Question 38350
<pre><font size = 4><b>Express the following as a complex number in the form a + ib:

The complex conjugate of i<sup>4n-3</sup>, with n <font face = "symbol">Î</font> Z

By definition we know that i<sup>2</sup> = -1

We know that i<sup>4n</sup> = i<sup>2·2·n</sup> = ((i<sup>2</sup>)<sup>2</sup>)<sup>n</sup> = ((-1)<sup>2</sup>)<sup>n</sup> = (1)<sup>n</sup> = 1

Since i<sup>4</sup> = 1

i<sup>4n-3</sup> = i<sup>4n-3</sup>i<sup>4</sup> = i<sup>4n-3+4</sup> = i<sup>4n+1</sup> = i<sup>4n</sup>i<sup>1</sup> = 1·i = i

Now we want the conjugate of i

The conjugate of A + Bi is A - Bi and vice-versa

We write i as 0 + i, so its conjugate is 0 - i, or -i, so

Answer = -i

Edwin
AnlytcPhil@aol.com</pre>