Animated Solution for Mathematics - Complex Numbers: Two number $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1}+i^{k_2}$, $(i=\sqrt{-1})$ is non-zero, equals

The imaginary unit $i=\sqrt{-1}$ follows a distinct cyclic pattern. By calculating the powers, we observe: $i^1=i$, $i^2=-1$, $i^3=-i$, and $i^4=1$.

This cycle repeats indefinitely, meaning for any natural number $k$, the value of $i^k$ is restricted to the set $\{i,-1,-i,1\}$. Geometrically, these points represent a rotation on the complex plane, forming a square inscribed in the unit circle.

We are tasked with finding the probability that the sum $S=i^{k_1}+i^{k_2}$ is non-zero. Rather than calculating all non-zero outcomes, we utilize the complement rule.

The sum is zero when:

Geometrically, this condition implies that the two points must be diagonally opposite on the unit circle. The pairs that satisfy this condition are $(i,-i)$ and $(1,-1)$.

Since $k_1$ and $k_2$ are chosen randomly, each of the four values $\{i,-1,-i,1\}$ has a probability of $\frac{1}{4}$ of being selected. For any fixed value of $i^{k_1}$, there are four equally likely possibilities for $i^{k_2}$.

Out of these four possibilities, exactly one will result in the additive inverse of $i^{k_1}$. Therefore, the probability that the sum is zero is:

Using the complement rule, the probability that the sum is non-zero is given by:

Substituting our calculated value:

The final probability is $\frac{3}{4}$. By identifying the cyclic symmetry and applying the complement rule, we have efficiently navigated the problem to reach the solution.