Quick questions. (1 Viewer)

unity-

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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

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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

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Have a look at the syllabus or a textbook.
 

lolcakes52

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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

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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-

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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 :D

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

RealiseNothing

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ok :D

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

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Related:

Think about what (-1)^pi is and you will notice an often glossed over gap in the high school definitions of powers.
 
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I think they only define integer or rational powers?
 

seanieg89

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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.
 

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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

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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

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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.
 

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