- #1

Tsunoyukami

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When using mathematical induction to show that a statement is true we first show that it is true for the lowest value (usually 0 or 1) and then assume it holds true for some value k > 0 (or 1). Then we show that if the statement is true for the value (k+1) that it is true for all values, correct?So here is a question I must complete for homework from Stephen D. Fisher's

*Complex Variables, 2e*:

"12. Let ##z_1, z_2, ..., z_n## be complex numbers. Establish the following formulas by mathematical induction:

a) ##|z_1 z_2 ... z_n| = |z_1| |z_2| ... |z_n|##" (Section 1.1, page 9).

Here is my attempt at a solution:First we show that this statement is true for n = 1 (which is obvious) - we find:

##|z_1| = |z_1|##

Next we assume that this statement holds true for some k > 1; that is that the following is true:

##|z_1 z_2 ... z_k| = |z_1| |z_2| ... |z_k|##

Lastly we must show that for some value, (k+1), the statement ##|z_1 z_2 ... z_n| = |z_1| |z_2| ... |z_n|## is true. So:

##|z_1 z_2 ... z_k z_{k+1}| = |z_1| |z_2| ... |z_k| |z_{k+1}| (1)##

To me this seems obvious is I can write:

##|z_1 z_2 ... z_k z_{k+1}| = |z_1 z_2 ... z_k| |z_{k+1}| (2)##

##|z_1 z_2 ... z_k| |z_{k+1}| = |z_1| |z_2| ... |z_k| |z_{k+1}| (3)##

By using the assumption that it holds true for k > 1. Is this a valid step to make? Can I go from (1) to (2), from which point (3) readily follows? Or am I missing something?

Any assistance is much appreciated; please provide me confirmation or a hint to lead me in the right direction - thanks!