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I am looking at this proof (in the image) from a textbook for engineering mathematics:

enter image description here

I don't understand this proof, and my reasoning is as follows:

Known True Statements:

$z=r(cos\theta + isin\theta)$

$z_1z_2 = r_1r_2[cos(\theta_1 + \theta_2) + isin(\theta_1 + \theta_2)]$

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[cos(\theta_1 - \theta_2) + isin(\theta_1 - \theta_2)]$

$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}$

The line in the proof saying "When expressed in terms of Euler's formula, this becomes..." seems to me to be equivalent to:

IF $e^{i\theta} = cos\theta + isin\theta$, then

$e^{i\theta_1}e^{i\theta_1} = (cos\theta_1+isin\theta_1)(cos\theta_2+isin\theta_2)$

$e^{i\theta_1}e^{i\theta_1} = cos(\theta_1+\theta_2) + isin(\theta_1+\theta_2) = z_1z_2,$ where $r_1=r_2=1$

and:

$\frac{z_1}{z_2} = cos(\theta_1-\theta_2) + isin(\theta_1-\theta_2) = \frac{z_1}{z_2},$ where $r_1=r_2=1$

Therefore if Euler's formula is true, then it can be shown that $z = re^{i\theta}$, and since $z=r(cos\theta + isin\theta)$, it is finally shown that $e^{i\theta} = cos\theta + isin\theta$

I don't understand this proof because at the step when the proof says "When expressed in terms of Euler's formula this becomes...", I interpret this as meaning we assume the statement to be true. Am I correct in this assumption?

If that assumption is correct, is the proof circular because it assumes the statement is true in order to prove the statement?

Thanks very much!

masiewpao
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    I don't think this is intended to be a proof of Euler's formula in the first place – Wouter Jun 25 '19 at 18:18
  • I have just realised that would make sense... What's throwing me off is the part about the justification of the definition. – masiewpao Jun 25 '19 at 18:20
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    What's throwing me off is the part about the justification of the definition --- Things like this used to bother me a lot also, especially in non-math but supposedly heavily mathematical and precise subjects such as physics and engineering. I often had trouble determining which author assertions were statements whose validity I was supposed to be able to rigorously deduce from prior material in the text, which author assertions were statements whose validity I was supposed to provide non-rigorous algebraic manipulation arguments (or heuristic support for), (continued) – Dave L. Renfro Jun 25 '19 at 18:47
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    and which author assertions were simply assertions of the FYI variety. In your situation, I think the author's use of "justification" was not a good word choice. Better would have been to use something like "support" or "reasonableness" or something similar. Also, now that I'm looking at it, the author uses "The justification" when he really meant "A justification", since there is certainly more than one justification (just use different words or something). Or the author should have written "The following justification ...". – Dave L. Renfro Jun 25 '19 at 18:54
  • I'm glad to know I am not the only one! Especially in this case, I didn't include a sentence above the text in the image which stated "In (some section)... we obtained the result..." which caused me to immediately assume this result was being 'obtained' somehow in the book. – masiewpao Jun 25 '19 at 19:18

2 Answers2

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This is not a proof of Euler's formula. Instead it is stating Euler's formula and using it to explain some otherwise difficult calculations that then become easier.

For example, we don't need to multiply out the real parts of two complex numbers, instead we can easily say $$ z_1z_2=r_1r_2e^{\theta_1 + \theta_2} $$

the same is done for dividing complex numbers in this example

wjmccann
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If you put, formally, $e^{jx}=\cos x + j \sin x$ all the formulae work nicely (mathematicians use $i$ where physicists use $j$, by the way). That justifies using the expression.

However, this does not illuminate the particular value of $e$ - any number would give the same parallel between formulae considered formally. That is because the exponents add when expressions are multiplied and the formulae for the sine and cosine only depend on addition (or subtraction) of angles.

To identify $e$ with a particular constant you first need to give meaning to the expression $e^{jx}$, and then show that this meaning makes the equation work for Euler's constant $e$. There are various ways of doing this.

Note that the formula also depends for its beauty on the units used to measure the angle - radians must be used. There is a similar (but messier) formula for degrees and similarly for other angle measures.

The explanation given does not make any of this explicit, or prove it. What it does is show that the formula potentially makes sense.

Mark Bennet
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