Learn the fundamentals of polynomials & complex numbers for absolute beginners (Polar form, Euler’s form, De Moivre’s)
Just like everything in this universe is made up of atoms which in turn is made up of sub-atomic structures like electrons, protons and neutrons, everything in the mathematical universe is made up of complex numbers. In that sense, complex numbers are the most fundamental elements of the mathematical universe. Everything we know of numbers since elementary school also applies to complex numbers. But everything we know of complex numbers doesn’t apply to real numbers. This is because, complex numbers consists of imaginary numbers (that we do not know of) in addition to real numbers (that we know everything of). So, learning complex numbers is fun. We revisit everything we know of and generalize the result and understanding so that it applies all the numbers i.e. real and imaginary.
What you’ll learn
- Conceptual understanding of why we need complex numbers.
- Solving a polynomial equation.
- Undertake Operations of complex numbers like Addition, Multiplication & Division.
- Find zeros, roots and factors of a polynomial function.
- Graphically understand the dynamics of polynomial equation.
- Complex Conjugates.
- Understand the Polar form of a complex number.
- Euler’s Form.
- De Moivre’s Theorem.
- Roots of Complex Numbers.
- Cube Roots of Unity.
Course Content
- Introduction to complex numbers and polynomials –> 2 lectures • 12min.
- Definition, Operations & Equality –> 4 lectures • 29min.
- Complex Conjugates & Properties –> 3 lectures • 18min.
- Real Polynomials & Properties –> 3 lectures • 21min.
- Zeros, Roots & Factors –> 2 lectures • 13min.
- Polynomial Theorems –> 2 lectures • 13min.
- Graphing Real Functions –> 4 lectures • 13min.
- Modulus of a Complex Number –> 2 lectures • 14min.
- Argument & Polar Form –> 4 lectures • 18min.
- Solved Problems –> 2 lectures • 20min.
- Euler’s Form –> 2 lectures • 7min.
- De Moivre’s Theorem –> 2 lectures • 9min.
- Application of De Moivre’s Theorem –> 3 lectures • 18min.
Requirements
Just like everything in this universe is made up of atoms which in turn is made up of sub-atomic structures like electrons, protons and neutrons, everything in the mathematical universe is made up of complex numbers. In that sense, complex numbers are the most fundamental elements of the mathematical universe. Everything we know of numbers since elementary school also applies to complex numbers. But everything we know of complex numbers doesn’t apply to real numbers. This is because, complex numbers consists of imaginary numbers (that we do not know of) in addition to real numbers (that we know everything of). So, learning complex numbers is fun. We revisit everything we know of and generalize the result and understanding so that it applies all the numbers i.e. real and imaginary.
Any given function has at least a solution. This is a mathematical way of saying that the graph of the function cuts the X-axis at least once. A simple way to put it is ‘the number of times a function cuts the X-axis, the same number of Xs will satisfy the given equation. When a given X satisfies a given equation, it is called the root of the equation. But it turns out that there are functions which do not seem to cut X-axis for any value of X. So for those functions we do not have a root or solution. It was insulting for the mathematicians to know that a function may not have a solution. To get around this, they invented complex numbers. Now with the introduction of complex numbers, even those functions which had no solution, seem to have a solution. But this solution is called a complex solution because it is in the form of a complex number (which consists of real and imaginary parts). Although a complex number consists of imaginary part (and therefore not only real part), the notion that there is a complex solution is actually contributing to newer real avenues of discovery and research. This is the beauty of complex number.
After the completion of the course, you will also see and do problems involving how an imaginary number raised to a power of another imaginary number also might lead to a real number.
If you want to know the fact of this matter, please enroll in the course where we have covered everything there is to know on the topic.