Properties of the modulus of the complex numbers

5.3. Properties of the modulus
      5.3.1 Proof of the properties of the modulus

5.3. Properties of the modulus

Triangle Inequality: Proof 1. |z1 + z2||z1| + |z2| Proof 2. |z1 + z2||z1| - |z2| Proof 3. |z1 - z2||z1| - |z2| Proof

4. |z1 + z2 + z3||z1| + |z2| + |z3| Proof

5. |z1z2| = |z1||z2| Proof

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5.3.1 Proof of the properties of the modulus

Proof of the Triangle Inequality #1:

1. |z1 + z2||z1| + |z2|

|z1 + z2|= .
|z1| + |z2|= .

We have to prove that

is true.
Square both sides.

.

2x1x2 +2y1y2

Square both sides again.
2x1x2y1y2 x1²y2² + y1²x2² and we get
0(y1x2 - x1y2)².
It is true because x1, x2, y1, y2 are all real.

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Proof of the Triangle Inequality #2:

2. |z1 + z2||z1| - |z2|

We have to prove that

is true.
Square both sides.

- .

2x1x2 +2y1y2 -.

Multiply both sides by (-1/2).

-(x1x2 +y1y2) .

Square both sides.
- x1²x2² - 2x1x2y1y2 - y1²y2² x1²x2² + x1²y2² + y1²x2²+ y1²y2² and we get
0 (y1x2 + x1y2)² + 2x1²x2² + 2y1²y2².
It is true because x1, x2, y1, y2 are all real, and squares of real numbers are 0.

Proof of the Triangle Inequality #3:

3. |z1 - z2||z1| - |z2|

We have to prove that

is true.
Square both sides.

- .

-2x1x2 -2y1y2 -.

Multiply both sides by (-1/2).

(x1x2 +y1y2) .

Square both sides again.
2x1x2y1y2 x1²y2² + y1²x2² and we get
0(y1x2 - x1y2)².
It is true because x1, x2, y1, y2 are all real.

4. |z1 + z2 + z3||z1| + |z2| + |z3|

Proof: By the triangle inequality,
|z1 + (z2+z3)||z1| + |z2+z3||z1| + |z2| + |z3|

5. |z1z2| = |z1||z2|

Proof: |z1z2| = |(x1+y1i)(x2+y2i)| =
= = |z1||z2|.

by Tetyana Butler