Operations with complex numbers in binomial form
You already know what a complex number is and where it lives. Now it is time to operate with them: adding, multiplying and dividing. The good news is that there are no new rules to memorise: everything follows from treating $a+bi$ as an ordinary binomial and remembering one extra fact, $i^2=-1$.
Addition and subtraction: real part with real part
Carry out the following addition and subtraction operations with complex numbers: given $z_1 = 3 - 4i$, $z_2 = -1 + 2i$ and $z_3 = 5i$, calculate the result of the…
Carry out the following addition and subtraction operations with complex numbers: given $z_1 = 3 - 4i$, $z_2 = -1 + 2i$ and $z_3 = 5i$, calculate the result of the expression $z_1 - z_2 + z_3$ in Cartesian form.
Given the complex number $z = -7 + 8i$, determine its conjugate $\bar{z}$ and its opposite $-z$. Then, perform the operation $z + \bar{z}$ and check algebraically that…
Given the complex number $z = -7 + 8i$, determine its conjugate $\bar{z}$ and its opposite $-z$. Then, perform the operation $z + \bar{z}$ and check algebraically that the result is a real number.
Multiplication: like a binomial, with $i^2=-1$
Multiply the following complex numbers expressed in Cartesian form and simplify the result by grouping the real part and the imaginary part: $(2 + 5i)(-3 + i)$.
Multiply the following complex numbers expressed in Cartesian form and simplify the result by grouping the real part and the imaginary part: $(2 + 5i)(-3 + i)$.
Calculate the value of the polynomial $P(z) = z^2 - 4z + 5$ when evaluated at the complex number $z = 2 + i$.
Calculate the value of the polynomial $P(z) = z^2 - 4z + 5$ when evaluated at the complex number $z = 2 + i$.
Division: multiply by the conjugate of the denominator
The problem with dividing complex numbers is that you do not know what it means to "get rid of the $i$ on the bottom". The solution reuses something you already know from rationalising surds: multiply top and bottom by the conjugate, so that the denominator becomes real.
Perform the following division of complex numbers by multiplying the numerator and denominator by the conjugate of the denominator. Express the final result in the form…
Perform the following division of complex numbers by multiplying the numerator and denominator by the conjugate of the denominator. Express the final result in the form $a + bi$:
$\frac{4 - 3i}{1 + 2i}$
Determine the value of the real parameter $k$ so that the result of the following quotient is a pure real number (that is, its imaginary part is zero):…
Determine the value of the real parameter $k$ so that the result of the following quotient is a pure real number (that is, its imaginary part is zero):
$\frac{k + 2i}{3 - 4i}$
Quadratic equations with complex solutions
When the discriminant $b^2-4ac$ of $ax^2+bx+c=0$ is negative, the formula asks you for the square root of a negative number: you no longer have to say "there is no real solution"; you take that root using $i$ and the problem is solved in $\mathbb{C}$.
Determine the values of the real numbers $x$ and $y$ such that the following equality between complex numbers holds: $(x + 2) + (y - 3)i = 5 + 4i$
Determine the values of the real numbers $x$ and $y$ such that the following equality between complex numbers holds:
$(x + 2) + (y - 3)i = 5 + 4i$
Solve the following quadratic equation in the set of complex numbers $\mathbb{C}$ and express its two solutions: $x^2 + 25 = 0$
Solve the following quadratic equation in the set of complex numbers $\mathbb{C}$ and express its two solutions:
$x^2 + 25 = 0$
Application: impedance in alternating current
In electronics, the impedance of a component is represented as a complex number $Z=R+Xi$ ($R$ = resistance, $X$ = reactance). Combining components in a circuit means adding, multiplying or dividing their impedances using exactly the rules you have just learnt.
In an alternating current electrical circuit, two components are connected in parallel. Their complex impedances are given by $Z_1 = 4 + 3i \ \Omega$ and…
In an alternating current electrical circuit, two components are connected in parallel. Their complex impedances are given by $Z_1 = 4 + 3i \ \Omega$ and $Z_2 = 2 - i \ \Omega$. Calculate the equivalent impedance of the circuit, expressed in Cartesian form, using the physics formula:
$Z_{eq} = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2}$
Perform the combined operations indicated below, respecting the order of operations, and express the final result in Cartesian form:…
Perform the combined operations indicated below, respecting the order of operations, and express the final result in Cartesian form:
$\frac{(3 - i)(1 + 2i)}{2 + i} - (1 - 3i)$