MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

De-Moivre's Theorem

If 1, α1, α...

Question

If $1, α_{1}, α_{2}, α_{3}, . . . . ., α_{n - 1}$ are the nth roots of unity then show that $n = (1 - α_{1}) (1 - α_{2}) (1 - α_{3}) . . . . . (1 - α_{n - 1})$ .

Open in App

Solution

Given n, nth roots of unity are
$1, α_{1}, α_{2}, α_{3}, . . . . ., α_{n - 1}$
Let $x = (1)^{1 / n}$
$\Rightarrow x^{n} - 1 = 0$
$\Rightarrow x^{n} - 1 = (x - 1) (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . . (x - α_{n - 1})$
$\Rightarrow \frac{x^{n} - 1}{x - 1} = (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . (x - α_{n - 1})$
Taking $lim x \to 1$ on both sides, we have
$lim x \to 1 \frac{x^{n} - 1}{x - 1} = lim x \to 1 (x - α_{1}) (x - α_{2}) (x - α_{3}) . . . . . (x - α_{n - 1})$
$\Rightarrow n = (1 - α_{1}) (1 - α_{2}) (1 - α_{3}) . . . . (1 - α_{n - 1})$

Suggest Corrections

thumbs-up

0

Similar questions

1, α_{1}, α_{2}, α_{3}, . . ., α_{n - 1}

are the nth roots of unity, then the value of

1. α_{1} . α_{2} . α_{3} . . . α_{n - 1}

1, α_{2}, α_{3} . . . . . . . α_{n - 1}

n^{t h}

roots of unity then evaluate the following :-

(3 + α_{1}) (3 + α_{2}) . . . . . . (3 + α_{n - 1})

1, α_{1}, α_{2}, α_{3}

are the fourth roots of unity, then the value of

(1 + α_{1}) (1 + α_{2}) (1 + α_{3})

α_{1}

α_{2}

α_{n - 1}

are the nth roots of unity then

(1 - α_{1}) (1 - α_{2}) . . . (1 - α_{n - 1}) =

If 1, $α$ , $α^{2}$ , $α^{3}$ ,............, $α^{n - 1}$ are roots of unity.What is the value of If 1 $\times$ $α$ $\times$ $α^{2}$ $\times$ $α^{3}$ , $\times$ ............, $α^{n - 1}$ .

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

De-Moivre's Theorem

MATHEMATICS

Watch in App

explore_more_icon

Explore more

De-Moivre's Theorem

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon