Express the following complex numbers in the standard form a + ib : \

1.	Express the following complex numbers in the standard form a + ib : \(\frac{2+3i}{4+5i}\)
Answer» 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 \(\frac{23}{41}\) , \(\frac{2}{41}\) .

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

Answer»

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 \(\frac{23}{41}\) , \(\frac{2}{41}\) .