1.2 Q-10

Exercise 1.2 — Question 10

Problem Statement

If $z=a+bi$, find $\mathrm{Re}(iz)$ and $\mathrm{Im}(iz)$. Also state the condition for $z$ to be purely real in terms of its conjugate $\bar{z}$.

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Part (i): Finding $\mathrm{Re}(iz)$ and $\mathrm{Im}(iz)$

Given $z=a+bi$, multiply by $i$:

$iz=i(a+bi)=ai+b{i}^2$

Since $i^2=-1$:

$iz=ai+b(-1)=-b+ai$

This is now in standard form $x+yi$ where $x=-b$ and $y=a$. Therefore:

$\mathrm{Re}(iz)=-b=-\mathrm{Im}(z)$

$\mathrm{Im}(iz)=a=\mathrm{Re}(z)$

Conclusion: Multiplying a complex number by $i$ swaps its real and imaginary parts, negating the new real part.

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Part (ii): Condition for $z$ to be Purely Real

A complex number $z=a+bi$ is purely real when its imaginary part equals zero, i.e., $b=0$.

Recall the conjugate: $\bar{z}=a-bi$.

Now compare $z$ and $\bar{z}$:

$z=\bar{z}⟺a+bi=a-bi⟺2bi=0⟺b=0$

Therefore: $z$ is purely real $⟺$ $z=\bar{z}$.

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Key Results