# Thread: write the expression in standard form

4/(5+i)

2. ## Re: write the expression in standard form

Are you familiar with the conjugate of a complex number? The conjugate of $\displaystyle a+bi$ is $\displaystyle a-bi$. Multiply top and bottom of your expression by the conjugate of the denominator. Multiplying any complex number by its conjugate will give a real number, so you are guaranteed to get a real number in the denominator.

3. ## Re: write the expression in standard form

Originally Posted by asilvester635
4/(5+i)
Here is a very useful fact: $\displaystyle z\ne 0,~~\frac{1}{z} = \frac{{\overline z }}{{{{\left| z \right|}^2}}}$

4. ## Re: write the expression in standard form

If all else fails, here's how it's done:

Spoiler:
4/(5 + i) x (5 - i)/(5 - i) -------- (Multiplying top and bottom by conjugate)
= 4(5 - i)/((5 + i)(5 - i))
= (20 - 4i)/(25 - i2) -------- (We know that i2 is equal to -1, therefore 25 - (-1) = 26)
= (20 - 4i)/26
= (10 - 2i)/13 -------- (Dividing throughout by 2)
= (10/13) - (2/13)i -------- (Rearranging it into standard form, i.e., a + bi)

Apologies for the formatting.

5. ## Re: write the expression in standard form

Hi Plato,
I don't understand why this is a useful fact.
(I don't know if this is relevant but the original post posed no problem for me)
Thanks.

6. ## Re: write the expression in standard form

Originally Posted by Melody2
I don't understand why this is a useful fact.
(I don't know if this is relevant but the original post posed no problem for me)
It gives a one step solution:
$\displaystyle \frac{4}{5+i}=\frac{20}{26}-\frac{4}{26}i$

7. ## Re: write the expression in standard form

I'm sorry but I still don't understand what you have done.
Maybe I don't understand the notation.
I really want to understand.
Thanks

8. ## Re: write the expression in standard form

Originally Posted by Melody2
I'm sorry but I still don't understand what you have done.
Maybe I don't understand the notation.
I really want to understand.
$\displaystyle \frac{1}{a+bi}=\frac{\overline{a+bi}}{a^2+b^2}= \frac {a}{a^2+b^2}-\frac{b}{a^2+b^2}i$

$\displaystyle \overline{z}$ is almost the universal notation for the conjugate of $\displaystyle z$.

9. ## Re: write the expression in standard form

Thanks Plato,

It was $\displaystyle {{\left| z \right|}^2}$ that threw me.

So if z=5+6i then $\displaystyle {{\left| z \right|}^2}$ = $\displaystyle {5^2+6^2}$

I assume that this is also standard notation? Or is there some logic to it that I am missing?

This is the first time I have tried to use Latex. I copied yours. I am just a little pleased with myself.

Thanks.

10. ## Re: write the expression in standard form

Originally Posted by Melody2
It was $\displaystyle {{\left| z \right|}^2}$ that threw me.
So if z=5+6i then $\displaystyle {{\left| z \right|}^2}$ = $\displaystyle {5^2+6^2}$
I assume that this is also standard notation? Or is there some logic to it that I am missing?
$\displaystyle |a+bi|=\sqrt{a^2+b^2}$ therefore $\displaystyle |a+bi|^2=a^2+b^2=(a+bi)(a-bi)=(a+bi)\overline{(a+bi)}$.