A nice #shorts trick for complex numbers.

#shorts

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[זאבגלברד]

In this case squaring 1+i gives 2i ... So we can solve it even quicker ... Thank you for so many great contant.

[rajk]

Just do (1+i)^2 = 2i and raise it to 10th = 2^10 . i^10 = -1024

[bensosfrequents]

I remember this in my complex analysis class

[johnloony]

I don’t know where the first line came from. I don’t know where the second line came from. I just did it by “add the arguments, multiply the moduli”. An angle of 45 degrees, multiplied by 20, equals 180 degrees (a negative real number with no imaginary part); a modulus of root2 ^ 20 equals 1024.

[ryurai]

"haha i like your funny words, magic man"

[davidseed]

the first step is pulled out of the air. and is complicated.
Clearer i think to just convert
1+i into polar form (r, θ). use the argand diagram to explain convertion.
(r, θ)=(sqrt2, π/4)
r=sqrt2 so r^20=1024
θ=π/4, so exp(iθ)^20 = exp(i5π)= exp(iπ)=-1
Αns=-1024

[vladimirrodriguez]

Great resolution. Keep your math challenges, they are very didactic and instructive

[nuclo]

Michael please a video on groups actions thats mindblowing

[mahajan]

We can also do (1+I)²)^10 than it will simply become 2i raise to power 10 ans i raise to power 10 is -1 and it will be -2 ^10

[timseytiger]

And there was me thinking this was going to be a compound interest question.

[alejrandom]

(1+i)^n forms a nice spiral of isoceles right triangles

[mrminer]

Michael Penn doing one breath videos . . . NICE!
Also, now we know at least one 20th root of -1024, a fact which will amuse people at parties.

[inigolarraza]

You just transform the complex number into its polar form, which allows to easily calculate any power or root of any complex number

[cielblue]

Here is the solution those who don't learn de Moivre's theorem or Euler formula .

(1+i)^2 = 2i ==> (1+i)^{20} = (2i)^{10} = 2^{10} ＊ i^{10} = 2^{10} ＊( i^2)^5 = 2^{10} ＊ (-1)^5 = - 1024 ^^
2^{10} = 1024 should be memorized !

[kairostimeYT]

People keep mentioning that we could square it but squaring it is literally (sqrt(2))^2*exp(2i*pi/4).

Basically the same thing. One involves algebra in argand plane whereas other involves algebra in polar plot, I'd say.

[yroldgosu-gosu-sin]

Why does this seem too complicated? 1 + i becomes (Sqrt(2)e^(i*pi/4))^20.

Then Sqrt(2)^20 becomes 2^10 and e^(i*pi/4)*20 becomes e^(5pi*i) becomes -1 and you have -2^10

[Aseronios]

I would just make the 20 power 2×10 and calculate (1+i)^2 and then calculate the 10 power:
[(1+i)^2]^10=(2i)^10=2^10 × -1=-2^10

[keypo]

hehe I solved it in my head without algebra because of demoivre's formula

[changjeffreysinto]

Literally the only trick for taking complex numbers to a power

[johnny_eth]

In short, always convert your complex numbers to polar form when doing power operations.