Squares and roots of fractions.

[Andropov]

Can someone intuitively explain the logic behind a one half root or x to the power of one half or a third or whatever? What I mean is X^(1/2) and the like.

 

[Cogwulf]

x^(1/2) is simply another way of writing a square root. x^(1/3) is a cube root, etc.

And in the same way, a half root is the same as x^2

So when you have a fraction as a root or power, you simply use the opposite of the function using the bottom number.

 

[Andropov]

You haven't done what I asked. I want an intuitive explanation of the concept. How is it possible to exponentiate something to a fraction? Say it's x to the power of 2 and a half; Do you square it and then add half of x to the power of one? No, that wouldn't make sense.

 

[Andropov]

That sounded sort of rude. Sorry I guess?

 

[Bird]

I'm having a hard time understanding
what you mean by "intuitively".

 

[Cogwulf]

There isn't a mathmatical explanation. I used to think there must be, but there just isn't.

Using fractions as powers is simply mathematical shorthand for a root. 4^(1/2) just means the square root of four.


edit: ignore what I wrote, it's all bollocks.

 

[EyeSeeCold]

You haven't done what I asked. I want an intuitive explanation of the concept. How is it possible to exponentiate something to a fraction? Say it's x to the power of 2 and a half; Do you square it and then add half of x to the power of one? No, that wouldn't make sense.

 

[WittyUsername]

I dont know how to explain you 'intuitively' but its because [(x)^1/2]^2=x, hence x^1/2 is the square root of x.

Basically just another way of representing roots.

 

[Artsu Tharaz]

It's just a definitions thing, designed to be consistent with the exponentiation rules of integral powers (like how we define negative exponents).

so, y=x^{Text} satisfies y^{Text} = x, and so y is the positive (by definition) square root of x.

Keep in mind that in mathematics, things are just based on definitions, which are designed to work nicely and satisfy intuitive requirements. In this case it's "a number raised to a fractional power, when raised to the integer corresponding to the reciprocal of this power, will give us back that number, by following our usual exponentiation rules. Hence, we can define such a number in this way". General rational powers are defined similarly, and irrational powers are defined in terms of limiting sequences of rational powers, and then it makes sense to talk about an exponential function of all the real numbers (well, this isn't entirely true). Thennn we can also define complex exponents and so on.

Saying it for a third time: it's a definition which is designed to be consistent with our more simple intuitive concepts.

extension problem: what is a reasonable way to define a non-integer factorial?

 

[WittyUsername]

^I think there's Gamma function for that. Its not taught in high school so I dont know much.

 

[Latro]

You haven't done what I asked. I want an intuitive explanation of the concept. How is it possible to exponentiate something to a fraction? Say it's x to the power of 2 and a half; Do you square it and then add half of x to the power of one? No, that wouldn't make sense.

Consider the exponent rules:
(a^b)*(a^c) = a^(b+c)
(a^b)^c = a^(b*c)

So for example:
sqrt(a)^2 = a
makes sense in fractional exponents, because:
(a^(1/2))^2 = a^((1/2)*2) = a^1 = a

As for noninteger factorial, there is a vast infinity of ways to interpolate the factorial function, but the "best" one, property-wise, is the Gamma function. Unfortunately there is the frustrating convention that Gamma(n+1) = n!

The exponentiation rules derive from the meaning of exponentation with integer exponents; then we allow them to be fractional using those same rules. They can also be irrational, again using those same rules.

 

[Artsu Tharaz]

Its not taught in high school so I dont know much.

"I haven't been taught this so I don't know it"

It's maths. You don't need to be taught it, just try and figure it out. It doesn't matter if you get no where, either.

How is it possible to exponentiate something to a fraction?

Squaring is: multiplying 1 by a number twice. Giving an exponent of 1/2 means multiplying 1 by that number half a time. Doing this twice, then, results in the number.

And come on, figure these things out for yourself.

 

[WittyUsername]

^ whats the point if I get nowhere? Plus I like to study things only when I have a solid foundation for a particular level. I dont even have the pre-requisite knowledge t
o grasp the concepts of 'mathematical analysis'


I know I can learn it myself but I'm comfortable with the conventional pace of my school. My interest levels are not high enough to force me to start studying it by myself.

 

[Artsu Tharaz]

^ whats the point if I get nowhere?

You think that the process of looking for the solution to a mathematical, or any problem is useless in itself if you don't get the answer? Especially given that this is the type of problem where the main point would be to get you thinking, and not in the answer itself (which you could just look up) I have a hard time comprehending this.

Plus I like to study things only when I have a solid foundation for a particular level.

I'm sure you know what a factorial is.

Not that this is about you, anyway. It's the principle, the idea.

 

[WittyUsername]

Well I didn't mean to say that its useless if I dont find the answer. (Whats the 'answer' you seem to refer anyway?) I meant that I dont find the point of studying something which I'm not ready for. (gamma function doesn't just involve factorial) Whats the point of such brittle knowledge?

I also clarified in my post that I'm not interested in studying it. I looked at your post and happened to know the name of the concept which was related to your question. I'm sorry if my excuse for not knowing it in depth wasn't fit enough.

 

[Agent Intellect]

The way I always remembered fraction exponents is that the numerator of the exponent acted like a normal exponent, then the denominator acted like the root.

So, for instance, if you have 10^(2/3) then you square 10 to get 100 (eg the 2 in the exponents numerator), then take the cube root of 100 to get ~4.6416 (eg the 3 in the exponents denominator). You'll get the same thing if you just take 10^(2/3).

So, in general:

If:
A^(m/n) = B;

Then:
A^m = x
nroot(x) = B

 

[walfin]

You haven't done what I asked. I want an intuitive explanation of the concept. How is it possible to exponentiate something to a fraction? Say it's x to the power of 2 and a half; Do you square it and then add half of x to the power of one? No, that wouldn't make sense.

(a^b)*(a^c) = a^(b+c)

A specific "intuitive" example for you might be x^2.5=x^2*x^0.5

So you square it then multiply that by the square root.

Mutatis mutandis for x^2.333 etc.

 

[Reluctantly]

Yeah, I think it's mostly a definition thing as well.

If you consider that the exponent is >1 (and a whole number), then it makes sense that we are multiplying a number a certain amount of times by itself.

But if you consider that the exponent is <1, but >0, then we will not be multiplying a number by itself a number of times, but in effect dividing itself a number of times. Graphically speaking, then we would use a smaller number (bigger denominator) exponent to represent the number being divided by itself as a smaller result than with an exponent with a smaller denominator.

This would probably make a lot more sense if someone would be so kind as to create a graphical representation of why this makes sense.

 

[Dr. Manhattan]

See a pattern.

For the first 3 lines it's important to keep in mind X has the exponent of 1. Since X^1=X

What you generally do is when you have numbers that can be broken down/simplified or problems where there are both fractional exponents and roots you make them all one or the other to simplify the expression.

 

[GYX_Kid]

lolwut
he was trolling you guys.

 

[Cogwulf]

lolwut
he was trolling you guys.

It was a good question

 

[Artsu Tharaz]

lolwut
he was trolling you guys.

The guy is brilliant.

But yes it is still a good question, and examining it can give rise to many other insights into mathematics.

What's the half derivative of x^n?

 

[walfin]

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

 

[Dr. Manhattan]

Intuition is a useful and important extra mode to assess the quality/value/usefulness of mathematics, not its correctness. It helps you to understand how to apply mathematics, how to choose between different ways of doing something, how to remember the main facts without having to remember all details, how to organize a subject, whether you should look at a particular piece of research or at the details of a proof, etc.. Typically, the more (good) intuition you have about a topic the less complex the matter appears, and the easier it is to quickly see what goes on.

But different mathematicians may have very different intuitions for the same subject matter; so this is a personal and somewhat subjective thing. That's why some writers (e.g., Bourbaki) minimize the amount of intuitive guidance - to be readable, such material must be very well organized then.

 

[Agent Intellect]

Squares and roots of fractions are easy.

When someone finds an easy way of factoring polynomials of degree 3 and higher, I'll be happy. Especially when it includes complex numbers.

 

[Oblivious]

I have not tried, but perhaps you could try reversing expansion using pascal's triangle?

I know pascals triangle makes it trivial to figure out the coefficients of any n degree polynomial when you want to expand them, but I have not tried that for factoring.

Basically the idea is that if you can arrange it into the pattern laid out by the triangle, you know what the factored polynomial will look like. I imagine that could be restrictive.

 

[Latro]

Squares and roots of fractions are easy.

When someone finds an easy way of factoring polynomials of degree 3 and higher, I'll be happy. Especially when it includes complex numbers.

Every polynomial takes the form:
an x^n + ... + a1 x + a0 = an (x-r1)(x-r2)...(x-rn)
where r1,...,rn are the roots of the polynomial counting multiplicities and complex roots. The roots of a cubic are given by the cubic formula, which is horribly complicated by comparison to the quadratic formula. The roots of a quartic are given by the quartic formula, which is horribly complicated even by comparison to the cubic formula. The roots of a quintic or higher cannot in general be expressed using only addition, multiplication, division, and roots; this is the Abel-Ruffini theorem.

Approximate factorizations, however, are relatively trivial using Newton's Method and a computer. Complex roots are somewhat harder to find, though.

 

[Artsu Tharaz]

Intuition is a useful and important extra mode to assess the quality/value/usefulness of mathematics, not its correctness. It helps you to understand how to apply mathematics, how to choose between different ways of doing something, how to remember the main facts without having to remember all details, how to organize a subject, whether you should look at a particular piece of research or at the details of a proof, etc.. Typically, the more (good) intuition you have about a topic the less complex the matter appears, and the easier it is to quickly see what goes on.

But different mathematicians may have very different intuitions for the same subject matter; so this is a personal and somewhat subjective thing. That's why some writers (e.g., Bourbaki) minimize the amount of intuitive guidance - to be readable, such material must be very well organized then.

Intuition is what allows new insights in maths to occur. It's a shame that modern mathematics education does so little to foster intuitive understanding. (and something I hope one day to remedy)