# 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.

### 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