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What is property of modulas. |
| Answer» Proof of the properties of the modulusProof of the Triangle Inequality #1:1. |z1 + z2|<=|z1| + |z2||z1 + z2|= square 1. |z1| + |z2|= square 2.We have to prove that square 1square 2 is true. Square both sides. equation square 3.2x1x2 +2y1y2 <= square 4Square both sides again. 2x1x2y1y2 <= x12y22 + y12x22 and we get0<=(y1x2 - x1y2)2.It is true because x1, x2, y1, y2 are all real.Proof of the Triangle Inequality #2:2. |z1 + z2|>=|z1| - |z2|We have to prove that square 1square 5 is true. Square both sides. >= - square 4.2x1x2 +2y1y2 >=-square 4.Multiply both sides by (-1/2).-(x1x2 +y1y2) <=square 1.Square both sides. - x12x22 - 2x1x2y1y2 - y12y22<= x12x22 + x12y22 + y12x22+ y12y22 and we get0<= (y1x2 + x1y2)2 + 2x12x22 + 2y12y22.It is true because x1, x2, y1, y2 are all real, and squares of real numbers are >=0. | |