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Week 1: Long Day Complex Numbers in Rectangular Form

The first semester of Advanced Engineering Mathematics 1 is a coherent course in linear algebra and linear differential equations. Complex numbers and elementary complex functions are an important part of the basis for this theory. Therefore the first few weeks of the course are dedicated to the study of complex numbers. Today we will introduce the complex numbers as a new set of numbers belonging to the well-known family of number sets and we will work out how to write a complex number in its so-called rectangular form.

First and foremost it is important to practise computations with complex numbers to become familiar with them so they won’t be dreaded when they show up in future mathematical topics.

Today’s Key Concepts

Complex numbers as ordered pairs of real numbers. The elementary arithmetic operations: addition, subtraction, multiplication and division. The imaginary unit $\,i\,.$ The rectangular form $\,z=a+ib\,.$ Real parts $\,\mathrm{Re}(z)\,$ and imaginary parts $\,\mathrm{Im}(z)\,$. Complex conjugated numbers $\,\bar z\,.$ Absolute value of complex numbers $\,|z|\,.$ The complex number plane and graphical visualization of numbers. The number sets $\,\mathbb N\,,$ $\,\mathbb Z\,,$ $\,\mathbb Q\,,$ $\,\mathbb R\,$ and $\,\mathbb C\,$ (natural, integer, rational, real and complex numbers).

Preparation and Syllabus

Today’s topics are based on eNote 1 Complex Numbers, sections 1.1 to 1.4.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Exercises

  1. The Number i
  2. The Complex Number Plane
  3. Basic Computations
  4. Complex Conjugation
  5. Absolute Values
  6. The Real Criterion
  7. The Magnitudes of Rational Numbers (Advanced)
  8. Ordering of Complex Numbers (Advanced)