Express the following complex numbers in the standard form a + ib : 2+3i/4+5i

Question

Express the following complex numbers in the standard form a + ib : \(\frac{2+3i}{4+5i}\)

Answer 1

Given:

⇒ a +ib =  \(\frac{2+3i}{4+5i}\)

Multiplying and dividing with 4-5i

⇒ a +ib =  \(\frac{2+3i}{4+5i}\) x  \(\frac{4-5i}{4-5i}\)

⇒ a +ib =  \(\frac{2(4-5i)+3i(4-5i)}{(4)^2-(5i)^2}\)

⇒ a +ib = \(\frac{8-10i+12i-15i^2}{16-25i^2}\) 

We know that i²=-1

⇒ a +ib = \(\frac{8+2i-15(-1)}{16-25(-1)}\) 

⇒ a +ib =  \(\frac{23+2i}{41}\) 

∴ The values of a, b are \(\frac{23}{41}\) , \(\frac{2}{41}\) .