in this case, the square root is there to convert the spread to the same units as the data, if you get my drift.

Thing is, I don't believe there's any single explanation for why the square root exists. It's even harder to explain when you write it as a power value of half. The square root itself is but one of many functions -- but I'm no math major, so if I'm wrong I will stand corrected.

And to go back to my first post, when you have a complicated function with a square root here and there, it's more than just sqrt y = x, where x*x = y. What if you have sqrt x + sqrt (x+4) = y? The sqrt provides a simplification of sorts in cases like these. If you have a graphing calculator, all the better, because then you can see where all the possible answers lie for every inputted parameter of y.
Post edited October 04, 2012 by lowyhong
Posted October 04, 2012
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Nirth: I'm not sure which univocal sense but I think 1/0 is the closest one, because it's not -0 or just 0 as it's defined in some situations. Speaking of which, x/0, is that defined? Is it the same as Tan 90°?

I gather that what I'm after is sort of a mix answer from mathematical and philosophical perspectives but I'm not sure.
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stonebro: x/0 would work out to be infinity, which is basically the point at which mathematics turns into philosophy.

http://en.wikipedia.org/wiki/Infinity_%28mathematics%29#Mathematics

The value of tan(90) would be a slightly different scenario. The graph of y = tan(x) is not defined for x = 90. So, the value y = tan(90) is not infinity. In that respect there is a clear difference between tan(90) and 1/0. The first is not defined, the second is defined (as infinity).

Have fun with maths! :-)
Interesting, I will read up on that, thanks. Question, why isn't tan 90 defined? When you close in on it as in tan (89.999) and so on the number becomes larger and larger effectively comes closer to infinity.
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General discussion (archive)A math question(54 posts)(54 posts)
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