Logic

Ryan$

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Actually, the approximation takes no time at all; it's just ignoring the Y, right?

And the decision whether to do that is not hard; but it depends on context. If a 1% error in your answer is acceptable, and Y is less than 1% of Z (actually, of their sum), then it's acceptable to ignore it. Unfortunately, you haven't told us the context of the video, so we have no idea what conditions led to the statement that Y can be ignored.

As for X, the idea is that multiplication does not change the relative error. The relative (percent) error in Z relative to Z+Y is the same as the relative error in XZ relative to X(Z+Y), because relative error is proportional; the X can be canceled. So, again, the decision whether to ignore a small number is easy: you only need to compare to the other addend, and don't have to consider any other parts of the calculation.

But by not clearly showing the specific problem you are asking about, you have complicated things. What was actually said in that video? What were they doing? How did they know that Y was negligible?
Actually, the approximation takes no time at all; it's just ignoring the Y, right?

And the decision whether to do that is not hard; but it depends on context. If a 1% error in your answer is acceptable, and Y is less than 1% of Z (actually, of their sum), then it's acceptable to ignore it. Unfortunately, you haven't told us the context of the video, so we have no idea what conditions led to the statement that Y can be ignored.

As for X, the idea is that multiplication does not change the relative error. The relative (percent) error in Z relative to Z+Y is the same as the relative error in XZ relative to X(Z+Y), because relative error is proportional; the X can be canceled. So, again, the decision whether to ignore a small number is easy: you only need to compare to the other addend, and don't have to consider any other parts of the calculation.

But by not clearly showing the specific problem you are asking about, you have complicated things. What was actually said in that video? What were they doing? How did they know that Y was negligible?
It was given that Y was negligible ..

my problem is that .. who said that relative error wouldn't change if we continue with other operands/elements?(in my case multiplication .. ) ?! ..may please gimme an example for that? thanks alot! exactly that's my problem, why wouldn't relative error change if we continue to other elements/operands ?!
 

JeffM

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It was given that Y was negligible ..

my problem is that .. who said that relative error wouldn't change if we continue with other operands/elements?(in my case multiplication .. ) ?! ..may please gimme an example for that? thanks alot! exactly that's my problem, why wouldn't relative error change if we continue to other elements/operands ?!
You are trying to generalize from one specific case. I gave you the definitions for both the relative and absolute cases. Those definitions are what is general. Then you must apply them individually to each specific case.

We keep telling you to give specific cases. I have a feeling that you have tried to deduce a general rule that y is always relevant and x is always irrelevant to an approximation. There is no such general rule.
 

Dr.Peterson

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my problem is that .. who said that relative error wouldn't change if we continue with other operands/elements?(in my case multiplication .. ) ?! ..may please gimme an example for that? thanks alot! exactly that's my problem, why wouldn't relative error change if we continue to other elements/operands ?!
In math, the issue is not "who said?". We don't go by authority. If you question what you're told, you can check it out (that is, make up an example for yourself!), or examine the reasons that were given. I showed you a reason:
As for X, the idea is that multiplication does not change the relative error. The relative (percent) error in Z relative to Z+Y is the same as the relative error in XZ relative to X(Z+Y), because relative error is proportional; the X can be canceled. So, again, the decision whether to ignore a small number is easy: you only need to compare to the other addend, and don't have to consider any other parts of the calculation.
Did you try working through what I said here, seeing why the X has no effect? Did you try working through an example (which has been mentioned in this thread long ago)? You're not going to understand anything fully if you don't do the thinking for yourself.

Now, in general, if there is something other than multiplication involved, it might turn out that relative error is changed. No one here (and presumably not the person in the video) has claimed any broad generalization. In order to discuss how relative error is affected by various functions, we'd have to look at error propagation and calculus. But that's not part of your question.

Now, your questions might be very much worth discussing in the context of the video (which you have told us nothing about). It may be that there are things that weren't said that should have been, or even things that were done that are not fully justified. But if you can't show us the video, then you should just be writing to its author, rather than to us, asking what his justifications were. Don't be looking for extreme generalizations that don't exist.
 

Otis

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… X is neglected relative to Y , so my problem how do I do mental Effort on EXP(LOG(EXP(LOG(EXP(x-y))))))) …
The same way you were shown in posts #3 and #4: Experiment, by substituting numbers for X and Y and evaluating to compare results. Choose values for X that are relatively small compared to Y.

Your expression simplifies to e^(X-Y), so you may experiment using that simpler version.

Are you willing to try?

\(\;\)
 

Ryan$

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In math, the issue is not "who said?". We don't go by authority. If you question what you're told, you can check it out (that is, make up an example for yourself!), or examine the reasons that were given. I showed you a reason:

Did you try working through what I said here, seeing why the X has no effect? Did you try working through an example (which has been mentioned in this thread long ago)? You're not going to understand anything fully if you don't do the thinking for yourself.

Now, in general, if there is something other than multiplication involved, it might turn out that relative error is changed. No one here (and presumably not the person in the video) has claimed any broad generalization. In order to discuss how relative error is affected by various functions, we'd have to look at error propagation and calculus. But that's not part of your question.

Now, your questions might be very much worth discussing in the context of the video (which you have told us nothing about). It may be that there are things that weren't said that should have been, or even things that were done that are not fully justified. But if you can't show us the video, then you should just be writing to its author, rather than to us, asking what his justifications were. Don't be looking for extreme generalizations that don't exist.

what do you mean by "relative error" here I didn't understand that term
 

JeffM

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what do you mean by "relative error" here I didn't understand that term
For goodness' sake ryan, do you bother to try understanding what is written in response to your questions?
The term is defined in post 9. Dr. P used it in post 11. Suddenly, in post 19, you notice that you have not been aware of the topic being discussed. It is possible that the video was unclear, but, as usual, you do not give a link or anything that gives specific context.
 

Ryan$

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Hi guys, I'm struggling something which I don't know why I find it hard but I hope I will get help by you guys and convince me to understand it properly.

my problem is this, sometimes when I solve a questions, I face like this:
lets assume I was solving a question, I arrived to Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2 , forget from the equation itself exactly from the right side ..
what's confusing me I know that's sin(x)^2+cos(x)^2 =1, but I'm not assigning that as 1 although I know it's 1 because in my equation above (Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2) didn't tell me that I can use it, how should I know if it's allowed in my equation above to assign or not assign "1" instead of "Sin(x)^2 + Cos(x)^2" ?!
 

MarkFL

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No matter what the angle \(x\) is, it is an identity that:

\(\displaystyle \sin^2(x)+\cos^2(x)=1\)
 

Dr.Peterson

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Since it is always true that sin^2(x) + cos^2(x) = 1, you can always replace it with 1. Always! You need no additional information to make that permissible.

Why would you even imagine you couldn't??
 

Ryan$

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Since it is always true that sin^2(x) + cos^2(x) = 1, you can always replace it with 1. Always! You need no additional information to make that permissible.

Why would you even imagine you couldn't??
I have no idea why I imagine that I couldn't , but I frankly face that problem, I know there's identity sin^2(x) + cos^2(x) = 1 , but ! while solving like if I have another problem like
Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2 .. so here i STUCK although I know that there's
sin^2(x) + cos^2(x) = 1 but the question why I could assign that?! here what's exactly I'm facing while solving question ...

maybe because sin^2(x) + cos^2(x) = 1 it's not given explicitly like sin^2(x) + cos^2(x) = 1 ..... so I get mislead while it's given with other equation like Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2....how can I solve that problem of thinking ?!!
 

JeffM

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I have no idea why I imagine that I couldn't , but I frankly face that problem, I know there's identity sin^2(x) + cos^2(x) = 1 , but ! while solving like if I have another problem like
Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2 .. so here i STUCK although I know that there's
sin^2(x) + cos^2(x) = 1 but the question why I could assign that?! here what's exactly I'm facing while solving question ...

maybe because sin^2(x) + cos^2(x) = 1 it's not given explicitly like sin^2(x) + cos^2(x) = 1 ..... so I get mislead while it's given with other equation like Sin(x)^2 + Cos(x)^2 = X^2+Y^2+Z^2....how can I solve that problem of thinking ?!!
Again, you make up some problem to confuse yourself.

expression one = expression two.

All that means is that the two expressions represent the same numeric value.

\(\displaystyle cos^2(x) + sin^2(x) = \text {some expression.}\)

But no matter what x is \(\displaystyle cos^2(x) + sin^2(x) = 1.\)

So \(\displaystyle cos^2(x) + sin^2(x)\) and 1 are just different names for the exact same numeric value.

And because you have stipulated that \(\displaystyle cos^2(x) + sin^2(x) = \text {some expression.}\),

then some expression is just a third name for the exact same numeric value, namely 1.

The equal sign simply means "has the same numeric value."

You asked this exact same question using x, y, and z weeks ago.

Here is a BASIC rule of algebra: \(\displaystyle x = y \text { and } y = z \implies x = z.\)

Learn it and move on.
 

Ryan$

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thanks alot but for instance lets assume I have like this
np*(5+y)
and np is equal to ni^2/C
then I know that's np=ni^2/C BUT once I solve an equation like this np*(5+y) it rings in my head that
np = ni^2/C , but I don't assign it because I ask myself who said that I'm allowed to "assign" .. yea they are equal but who said that I could assign in the equation np*(5+y) ..

I hope I explained my problem, you know it rings in my head that np=ni^2/C .. but I'm not assigning it although it rings in my head ni^2/C .. idk what's that approach? am I facing that problem alone? or maybe my iq isn't ... ? but it rings for me that np=ni^2/C .. but not assigning that in my equation ..
 

JeffM

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thanks alot but for instance lets assume I have like this
np*(5+y)
and np is equal to ni^2/C
then I know that's np=ni^2/C BUT once I solve an equation like this np*(5+y) it rings in my head that
np = ni^2/C , but I don't assign it because I ask myself who said that I'm allowed to "assign" .. yea they are equal but who said that I could assign in the equation np*(5+y) ..

I hope I explained my problem, you know it rings in my head that np=ni^2/C .. but I'm not assigning it although it rings in my head ni^2/C .. idk what's that approach? am I facing that problem alone? or maybe my iq isn't ... ? but it rings for me that np=ni^2/C .. but not assigning that in my equation ..
Just stop making up these goofy examples. They frequently make no sense.

np * (5 + y) is an expression. It is NOT an equation. An algebraic expression has little meaning on its own.

"Ringing in your head" and "assigning" make no sense except perhaps in describing to yourself your own pschology. We are not psychiatrists. If two expressions have the same numeric value, it should be clear that it makes absolutely no numeric difference which you use. You can use whichever is more convenient for your purposes.
 

Harry_the_cat

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Here's a hypothetical question for you Ryan. Do these sentences mean the same thing?
1. I find mathematics difficult.
2. I find mathematics hard.

Yes they do, because we know that the word "difficult" and the word "hard" are synonyms, ie have the same meaning in this context. So we can interchange them because we know they mean the same thing. The meaning of the sentence doesn't change. We could say difficult = hard.

It's the same deal in maths. If we know that sin^2(x) + cos^2(x) = 1, which it ALWAYS does, then we can interchange them. Wherever we see sin^2(x) + cos^2(x) in an equation or an expression, we can replace it with 1.
 

Jomo

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You need to understand what equal means.
If klegjagkj= hjsaoaf then whenever you see klegjagkj you can replace it with hjsaoaf and whenever you see hjsaoaf you can replace it with klegjagkj.

Now in a given problem you are told for example that x+y = 9. So in this problem if you see x+y you can replace it with 9 and if you see 9 you can replace it with x+y.

Now x+y=9 is NOT an identity equation as it is NOT always true. However you know that 2+3=5. So if in your work you have 2+3 I bet you would replace it with 5. Now just like 2+3=5 we have sin^2(x) + cos^2(x) = 1 ALWAYS. So you can replace sin^2(x) + cos^2(x) =with 1 without any real thinking just as you would replace 2+3 with 5.

Just understand that just because you know very well that 2+3=5 and maybe don't really know why sin^2(x) + cos^2(x) = 1 they are still identities non-the-less.

So much is 7 + 4(sin^2(x) + cos^2(x))
 
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