IB Math HL – Complex Numbers
1. Complex Numbers, Equality of complex numbers – IB Mathematics HL
For what values of x and y, the following complex numbers are equal
and
Solution
Two complex numbers are equal if their corresponding real parts and imaginary parts are equal.
Therefore, and
and
2. Complex Numbers, Operations with complex numbers – IB Mathematics HL
How can we find given that
and
Solution
The sum (or the difference) of two complex numbers is the complex number which its real part is made up of the sum (or the difference) of their real parts and its imaginary parts are made up of the sum (or the difference) of their imaginary parts.
Therefore,
The multiplication is performed as usual and using the fact that we have the following:
and
3. Complex Numbers, Division of Complex Numbers – IB Mathematics HL
How can we find given that
and
Solution
When dividing two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator such as the product in the denominator to become a real number.
Thus, about this question we have that
4. Complex Numbers, Cartesian to polar form – IB Mathematics HL
How can we express the complex number in polar form?
Solution
The argument of a complex number
is given by the formula:
and the modulus
and the polar form will be
in this question
and the modulus is given by the following formula:
5. Complex Numbers, Cartesian to polar form – IB Mathematics HL
How can we express the complex number in polar form?
Solution
In this case the argument of the complex number
is
and the modulus is
So, the complex number can be written in polar form as follows:
6. Complex Numbers, the Cartesian form of Complex Numbers – IB Maths HL
How can we express the following complex number in polar form?
Solution
Complex Numbers, the Cartesian form of Complex Numbers – IB Maths HL
7. De Moivre’s Theorem – IB Mathematics HL
How can we find the following complex number
using De Moivre’s Theorem?
Solution
De Moivre’s Theorem – IB Math HL
The De Moivre’s theorem is given by the following formula
Before apply De Moivre’s theorem we must convert the Cartesian form to polar form
Therefore,
8. Complex Numbers, Quadratic over the complex field – IB Mathematics HL
How can we solve the following quadratic equation over the complex field?
SOL
The Discriminant (Δ) of the quadratic equation will be
So, the two roots will be given by the following formula