Complex Number

KaurexO_o SG

Hving much difficulty with this question >.<

By letting y=z^2. Find the four roots of the equation z^4-z^2+10.

wee_ws SG

Hi,

Please specify the difficulty. Thanks.

Cheers,
Wen Shih

CaiHongRainx SG

Cant be solved? Im taking Os this year.

wee_ws SG

Hi,

I found the roots of z^4-z^2+10 = 0 via GC and they are not nice values. Cambridge exam setters typically write questions where answers are student-friendly.

We consider a simpler equation

z^4 + 4 = 0 and let y = z^2, giving us

y^2 = -4
=> y = +/- 2i.

Let z = x + iy, so that z^2 = (x^2 - y^2) + 2xyi.

For y = 2i, we compare real and imaginary parts and arrive at

x^2 - y^2 = 0 -- (1)

2xy = 2 -- (2),

from which we are able to solve for x and y easily.

For y = -2i, we apply the same method.

The four roots (with nice values) can then be obtained.

Thanks.

Cheers,
Wen Shih

wee_ws SG

Hi,

Here is another example (with nice root values and detailed explanations), which is closer to the original problem that was proposed: Solve z^4 - 4z^2 + 16 = 0, by considering the substitution y = z^2.

First, the equation becomes y^2 - 4y + 16 = 0. By completing the square,

(y - 2)^2 + 12 = 0
=> y = 2 +/- 2*sqrt(3) i.

Let z = x + iy, so that z^2 = (x^2 - y^2) + 2xyi.

For y = 2 + 2*sqrt(3) i, we compare real and imaginary parts to obtain two equations:

x^2 - y^2 = 2 -- (1)

2xy = 2*sqrt(3) => xy = sqrt(3) => y = sqrt(3) / x -- (2).

Substituting (2) into (1),

x^2 - 3/x^2 = 2
=> x^2 - 2 - 3/x^2 = 0
=> (x - 3/x)(x + 1/x) = 0
=> x = +/- sqrt(3), so y = +/- 1.

Two roots are z = sqrt(3) + i and z = -sqrt(3) - i.

For y = 2 - 2*sqrt(3) i, the same method applies for us to find x, y values.

One will get the roots z = sqrt(3) - i and z = -sqrt(3) + i.

Thanks.

Cheers,
Wen Shih

Domo Kun SG

Cant be solved? Im taking Os this year.

O levels no complex numbers.

frekiwang SG

Obviously A level question, KaurexO_o never states the question is for O.

However, the question is incomplete as the equation has no equal sign. It is a typo error so please fix it before asking for help.

eagle SG

Obviously A level question, KaurexO_o never states the question is for O.

However, the question is incomplete as the equation has no equal sign. It is a typo error so please fix it before asking for help.

Hi,

the question is complete. By asking the student to find the roots of an equation, it implies that we are setting it equal to zero.

I agree with Mr Wee that the answers are not pretty at all.

KaurexO_o SG

Sorry, it should have been z^4-z^2+1=0.

But thanks for the help!!

Nope, it's neither 'A's nor 'O's. They're from my tutorials in ntu.

eagle SG

Hi Kaurex

The question can be done simply by

(z^2 )^2 - z^2 +1 = 0

Hence, z^2 = [1 +- sqrt (-3)] / 2

Converting to polar form gives us
z^2 = e^(iπ/3) or e^(-iπ/3)

Hence
z = e^(iπ/6), -e^(iπ/6), e^(-iπ/6), -e^(-iπ/6)
Or
z = e^(iπ/6), e^(-i5π/6), e^(-iπ/6), e^(i5π/6)

Domo Kun SG

Complex numbers are applicable in what fields ?

eagle SG

Complex numbers are applicable in what fields ?

One example would be the field of electrical engineering, in the RLC circuits.
RLC stands for Resistors, Inductors and Capacitance.
Such a circuit consists of its own natural frequencies, and can be attached to antennas to latch on to radio signals. There will be other circuits to further remove the carrier frequency as well as reduce white noise before the final signal is processed, amplified and sent to the speakers.

Another example would be more advanced differentiation and integration of trigonometric functions. This is used in higher levels of mathematics and physics.

Domo Kun SG

At least can apply the things we learnt in school is good enough.

eagle SG

At least can apply the things we learnt in school is good enough.

Everything in school can be applied in certain fields one way or another. The other things you learn are there to help make a more complete understanding. Example, to learn complex numbers, you would need to know trigonometry as well.

However, these should not be the main things you are learning from school. Students have to realise that knowledge is never sufficient. What they should pick up from schools are soft skills like teamwork, time management, leadership skills, communication skills, stress management, etc, on top of their school work. These are important skills that will last a lifetime.

KaurexO_o SG

Thanks for the help, eagle.

Been really out of touch with my school work since my army days. Not saying that i had worked hard in my JC days also. So my foundation isn't exactly strong.

Domo Kun SG

Yes, good advice.