Prove by induction that:
I seem to understand the components of this problem as I feel comfortable with the inductive proof process and I can solve for the modulus of complex numbers, e.g., if then . I also see that the two shorter sides of a triangle must be at least the length of the longest side and, if they are not parallel with it, then their sum must be greater than the longest side etc. However, I am not sure how to start with this inductive proof. How do I test the base case if z represents a complex number and I don't know the formula for it? How do I sub in a value for it?
How do I start? where is the gap in my understanding of these topics.
I am a little unsure of the notation I guess; how do I treat the expressions? Do I let them equal to or something? I don't see how I could test the base case if they are just variables and they are non negative numbers (natural).
I wasn't talking about your notation. I meant the terms. If they are just symbols then squaring both sides would give on both sides so then they are equal and the base case has held true? I don't know what they represent.
I can see that it should be true as when is positive and is negative, then the LHS would be the absolute value of the difference and the RHS would still be the sum of the two absolute values and therefore the RHS would be greater. Obviously if both terms are positive then they are equal. But, again, I don't know what the terms represent in this example.
Are you proving for complex numbers or not?
If is complex then it is neither positive nor negative
If these are real numbers the proof is easy.
If they are complex then the base case is difficult.
To avoid subscripts suppose that are complex.
. Expand that.