Two Methods | Can you solve this Complex Problem? | Math Olympiad Preparation

[@sadafkhanam7220]

great solution 👍 you are amazing teacher 🤗

[@PreMath]

Glad you think so! Thanks for your feedback! Cheers! You are the best, Sadaf Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@張茗茗-y9i]

Set x=cos(c)+i sin(c)=exp(ic), and c=π/3 (by Taylor Series for exponential-trigonal complex function at c=0) x^7777=exp(i*7777c)=exp(i*(2592+1/3)*π)=exp(i*π/3)=x

[@SuperYoonHo]

WOW! That was brilliant teacher! You are awesome! God bless you for your good work🙏🙏

[@HappyFamilyOnline]

Another great video👍 Thanks for sharing😊

[@bigm383]

Two great methods!😀👍

[@BlipBlop7703]

Amazing work and amazing math fact to appreciate.

[@PreMath]

Excellent! Glad you think so! Thanks for your feedback! Cheers! You are awesome, Patrick. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@davidfromstow]

Brilliant question and explanation - thank you

[@PreMath]

Glad you enjoyed it! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@nirupamasingh2948]

Sir l liked both methods. Wish to see more of typical complex number problems.

[@PreMath]

Excellent! Thanks for your feedback! Cheers! You are awesome, Niru. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@dreael]

Here we have the classic unit vector on the complex plane: x=(1+sqrt(3)*i)/2 represents a 60° vector. So x^n -> x^(n+1) will rotate this vector 60° counterclockwise. n=6 means that the 360° position is the 0° position again, which implies x^6=1 Since 7777=1296*6+1 this means that our complex plane vector rotates 1296 times a whole 360° turn an then a single 60° turn again which results into the solution as shown in the video.

[@raymondarata6549]

I did the problem the same way. It is a quick and eloquent solution.

[@KipIngram]

Note that this complex number has unit magnitude: sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2, and 2/2 = 1. So the result will also be a unit complex number. To raise a complex unit vector to a power, we just multiply its angle by the power and take that mod 360 degrees. Here our angle is arctan(sqrt(3)), which is 60 degrees. So, 60*7777 = 466,620, and 466,620 mod 360 is 60 again. So the result is identical to the argument: ((1+i*sqrt(3)/2)^7777 = (1+i*sqrt(3))/2

[@242math]

excellent presentation

[@marcello3621]

First, we have to put the number x in the trigonometrical form, then we solve the exponent: ρ = √(1/2)² + (√3/2)² ρ = √ 1/4 + 3/4 ρ = √1 *ρ = 1* Let's find cosine and sine of the x argument, in the complex plane, its afixes are seen in the first quadrant, so the argument is greater than 0° and less than 90°, then: sin arg x = √3/2 / 1 = √3/2 cos arg x = 1/2 / 1 = 1/2 Argument of x = 60° So, its trigonometrical form equals x = cos 60° + i sin 60° By the First Theorem of Moivre: x⁷⁷⁷⁷ = cos (60° × 7777) + i sin (60° × 7777) x⁷⁷⁷⁷ = cos 466 620° + i sin 466 620° To solve this is not that hard, just reduce to the first quadrant. 466 620°/ 360° = 1296,1666666... = 1296 + 0,1666... = *1296 + 1/6* It means that the angle rounded the cicle 1296 times and stopped in 1/6 of 360°, i.e., stopped in 60°. Then: x⁷⁷⁷⁷ = cos 60° + i sin 60° x⁷⁷⁷⁷ = 1/2 + i(√3/2) *x⁷⁷⁷⁷ = 1 + √3 i / 2* What a weird surprise! It's equal to x!

[@oenrn]

It is certainly not equal to x! 😜

[@-EaswarNaik]

Since power is odd.... for every odd power the answer is same ...simply solve it for power = 3 and ans is -1

[@PreMath]

Thanks for your feedback! Cheers! You are awesome, Naik. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@KAvi_YA666]

Thanks for video.Good luck sir!!!!!!!!

[@PreMath]

Thank you too You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@justabunga1]

There is a third method to figure out this problem, which is to use De Moivre's theorem (cis(x))^n=cos(nx)+isin(nx) where cis(x) is cos(x)+isin(x) and n is an integer and convert everything in polar form in the form re^(iθ)=cis(θ). We need to find when cos(x)=1/2 and sin(x)=√(3)/2. We know that the first equation is x= π/3+2πk and 5π/3+2πk and second equation is x=π/3+2πk and 2π/3+2πk where k is an integer. There Both equations have one answer in common, which is x=π/3+2πk where k is an integer, but we'll take x to be π/3 since all other angles are going to be the same when plugging the value from that question there. The value here in that question there can be converted as x=(cis(π/3))^7777. This equals to cos(7777π/3)+isin(7777π/3). In polar form, r=√((cos(7777π/3))^2+(sin(7777π/3))^2)=1 and θ=arctan(sin(7777π/3)/cos(7777π/3))=arctan(tan(7777π/3))=π/3. This would be x=e^((π/3)i). Raise both sides by 3, which is x^3=e^(πi)=-1. Therefore x^3=-1, or x=-1.

[@dan-florinchereches4892]

Why would you consider 5Pi/3? Since both terms of x are positive we can only accept first Quadrant angles. Sine of 5pi/3 is going to be negative, but valid as it is the conjugate of the given value for this specific problem. I would just verify the modulus of the number, the angle being 1/3pi and then divide 7777 by 6 to see how many Complete revolutions we got around the unity circle

[@justabunga1]

@@dan-florinchereches4892 there is a typo I made so far.

[@nerdherd281]

Great question again.

[@PreMath]

Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@SurajSingh-nx7yj]

Thanks Guruji 🙏

[@PreMath]

So nice of you, Suraj dear Keep shining like Suraj! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@mathsplus01]

Thanks Sir

[@TurquoizeGoldscraper]

I looked at x and realized it was e^(i * 1/3 * Pi), which leads to x^6 = e^(i * 2 * Pi) = 1. x ^ 7777 = x ^ (6 * 1296) * x ^ 1 = 1 ^ 1296 * x = x.

[@PreMath]

Thanks for sharing! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@janekfortepianek]

Great video as always

[@PreMath]

Glad you enjoyed Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@mathbynisharsir5586]

Amazing sir.

[@neowong7224]

What about Euler's formula? e^ix ＝ cosx+isinx. x＝3/pi cos(7777*3/pi)＝cos(2*pi*1296+3/pi)＝cos(3/pi), sine the same

[@명진-x4r]

awesome! thank you👍😄

[@seegeeaye]

A shortcut by the De Moivre Theorem: (cos(pai/3) + sin(pai/3))^7777 = cos(2592pai + pai/3) + sin(2592pai + pai/3) ) = 1/2 + ( i/2 ) sqrt 3

[@beeruawana6662]

Very nice 🙏🙏

[@PreMath]

Thank you! Cheers! You are awesome, Beeru. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@floxie8914]

How do I learn to do stuff like this myself. Im quite amazed.

[@Math-625]

Sir plz giving this type of question

[@msafasharhan]

Hello sir how to solve thissqr

[@69420guyhaha]

heres the problem i remember theres a thing called bodmas in math (i remember idk if its wrong) its short of base, order, division, multification (correct my spelling aaaaaaaaaaaaaaaaaa), addition, and subtraction now heres the problem, since it should be addition then subtraction, then it should be (a-b)^2 = (a^2-2ab)+b^2, not a^2-2ab+(b^2)

[@pranavamali05]

thnku

[@PreMath]

You are very welcome! Thank you! Cheers! You are awesome, Pranav. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@anoopshivhare7223]

Nice sir

[@PreMath]

So nice of you, Anoop Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@parthmadaan3399]

Simple if u apply omega cube root of unity it will be solved in 2 steps

[@sivaprasadrapeti8466]

👏👏👏👏👌👍

[@mva286]

It can easily shown that x^3 = -1. This leaadas to x^7776 = (x^3)^2592 = (-1)2592 = 1. Thus: x^7777 = x * x^7776 = x * 1 = x = (1 + i * sqrt 3)/2

[@Sspatel235]

Simple,Use De Moivre's formula.

[@matt-fitzpatrick]

I saw the huge exponent, and guessed the powers of x would be cyclical. (Good guess!) So I squared x, just to explore what happens. Then squared it again, and saw x^4 = -x. (Oops, I shouldn't have skipped x^3!) But I didn't think to reduce x^4 = -x to x^3 = -1. So I eventually got to x^7777 = x, but it was ugly. Your solutions were much nicer!

[@anirudhya110]

In our engineering entrance examination, you will get such problems to be resolved by less than 30secs.

[@AryanY_jee]

You can solve it easily by de moiver theorem

[@alcibiadesmarcialneto922]

X^3=-1 => x=-1 is that wrong? If not x^7777=-1.

[@natmirmira8329]

X=exp(i*pi/3) will yield the answer in 2 steps!

[@Denish_Uprety]

easy

[@PreMath]

Great!

[@jasimmathsandphysics]

I converted to polar form

[@PreMath]

Great, Jasim

[@xroonos3151]

Это очень просто. -1/2+i*sqrt(3)/2. Модуль этого числа равен 1, так что при перемножении этот вектор крутится по кругу с шагом в 60 градусов. 7777=1296*6(6*60=360)+1. Так что будет всего лишь 1 перемножение этого числа на само себя или данное число в квадрате. (1/2+i*sqrt(3)/2)^(6*N)=(1/2+i*sqrt(3)/2).