Complex Numbers
Complex number expression being integer — smallest n
MJAT_TS5_P2
Grade 12
Question:
Let $z=10+n^{1/4}i$ be a complex number ($n\in\mathbb{N}$). For the expression $z+(\text{Re}(z)+\text{Im}(z))(\text{Re}(z)-\text{Im}(z))$ to be an integer, the smallest positive integral value of $n$ equals $k$. The smallest digit in $k$ is:
Step-by-Step Solution
Key Concept: $z=10+n^{1/4}i$. $\text{Re}(z)=10$, $\text{Im}(z)=n^{1/4}$. Expression $=10+n^{1/4}i+(10+n^{1/4})(10-n^{1/4})=10+n^{1/4}i+(100-n^{1/2})=(110-\sqrt{n})+n^{1/4}i$. For this to be an integer: imaginary part $=n^{1/4}=0$ (impossible for $n\in\mathbb{N}$) OR the expression is interpreted as modulus or something else.
$k=1$, smallest digit $=\mathbf{1}$.
Correct Answer: 1