Complex Numbers for + Rep (2 Viewers)

[Utility]

Find z₁z₂ and z₁² if
z₁ = cis(π / 6) & z₂ = 3cis(-5π/6)


Answer: z₁z₂ = 3e^{-2iπ / 3} = 3/2 (-1 - √3i)
z₁² = e^{iπ / 3} = ½(1 + √3i)


Can someone please show working to reach this? π = pi.

 

[SpiralFlex]

 

[Utility]

Thank you!

 

[asianese]

Note that while it is good to know exponential polar form it isn't in the syllabus.

 

[Utility]

That's alright, I'll be using it all through Uni. anyway. Converting between Cartesian / Polar / Exponential will be useful.

Rep. seems to have disappeared since my last visit...

 

[Rixon]

Rep. seems to have disappeared since my last visit...

mods are dogs

 

[Skeptyks]

Note that while it is good to know exponential polar form it isn't in the syllabus.

Good to know, I was wondering about what the hell he did at the end

 

[D94]

Note that while it is good to know exponential polar form it isn't in the syllabus.

"cis" notation isn't in the syllabus. Using e notation is as valid, if not more, than cis notation. They are just notations we use to simplify working out (at least at a basic level).

In fact, if you were to leave your answer as 'cis' in the HSC, you should not gain any marks. cis should always be expanded back to cos(@) + isin(@).

 

[cutemouse]

In fact, if you were to leave your answer as 'cis' in the HSC

That depends on the question. Most markers will probably know what 'cis' means. It's very common in textbooks.

Euler's rule is definitely not in the current syllabus, although it probably should be. Also, most markers probably wont know euler's rule (compared to 'cis') as it is not that common in textbooks.

 

[D94]

That depends on the question. Most markers will probably know what 'cis' means. It's very common in textbooks.

Euler's rule is definitely not in the current syllabus, although it probably should be. Also, most markers probably wont know euler's rule (compared to 'cis') as it is not that common in textbooks.

Even if they know what it means, cis is not completely correct as a final answer - it must be fully expanded. Being common in textbooks doesn't justify a non-complete solution. 'e' notation is universally recognised, so using such notation is very valid - markers should not mark a correct solution incorrect if they see the use of e notation.

Anyway, this is all beside the point. The point is that 'cis' isn't actually in the syllabus, nowhere does it abbreviate cos(@) + isin(@) to cis(@), so it isn't something the syllabus brought on.

 

[Carrotsticks]

In fact, if you were to leave your answer as 'cis' in the HSC, you should not gain any marks. cis should always be expanded back to cos(@) + isin(@).

I think that's going a little bit too far. The HSC is there to mark your ability to apply what you (are supposed to have) learnt, not the notation of your final answer. For example, 1995 HSC Question 7 (a). If you had used Product notation instead of actually writing down 2^2 * 4^2 etc etc for the Wallis Product, then you would NOT have been deducted marks because of direct equivalence (yes, that term is actually used in the marking centre).

Leaving something as log(100)-log(10) would have a chance of being marked down (assuming the state average for that question is high enough) because the answer is not directly equivalent. Sure, it means the exact same thing as log(10) but the fact is that you didn't demonstrate KNOWLEDGE of log law simplification, which is why you get docked down. On the other hand, cos+isin is directly equivalent to cis, so no docking would be had. There's the 'transcription error' (denoted by a TE during HSC marking), but not quite applicable to writing cis instead of cos+isin. Usually, it is used if you had x+y, but wrote x-y in your next line of working out, causing you to acquire the incorrect answer.

 

[Sanical]

hah made me smile when I read rep in the title

anyway I haven't heard of anyone being marked down for writing cis

 

[D94]

I think that's going a little bit too far. The HSC is there to mark your ability to apply what you (are supposed to have) learnt, not the notation of your final answer. For example, 1995 HSC Question 7 (a). If you had used Product notation instead of actually writing down 2^2 * 4^2 etc etc for the Wallis Product, then you would NOT have been deducted marks because of direct equivalence (yes, that term is actually used in the marking centre).

Leaving something as log(100)-log(10) would have a chance of being marked down (assuming the state average for that question is high enough) because the answer is not directly equivalent. Sure, it means the exact same thing as log(10) but the fact is that you didn't demonstrate KNOWLEDGE of log law simplification, which is why you get docked down. On the other hand, cos+isin is directly equivalent to cis, so no docking would be had. There's the 'transcription error' (denoted by a TE during HSC marking), but not quite applicable to writing cis instead of cos+isin. Usually, it is used if you had x+y, but wrote x-y in your next line of working out, causing you to acquire the incorrect answer.

That wasn't said as well as I thought it to be. What I meant was that for your final answer, you shouldn't gain the whole mark. But don't quote me on this, I'm just relaying what MANSW said in a conference, which my teacher, a HSC marker, said.

I'm not totally convinced with your point about direct equivalence. We never see cis is equivalent to cos + isin, rather we see cis is equal to cos + isin. (as in, we only see two lines, not three lines).

Cis notation is merely shorthand, the full expanded version should be the final answer. Note that I said 'should' not 'will'. Its a technicality so some teachers and mathematicians might agree, some won't.

 

[asianese]

TBH it is just a bit of notation...

 

[Sanical]

That wasn't said as well as I thought it to be. What I meant was that for your final answer, you shouldn't gain the whole mark. But don't quote me on this, I'm just relaying what MANSW said in a conference, which my teacher, a HSC marker, said.

I'm not totally convinced with your point about direct equivalence. We never see cis is equivalent to cos + isin, rather we see cis is equal to cos + isin. (as in, we only see two lines, not three lines).

Cis notation is merely shorthand, the full expanded version should be the final answer. Note that I said 'should' not 'will'. Its a technicality so some teachers and mathematicians might agree, some won't.

what about in geometry when you use symbols for giving reasons, such as angle () symbol, triangle symbol etc. Where does it say you can use the therefore () symbol? In the HSC when a good 6/10+ students would write cis, they won't mark you down. Granted, 'cis' isn't a symbol but it won't be marked down. They'll treat it the same as if you were notating your reasoning in geometry as (ext. = sum of opp. int. )

 

[Carrotsticks]

I'm not totally convinced with your point about direct equivalence. We never see cis is equivalent to cos + isin, rather we see cis is equal to cos + isin. (as in, we only see two lines, not three lines).

The equivalence used there is not the mathematical definition of equivalence with the 'three lines' (btw it's called the Triple Bar), which is more appropriately used for say Partial Fractions when we equate coefficients of x, constants etc of two equivalent polynomials f(x) and g(x).

What they intend is "If their answer can be matched with ours, with at most one line of trivial computations, then there is direct equivalence". They would, of course, leave a bit of wiggle room in that statement.

So for example if they left their answer as 4/6 instead of 2/3, then that would fall under the DE category. However if they left their answer as my prior example with log(100)-log(10), then that would most likely not be accepted (and ofc like I said, depends on state average), because of the fact that there is more than 1 line of trivial working out to go from log(100)-log(10) = log(10).

Anyway, I have said enough on this topic of conversation. I try to advocate a degree of leniency when it comes to notation. The student should be deducted for saying something false or making a false assumption, as opposed to just writing a symbol that the marker doesn't like.

PS: Source of my information, I work with two HSC markers, one of which is at a senior position.

 

[D94]

Fair enough, it's an interesting moot point because at the end of the day, it's up to the marker.

What they intend is "If their answer can be matched with ours, with at most one line of trivial computations, then there is direct equivalence". They would, of course, leave a bit of wiggle room in that statement.

If we take a u-substitution, and our final answer is in terms of u, ie. all other computations are correct, show knowledge and reflect the difficulty of the question, but it is technically only one extra line of working out, would this be deemed a correct solution? I wouldn't think so - the correct solution would make the substitution so the answer is back in terms of x, say. But we have set u = whatever, so there is presupposition of truth in that statement, so why can't the final answer be left in terms of u?

This is a fair enough question - I just don't fully agree with that statement of yours.

 

[Shadowdude]

I wouldn't call substituting back into x a "trivial computation" though.