Operations with complex numbers in polar form
In binomial form, multiplying two complex numbers meant expanding four products. In polar form the same operation comes down to multiplying two numbers and adding two angles. That is why polar form is always the one used when you need to raise to a high power or find roots.
Product and quotient: moduli and arguments separately
Given the complex numbers $z_1 = 4_{30^\circ}$ and $z_2 = 3_{60^\circ}$, calculate their product $z_1 \cdot z_2$, expressing the result in polar form.
Given the complex numbers $z_1 = 4_{30^\circ}$ and $z_2 = 3_{60^\circ}$, calculate their product $z_1 \cdot z_2$, expressing the result in polar form.
Calculate the quotient $\frac{12_{150^\circ}}{3_{45^\circ}}$ and express the result in polar form.
Calculate the quotient $\frac{12_{150^\circ}}{3_{45^\circ}}$ and express the result in polar form.
Given the complex number $z = 3_{50^\circ}$, express its conjugate $\bar{z}$ and its opposite $-z$ in polar form.
Given the complex number $z = 3_{50^\circ}$, express its conjugate $\bar{z}$ and its opposite $-z$ in polar form.
De Moivre's formula: powers in polar form
Raising to a power means multiplying a number by itself several times. If you apply the product rule $n$ times, the modulus is multiplied by itself $n$ times (it becomes $r^n$) and the argument is added to itself $n$ times (it becomes $n\alpha$).
Using de Moivre's formula, calculate the power $(2_{40^\circ})^3$ and express the result in polar form.
Using de Moivre's formula, calculate the power $(2_{40^\circ})^3$ and express the result in polar form.
Calculate the value of $(1 - \sqrt{3}i)^5$. To do this, first transform the base into polar form, apply de Moivre's formula and give the final result in binomial form.
Calculate the value of $(1 - \sqrt{3}i)^5$. To do this, first transform the base into polar form, apply de Moivre's formula and give the final result in binomial form.
The inverse: reciprocal modulus, opposite argument
Find the multiplicative inverse of the complex number $z = 5_{120^\circ}$, expressing the final result in polar form.
Find the multiplicative inverse of the complex number $z = 5_{120^\circ}$, expressing the final result in polar form.
nth roots: $n$ solutions arranged in a polygon
An equation such as $x^2=4$ has two real solutions, $\pm2$. With complex numbers, $z^n = w$ has exactly $n$ solutions, and their affixes are always arranged symmetrically, forming the vertices of a regular polygon with $n$ sides centred at the origin.
Calculate the two square roots of the complex number $16_{120^\circ}$ and express the results in polar form.
Calculate the two square roots of the complex number $16_{120^\circ}$ and express the results in polar form.
Solve the equation $z^3 = -8i$. First convert the number $-8i$ to polar form before calculating the roots, and express the final results in both polar form and binomial…
Solve the equation $z^3 = -8i$. First convert the number $-8i$ to polar form before calculating the roots, and express the final results in both polar form and binomial form.
Solve the equation $z^4 + 81 = 0$ in the set of complex numbers. Express the solutions in polar form and represent them graphically in the complex plane (real axis…
Solve the equation $z^4 + 81 = 0$ in the set of complex numbers. Express the solutions in polar form and represent them graphically in the complex plane (real axis horizontal and imaginary axis vertical).