Two numbers k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1+ik2, (i=−1) is non-zero, equals
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Step-by-Step Solution
Step 1: Understand powers of i
The powers of i follow a cycle of four values: i, −1, −i, and 1. This cycle repeats for higher powers. We can determine the value of ik by finding the remainder when k is divided by 4.
Step 2: Analyze the sum ik1+ik2
The sum ik1+ik2 will be zero if and only if ik1 and ik2 are additive inverses of each other. This means they must be equal in magnitude but opposite in sign.
Step 3: Identify conditions for the sum to be zero
For the sum to be zero, the values of ik1 and ik2 must be opposite. There are four such pairs: (i, -i), (-i, i), (1,−1), and (−1,1). Each pair corresponds to specific remainders when k1 and k2 are divided by 4.
Step 4: Calculate the probability of the sum being zero
Since k1 and k2 are randomly chosen natural numbers, the probability that k leaves any specific remainder when divided by 4 (0, 1, 2, or 3) is 1/4. We sum the probabilities of the four cases where the sum is zero.
Step 5: Calculate the probability of the sum being non-zero
The probability that the sum is non-zero is the complement of the probability that the sum is zero. We subtract the probability of the sum being zero from 1.