The equation x

x

x = (cos 2kπ + i sin 2kπ)

= (cos (2kpi/n) + i sin (2kpi/n)) where k = 0 , 1, 2 , 3 , 4 , ……… , (n-1)

So each root of unity is cos[ (2kπ)/n] + i sin[(2kπ)/n] where 0 ≤ k ≤ n-1.

We know that complex numbers of the form x + iy can be plotted on the complex plane (Argand Diagram). If we compare each root of unity with x + iy , we get x = cos[(2kπ)/n] , y=sin[(2kπ)/n].

Now , we see that the values of x and y satisfy the equation of unit circle with centre (0,0) i.e. x

From above it can be concluded that the roots are located on the circumference of the unit circle with center at (0,0). If we join the points we always get a regular polygon of n sides.

^{n}= 1 has n roots which are called the nth roots of unity.x

^{n}= 1 = cos 0 + i sin 0 = cos 2kπ + i sin 2kπ [ k is an integer]x = (cos 2kπ + i sin 2kπ)

^{1/n}= (cos (2kpi/n) + i sin (2kpi/n)) where k = 0 , 1, 2 , 3 , 4 , ……… , (n-1)

So each root of unity is cos[ (2kπ)/n] + i sin[(2kπ)/n] where 0 ≤ k ≤ n-1.

We know that complex numbers of the form x + iy can be plotted on the complex plane (Argand Diagram). If we compare each root of unity with x + iy , we get x = cos[(2kπ)/n] , y=sin[(2kπ)/n].

Now , we see that the values of x and y satisfy the equation of unit circle with centre (0,0) i.e. x

^{2}+ y^{2}= 1.From above it can be concluded that the roots are located on the circumference of the unit circle with center at (0,0). If we join the points we always get a regular polygon of n sides.

find the special root of x^15-1=0.

ReplyDeletewhat is the sum of nth roots of unity?

ReplyDeletezero

DeleteFor any polynomial equation, the coefficient of x^1 term gives the sum of the roots. Since this is zero for every equation of the form x^n=1, the sum of a nth roots of unity is zero.

Delete*sum of all nth roots of unity is zero.

DeleteThe coefficient of X to the power n-1 divided by coefficient of X to the power n with a negative sign represents sum of roots which is zero.

DeleteThe coefficient of X to the power n-1 divided by coefficient of X to the power n with a negative sign represents sum of roots which is zero.

DeleteYes it is zero for any n

Delete( w^10+w^23) = ?

ReplyDeletewhere w represents omega

-1

DeleteAwesomeeee

ReplyDeleteCan u pls find the polar form of I^27

ReplyDeleteHow can I calculate cos {2*pi()/17}, from x^17 -1 = 0?

ReplyDelete