Quick questions. (1 Viewer)

[unity-]

1. Use logs and liate to find out what 0⁰ is. It's interesting.

2. I'm about to start MX2 at school. First topic is complex numbers, mechanics, conics, polynomials, etc.

Can someone please give me a first glimpse of what I'm about to face?

Unity.

 

[Carrotsticks]

1. Use logs and liate to find out what 0⁰ is. It's interesting.

2. I'm about to start MX2 at school. First topic is complex numbers, mechanics, conics, polynomials, etc.

Can someone please give me a first glimpse of what I'm about to face?

Unity.

There is a bit of debate as to what 0^0 is, much like "Is 0 a natural number?" Personally, I would say 0^0 = 1 because it makes many things convenient, prime example being the Binomial Expansion of (a+0)^n, and considering the first term of the expansion.

You can prove that the left and right limit of it is 1 using L'Hopital's Principle (in placement of LIATE I presume) by considering the function x^x and changing it using logs.

 

[iBibah]

Have a look at the syllabus or a textbook.

 

[lolcakes52]

There is a bit of debate as to what 0^0 is, much like "Is 0 a natural number?" Personally, I would say 0^0 = 1 because it makes many things convenient, prime example being the Binomial Expansion of (a+0)^n, and considering the first term of the expansion.

You can prove that the left and right limit of it is 1 using L'Hopital's Principle (in placement of LIATE I presume) by considering the function x^x and changing it using logs.

But the limit is a limit, or the value as it approaches zero. It is technically undefined and similar to how but if we can see that the expression is undefined.

 

[Carrotsticks]

But the limit is a limit, or the value as it approaches zero. It is technically undefined and similar to how but if we can see that the expression is undefined.

Yep I understand that just because left limit = right limit, doesn't mean that the point is actually defined. Easiest example being y=1/(1/x^2) being a parabola with a discontinuity at the origin.

Like I said, consider the Binomial Expansion, and you will see why it is most convenient to consider 0^0 = 1.

 

[unity-]

There is a bit of debate as to what 0^0 is, much like "Is 0 a natural number?" Personally, I would say 0^0 = 1 because it makes many things convenient, prime example being the Binomial Expansion of (a+0)^n, and considering the first term of the expansion.

You can prove that the left and right limit of it is 1 using L'Hopital's Principle (in placement of LIATE I presume) by considering the function x^x and changing it using logs.

ok

i dont know what the f*** L'Hopital's Principle is O_O

 

[RealiseNothing]

ok

i dont know what the f*** L'Hopital's Principle is O_O

Consider we want to find the limit as 'x' approaches either zero or plus/minus infinity of a fraction with a function as a numerator and a different function as a denominator, like shown:



Now what happens if both the numerator and denominator go to either zero or plus/minus infinity? ie:



The way around this is something called L'Hopital's rule (pronounced Lopital's, the 'h' is silent). What this rule states is that if the limit of a fraction is indeterminable (like above), then we can differentiate both the numerator and denominator and proceed to re-take the limit. Mathematically:



However I will note that this only works when the fraction is indeterminable after applying the limit given.

 

[seanieg89]

Related:

Think about what (-1)^pi is and you will notice an often glossed over gap in the high school definitions of powers.

 

[asianese]

I think they only define integer or rational powers?

 

[Carrotsticks]

I think they only define integer or rational powers?

Yep.

 

[seanieg89]

Even rational powers they don't define that clearly from what I remember. Eg (-1)^{1/2} is the square root of -1. But there are two square roots of -1, which one do we take? Etc.

 

[Carrotsticks]

I would have thought they assumed the principal root in that case, but I can see potential problems if its say the root of 1+i.

 

[seanieg89]

It should be something like that but there are a several possible conventions and they certainly don't touch upon this in high school.

In higher mathematics the most common convention would probably be to define

z^w = exp(w*Log(z)). Where Log(z)=log |z| + i*Arg(z) and -pi < Arg(z) =< pi.

This defines complex^complex uniquely for all z,w. But another possible and probably common convention would be to take arg in the interval 0 =< arg(z) < 2pi. And of course if we allow arg to be multi-valued we obtain the most general definition of complex powers.

 

[seanieg89]

It should be something like that but there are a several possible conventions and they certainly don't touch upon this in high school.

In higher mathematics the most common convention would probably be to define

z^w = exp(w*Log(z)). Where Log(z)=log |z| + i*Arg(z) and -pi < Arg(z) =< pi.

This defines complex^complex uniquely for all z,w. But another possible and probably common convention would be to take arg in the interval 0 =< arg(z) < 2pi. And of course if we allow arg to be multi-valued we obtain the most general definition of complex powers.