Find \\(\\operatorname{Im}(8-3i)\\).

Find \\(\\operatorname{Im}(-4+11i)\\).

Find \\(\\operatorname{Im}(6i)\\).

Find \\(\\operatorname{Im}(9)\\), treating \\(9\\) as a complex number.

Find \\(\\operatorname{Im}((2+3i)+(5-7i))\\).

Find \\(\\operatorname{Im}((1+i)(4-2i))\\).

Find \\(\\operatorname{Im}(5(2-9i))\\).

Find \\(\\operatorname{Im}(4e^{i\\pi/2})\\).

Find \\(\\operatorname{Im}(z+w)\\) if \\(z=7-2i\\) and \\(w=-10+5i\\).

Find \\(\\operatorname{Im}\\left(\frac{1}{1-i}\right)\\).

Verify \\(\\operatorname{Im}(z)=\frac{z-z^*}{2i}\\) for \\(z=-6+5i\\).

Explain why \\(\\operatorname{Im}(z+w)=\\operatorname{Im}(z)+\\operatorname{Im}(w)\\).

Find real \\(t\\) if \\(\\operatorname{Im}((t+2)+(3t-1)i)=8\\).

Find \\(t\\) if \\(\\operatorname{Im}((t+i)(2-i))=5\\).

For which real \\(t\\) is \\(\\operatorname{Im}((t+2i)^2)=8\\)?

For which real \\(t\\) does \\(\\operatorname{Im}\\left(\frac{t+i}{1+i}\right)=0\\)?

Find all real \\(t\\) such that \\(\\operatorname{Im}((t+i)(t-3i))=0\\).

A learner says \\(\\operatorname{Im}(2+5i)=5i\\). Diagnose the error.

Prove that \\(\\operatorname{Im}(cz)=c\\operatorname{Im}(z)\\) for real \\(c\\).

Give a counterexample to \\(\\operatorname{Im}(z_1z_2)=\\operatorname{Im}(z_1)\\operatorname{Im}(z_2)\\).