Two numbers \(k_1\) and \(k_2\) are randomly chosen from the set of natural numbers. Then, the probability that the value of \(i^{k_1} + i^{k_2}\) (where \(i = \sqrt{-1}\)) is non-zero equals:
Show Hint
Understanding the cyclic properties of complex numbers can significantly simplify probability calculations involving their powers.
Step 1: Understand the cyclic nature of \(i\). Note that \(i\) cycles every four powers: \(i^1=i, i^2=-1, i^3=-i, i^4=1\). Therefore, \(i^k\) repeats every four values. Step 2: Identify zero-sum pairs. Find pairs \((k_1, k_2)\) such that \(i^{k_1} + i^{k_2} = 0\). This occurs when \(i^{k_1}\) and \(i^{k_2}\) are additive inverses (e.g., \(i\) and \(-i\), \(1\) and \(-1\)). Step 3: Calculate the probability of a zero-sum pair. Given the four-cycle nature of \(i\), the probability of obtaining a zero-sum pair is \( \frac{1}{4} \) (as each value has exactly one additive inverse within the cycle). Step 4: Determine the probability of a non-zero sum. If the probability of a zero-sum is \( \frac{1}{4} \), then the probability of a non-zero sum is \( 1 - \frac{1}{4} = \frac{3}{4} \). Conclusion: The probability that the sum \(i^{k_1} + i^{k_2}\) is non-zero is \( \frac{3}{4} \).
