z = 2 + i: find real number d such that z + d/z is real

[oshea.emma]

HI THERE!
Have an interesting question for you that I can't seem to figure out.

We have the complex number z=2+i where i squared equals -1,
we're asked to find the REAL number d such that z+d/z (that's d over z) is real!

Is this solveable? Can anyone give me any pointers? THXXXXXXXXXX
:wink:

 

[stapel]

You say that you haven't been able to figure this out, which implies that you have made an attempt. What have you tried? How far did you get?

For instance, you would have started by plugging in "2 + i" for "z" in the expression "z + (d/z)" and simplified, rationalizing the denominator. Then you would have taken the imaginary part and, since the expression is supposed to be real, set this imaginary part equal to zero. What did that give you? What did you conclude?

When you reply, please include topics of recent study, as this will assist the tutors in providing hints that you will understand.

Thank you.

Eliz.

 

[pka]

We have the complex number z=2+i, where i squared equals -1, we're asked to find the REAL number d such that z+d/z (that's d over z) is real!

Thank you, thank you: "where i squared equals -1" that is the way it should be done.

\(\displaystyle \L
\begin{array}{l}
\frac{1}{z} = \frac{{\overline z }}{{|z|^2 }}\quad \Rightarrow \quad \frac{1}{{2 + i}} = \frac{{2 - i}}{5} \\
z + \frac{d}{z} = \left( {2 + i} \right) + \frac{{\left( {2 - i} \right)d}}{5} = \left( {2 + \frac{{2d}}{5}} \right) + i\left( {1 - \frac{d}{5}} \right) \\
\end{array}\)

What d makes the imaginary part equal 0?

 

[oshea.emma]

Hi! thanks for your answers!!
I wasn't sure what the 1/z formula was all about.
I solved it by getting a demonimator and then simplified it using conjurgates (this worked as i^2 equals -1) I then got 10+2d over 5 + (5-d over 5)i. Putting the i part equal to 0 gives me 5 for d.

The only thing i don't really understand is why do we put the i part equal to 0?

 

[pka]

The only thing i don't really understand is why do we put the i part equal to 0?

Any complex number is of the form \(\displaystyle z = a + bi\) and we say that a is the real part, \(\displaystyle Re(z) = a\), of z and b is the imaginary part, \(\displaystyle Im(z) = b\).
If the \(\displaystyle Im(z) = 0\) then z is a real number.