Temporalización: 7 horas

• 1. What is a complex number?

  2. Cartesian complex plane. Arithmetic of complex numbers.

  3. Polar complex plane. Operations with complex numbers in polar form.

  4. Roots of complex numbers.

  5. Graphic descriptions using complex numbers.

Introduction

Find out the solution of the following equations in the set they are given:

• x + 3 = 2 in N.

• 2x + 5 = 6 in Z.

• x² = 5 in Q.

• x² + 1 = 0 in R.

What is √-4 · √-9?

What is a complex number?

• i =√−1 ⇒ i² = −1. i is the imaginary unit. Then, √−4 = √4√−1 = 2√−1 = 2i.

• A complex number is a number z = a + bi, where a and b are real numbers. a is the real part of the complex number and b is the imaginary part of the complex number.

• Any complex number can be represented in a plane by this identification: a + bi = (a, b). 

• From now and on, any polonomical equation has as many solutions as the degree of it.

The Complex Plane

Division

That operation is made with the help of the conjugate of a complex number:

The conjugate of c + di is c − di. To make any division, you just first multiply by the conjugate of the denominator. Look what happen when you multiply a number by its conjugate: 

(a + bi)/(c + di) = (a + bi)(c − di)/(c + di)(c − di) ...

Activities

1. Represent the following complex numbers: 4, 5 - 3i, 1/2 + (5/4)i, -5i, √3i, 0, -1 -i.

2.  Solve the following equations

   1. z²+4=0

   2. z²+6z+10=0

   3. 3z²+27=0

   4. 3z²-27=0

3. Make three sums, three subtractions and three multiplications involving all the numbers in exercise 1. Try not to use 0 more than twice.

4. Worksheet on operations with complex numbers.

Polar Complex Plane

z = a + bi ⇒ r = 

Activities

1. Write in polar form: a) 1+√3i; b) √3+i; c) −1+i; d) 5 −12i; e) 3i; f) −5

2. Write in standard form: a) 530º ; b) 2135º ; c) 2495º ; d) 3240º ; e) 5180º ; f) 490º .

3. Given the following complex numbers Dados los complejos z1 = 2270º ; z2 = 4120º ; z3 = 3315º  ; calculate: a)  z1 · z2 ; b) z2 · z3 ; c) z1 · z3 ; d) z3 : z1 ; e) z2 : z1 ; f) (z1 · z3) / z2 ; g) (z1)² ; h) (z2)³ ; i)  (z3)⁴ .

Answer key (contains two errors): 

1. a) 260º ; b) 230º ; c) 2135º ; d) 13292.5º ; e) 390º ; f) 5180º

2. a) (5√3/2)+(5/2)i; b) −√2+√2i; c) −√2+√2i; d) (3/2) − (3√3/2)i; e) −5; f) 4i

3. a)  830º  ; b) 1275º ; c) 6225º ; d) 1.545º ; e) 2210º ; f) 1.5105º ; g) 4180º ; h) 640º ; i)  81180

Real life use of complex numbers

Complex numbers have a wide range of applications in various fields of science and engineering. Here is a list of some of the most notable applications:

1. Electronics and Electrical Engineering:

• Analysis of alternating current (AC) circuits.

• Design and analysis of electronic filters.

• Signal and system analysis.

2. Physics:

• Quantum mechanics.

• Fluid dynamics.

• Electromagnetism.

•  [Mecánica cuántica](https://es.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica)

•    - [Dinámica de fluidos](https://es.wikipedia.org/wiki/Din%C3%A1mica_de_fluidos)

•    - [Electromagnetismo](https://es.wikipedia.org/wiki/Electromagnetismo)

• 
  

3. Mathematics:

• Theory of differential equations.

• Analysis of series and sequences.

• Fourier and Laplace transforms.

•  [Teoría de ecuaciones diferenciales](https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_diferencial)

•    - [Análisis de series y secuencias](https://es.wikipedia.org/wiki/Serie_(matem%C3%A1tica))

•    - [Transformadas de Fourier y Laplace](

4. Civil and Mechanical Engineering:

• Analysis of vibrations and resonance.

• Study of forces and moments in structures.

5. Computer Graphics:

• Representation and manipulation of 2D and 3D graphics.

• Image processing and computer vision.

6. Control and Automation:

• Design of PID controllers.

• Stability analysis of control systems.

7. Information Theory and Communications:

• Modulation and demodulation of signals.

• Signal processing and digital communications.

8. Number Theory:

• Solving equations in the complex field.

• Analysis of functions of a complex variable.

9. Economics and Finance:

• Modeling complex economic systems.

• Time series analysis and market prediction.

Complex numbers are powerful tools that simplify and solve complex problems in these and other areas. 

Wrap up

Perform this activity to review all the contents of this unit.

Introductory problem

What is (sqrt(-4))(sqrt(-9))? a) 6 b) -6 c) 6i d) -6i

Links

• Números complejos en Geogebra.

• The most beautiful equation in Math.

• Euler's identity.

• When did imaginary numbers come out?.

• Complex numbers in Math is fun.

• The Mandelbrot set and the complex numbers.

• Fractals and the art of roughness (TED talk by Brenoit Mandelbrot).

• How to Plot the Mandelbrot Set By Hand.

• Magia vectorial.

How will the exam of this unit be?

It'll be a multiple choice test in which you will have to prove having the following skills:

• Simplifying expressions with negative roots.

• Solve any quadratic equation in ℂ.

• Basic operations with complex numbers in standard form (addition, substraction, multiplication, division and "small" powers).

• Drawing complex numbers in standard form and polar form.

• Identifying real and imaginary part of a complex number written or represented in any form.

• Calculating the modulus of a complex number.

• Converting complex numbers polar↔standard.

• Operations with complex numbers in polar form: multiplication, division and powers.

Saberes básicos implicados en la unidad

A. Sentido numérico.

MATE.1.A.2. Relaciones.

MATE.1.A.2.1 Conjunto de números: números racionales e irracionales. Los números reales. Logaritmos decimales y neperianos. Los números complejos como soluciones de ecuaciones polinómicas que carecen de raíces reales.

Criterios de evaluación

1.1. Manejar algunas estrategias y herramientas, incluidas las digitales, en la modelización y resolución de problemas de la vida cotidiana y de la ciencia y la tecnología, evaluando su eficiencia en cada caso.

2.2. Seleccionar la solución más adecuada de un problema en función del contexto -de sostenibilidad, de consumo responsable, equidad, etc., usando el razonamiento y la argumentación.