Mathematical Introduction: Quick introduction to complex numbers

Required materials

• paper and pencil

How to calculate with complex numbers

A complex number can be written as ${\displaystyle z=x+iy}$, where ${\displaystyle x}$ and ${\displaystyle y}$ are real numbers and ${\displaystyle i}$ satisfies ${\displaystyle i^2=-<!--\; -\; -->1}$. ${\displaystyle x}$ is called the real part of the complex number and ${\displaystyle y}$ the imaginary part. This may be written as: ${\displaystyle \mathrm{Re}<!--\; \; -->z=x}$ and ${\displaystyle \mathrm{Im}<!--\; \; -->z=y}$.

The basic calculation rules for complex numbers are:

1. Addition and subtraction: ${\displaystyle (x_1+i{y}_1)\pm <!--\; \pm \; -->(x_2+i{y}_2)=(x_1\pm <!--\; \pm \; -->x_2)(y_1\pm <!--\; \pm \; -->y_2)}$
2. Multiplication: ${\displaystyle (x_1+i{y}_1)(x_2+i{y}_2)=x_1{x}_2-<!--\; -\; -->y_1{y}_2+i(x_1{y}_2+x_2{y}_1)}$
3. Modulus (magnitude): ${\displaystyle \left|z\right|=\sqrt{x^2+y^2}}$

Example:

Let us do some calculations with the two complex numbers ${\displaystyle z_1=1+2i}$ and ${\displaystyle z_2=3-4i}$.
 

1. Addition and subtraction
   Calculate ${\displaystyle z_1+z_2}$ and ${\displaystyle z_1-z_2}$.
   Solution:
   ${\displaystyle z_1+z_2=(1+2i)+(3-<!--\; -\; -->4i)=4-<!--\; -\; -->2i}$
   ${\displaystyle z_1-<!--\; -\; -->z_2=(1+2i)-<!--\; -\; -->(3-<!--\; -\; -->4i)=-<!--\; -\; -->2+6i}$
2. Multiplication
   Calculate ${\displaystyle z_1{z}_2}$.
   Solution: ${\displaystyle z_1{z}_2=(1+2i)(3-<!--\; -\; -->4i)=11+i}$
3. Modulus
   Calculate ${\displaystyle \left|z_1\right|}$ and ${\displaystyle \left|z_2\right|}$.
   Solution: ${\displaystyle \left|z_1\right|=\sqrt{5}}$ and ${\displaystyle \left|z_2\right|=5}$

Complex numbers to determine probabilities

In quantum computing, you will often come across the following equation:
${\displaystyle |\mathrm{\psi <!--\; \psi \; -->}⟩=\mathrm{\alpha <!--\; \alpha \; -->}|0⟩+\mathrm{\beta <!--\; \beta \; -->}|1⟩}$.

This equation describes the superposition of the two basis states of a qubit. Qubits are the quantum bits with which a quantum computer carries out its calculations. In this lesson, we will treat this equation as a black box and just mention what each symbol means – for you and your students to get used to these terms and concepts. So, don’t be afraid if you don’t understand a thing at this point.

${\displaystyle \mathrm{\psi <!--\; \psi \; -->}}$ describes the properties of the whole system (in this case, the superposition of the two basis states of a qubit). The ket notation ${\displaystyle |⟩}$ may be read as “the state of” (see “Matrices and how they can be used in quantum computing” [Link]). ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$ are the two possible states a qubit can take.

Example: When flipping a coin, the two states the coin can take are “Head” or “Tail”.

And finally, ${\displaystyle \mathrm{\alpha <!--\; \alpha \; -->}}$ and ${\displaystyle \mathrm{\beta <!--\; \beta \; -->}}$ are two complex numbers called amplitudes. The squares of the modulus of these amplitudes, ${\displaystyle {\left|\mathrm{\alpha <!--\; \alpha \; -->}\right|}^2}$ and ${\displaystyle {\left|\mathrm{\beta <!--\; \beta \; -->}\right|}^2}$, represent the probabilities of finding the system in the state ${\displaystyle |0⟩}$ or in the state ${\displaystyle |1⟩}$, respectively, when performing a measurement.

Example: When flipping a coin (performing a measurement), there is a 50:50 chance that the coin lands on “Head”, and a 50:50 chance that it lands on “Tail”. In this case, the above equation can be written as:
${\displaystyle |\text{coin that is flipped}⟩=\frac{1}{\sqrt{2}}|\text{Head}⟩+\frac{1}{\sqrt{2}}|\text{Tail}⟩}$.

In this case, the amplitudes are real numbers. If you square them, you obtain ½, meaning that the probability of getting “Head” is ½ and the probability of getting “Tail” is ½ as well.

In general, the amplitudes have both a real and an imaginary part, in order to describe special behaviours observed in quantum physics.

Task for students: Quiz

You can download the quiz for students here:

• Quiz as docx
• Quiz as pdf

Below you find the same questions in an interactive format.
Use this link to send the interactive H5P task to your students.