Mathematics Database Programming Web Design Price List     Complex analysis studies the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of math of real numbers may break down when applied to complex numbers. Free Lessons Lesson 1 Complex numbers In this lesson: Forms and representations of the complex numbers; Modulus and arguments; Principal value of the argument. Lesson 2 Trigonometric and algebraic form conversion In this lesson: Complex numbers forms conversion; Examples of the conversion. Lesson 3 The algebra of complex numbers In this lesson: Arithmetic operations with complex numbers; Properties of the complex numbers; Geometric interpretation of addition & subtraction. Lesson 4 Geometric interpretation of multiplication In this lesson: The modulus and argument of the product; Multiplication of complex numbers as stretching - squeezing and rotation; Multiplying a complex number by imaginary unit i and by powers of i. Lesson 5 Division of the complex numbers In this lesson: Definition and notation of conjugates and reciprocals; Division as multiplication and reciprocation. Lesson 6 Powers and roots of complex numbers In this lesson: De Moire's theorem; Powers of complex numbers; n-th root of complex numbers. Lesson 7 Complex Exponential Function and Complex Logarithm Function In this lesson: Definition and notation; Complex logarithm function is a multi-valued function; Principal branch of the logarithm. Lesson 8 Complex Trigonometric Functions and Complex Inverse Trigonometric Functions In this lesson: A difference between the real and complex trigonometric functions; Relationship to exponential function; Identities; Derivatives and Indefinite integrals of inverse trigonometric functions. Lesson 9 Complex Hyperbolic Functions and Inverse Hyperbolic Functions In this lesson: The notations; Definitions; Derivatives and Indefinite integrals of inverse hyperbolic functions. Lesson 10 Complex Power Function In this lesson: Raising a complex number to a complex power; Derivatives and Indefinite integral of complex power function. Lesson 11 Complex Rational Functions In this lesson: Definition of the rational function; Möbius transformations; Fractional-linear function; Zhukovskii function. Properties of the modulus of the complex numbers Complex Analysis. FreeTutorial

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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.
2
x1x2
y1y2 x12y22 + y12x22 and we get
0 (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   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.
2
x1x2
y1y2 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|.

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by Tetyana Butler Top  Possibly the greatest paradox is that mathematics has paradoxes... Bernoulli's sophism Paradox of Bernoulli and Leibniz Paradox of even (odd) and natural numbers Paradox of Hilbert’s hotel Ross-Littlewood paradox Paradox of wizard and mermaid Paradox of enchantress and witch Paradox of Tristram Shandy Barber paradox Achilles and tortoise Contact us