do you agree with the decision to remove euler's formula from the syllabus? (3 Viewers)

[tywebb]

in the draft syllabus euler's formula was still there, but the syllabus which was released a few days ago here https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-2-11-12-2024/overview surprised everyone with the removal of euler's formula.

nesa gave no indication whatsoever they would remove it, and that it was in the draft was at best misleading. one might say a clayton's consultation.

euler's formula often simplifies things like proving de moivre's theorem, compound angle formulae, multiplying or dividing complex numbers, elegant forms of roots of complex numbers and multiple rotations in the complex plane.

yet it was deleted. do you agree that it should have been deleted?

 

[wollongong warrior]

Ridiculous. I bet the people who proposed this idea at nesa haven't even studied a maths-related degree in uni

 

[tywebb]

notice contrapositive is still in the syllabus

consider the following seemingly innocuous statement which would not get you banned on social media:

if euler's formula should be in the syllabus then the students are not stupid

this may elicit responses such as

thank you for that insightful and encouraging comment. i totally agree with you and can see why many others would agree with you too. i look forward to seeing many comments from you in the future

now consider the contrapositive

if the students are stupid then euler's formula should not be in the syllabus

that will definitely get you banned on certain social media and may elicit such responses as

you ought to be ashamed of yourself posting such horrendously offensive comments. i cannot fathom how anyone could possibly agree with you. you should get off the internet and stay off you numbskull. i never want to hear from you again!

 

[igor9]

 

[ShhQuietPlease]

Ridiculous, Eulers is the link between complex and exponentials. There are many things that can easily be solved with exponents than a binomial expansion ... NESA come on

 

[epicmaths]

I'm not sure actually. What is the upshot of learning Eulers' when you can already be very good with De'Moivres? @tywebb I would love some questions that really leverage Euler's that falls within the scope of the syllabus.

 

[Trebla]

Would have to agree with the above.

Firstly, the derivation of Euler’s formula is beyond the scope of the syllabus so to most students this looks like some random abbreviated notation (as an alternative to “cis”) that they are forced to just accept. Having it taught in that way without derivation quite frankly dilutes any appreciation of Euler’s formula.

Secondly, I have yet to see a HSC problem where Euler’s formula is actually necessary to solve it compared to just using standard polar form. If any exist, they would be extremely rare or beyond the syllabus.

Given it isn’t necessary for most HSC questions, I would rather have it excluded from the syllabus and taught/appreciated properly at university than have it taught in a half-baked way in high school.

 

[tywebb]

i certainly understand and respect that there will be a range of opinions on this issue from people with diverse backgrounds

perhaps a list of exercises on exponential form would be better suited in another thread for another day, but here are some comments from an esteemed professor from a uni who recently became aware of the change in the syllabus with a perspective from outside the school system:

I am very disappointed by the omission of Euler's equation and the powerful, and suggestive, notation it embodies. I am also extremely disappointed to see the teaching considerations encouraging (rather than discouraging) the use of cis(θ) notation.

Since it’s no longer a syllabus item its many elegant consequences are sadly impermissible.

I can understand the urge to abbreviate cos(θ) + i sin(θ), but surely introducing e^{iθ} is a more suggestive and powerful way to do it. Deducing De Moivre's theorem, multiple angle and angle sum formulas for sine and cosine from the exponential form is an elegant and almost irresistible piece of mathematical magic. It seems almost criminal to deny students such insights when they are so easily within their grasp.

There are better things to cut..

No problem about regarding cis as a function ℝ —> ℂ, but its only value lies in its name as an abbreviation for cos(θ) + i sin(θ), whereas the equation e^{iθ} = cos(θ) + i sin(θ) has profound significance and consequences.

For proving it, let f(θ) = cos(θ) + i sin(θ), show df/dθ = i f(θ), and f(0) = 1, which we know has solution f(θ) = e^{iθ}.

- De Moivre’s using (e^{iθ})^n = e^{inθ}

- deduce the multiple angle and angle sum/difference formulas for sine and cosine from, e^{i(θ+φ)} = e^{iθ} e^{iφ}.

- Also, makes, finding roots of unity easy.

As I said before, an elegant and almost irresistible piece of mathematical magic. It seems almost criminal to deny students such insights when they are so easily within their grasp. This is why Euler's formula is often regarded as one of the most beautiful famous, and powerful results in mathematics; it unified algebra and trigonometry.

I’m sure many E2 students would have read of it (certainly of the famous formula, e^{iπ} = -1). Seems a shame to have deprived them of reaching what might have been a "landmark" for where they are at mathematically.

 

[Canteen]

The above proof relies on the extension of calculus to complex numbers, which is not within the syllabus (although I'm sure most students won't hesitate in accepting it). The Cambridge ME2 Textbook provides a lengthier proof that is more within the confines of the syllabus.

At any rate, notation is a convenient way of writing and avoids the bastardisation of notation that is . It can easily be taught in a single lesson at school and doesn't require much practice to become familiar with. Its proposed exclusion is misguided in my opinion.