Given the reverse triangle inequality, mod(a+b+c) is greater than or equal to mod(a)- mod(b) - mod(c).

Suppose a, b, c are complex numbers such that mod(a)=4, mod(b)=10 and mod(c)=1. What is the smallest possible value that mod(a+b+c) can attain? Justify your answer.

I know the answer is 5, and that it relates to b's dominance over the over numbers but am struggling to explain how this is the case?

Could someone please shine some light on this for me please?

Thank you

  • $\begingroup$ What is mod(...)? $\endgroup$ – hmakholm left over Monica Oct 16 '15 at 15:57
  • $\begingroup$ The absolute value of... Apologies for my notation, I'm completely new to this! $\endgroup$ – B.Marsden Oct 16 '15 at 15:58
  • $\begingroup$ Actually, mod is the complex modulus... $\endgroup$ – Michael Burr Oct 16 '15 at 15:58

Step 1: $|a+b+c|\geq|b|-|a+c|$ by the reverse triangle inequality.

Step 2: $|a+c|\leq |a|+|c|$ by the triangle inequality.

Step 3: $|a+b+c|\geq|b|-|a+c|\geq|b|-|a|-|c|=10-4-1=5$.

Step 4: Find an instance where the equality is attained.

  • $\begingroup$ Thank you very much Michael, much appreciated! $\endgroup$ – B.Marsden Oct 16 '15 at 16:02

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