Calculatrice de nombres complexes
Effectuez des opérations sur les nombres complexes étape par étape
La calculatrice essaiera de simplifier toute expression complexe, en affichant les étapes. Elle effectuera l'addition, la soustraction, la multiplication, la division, l'élévation à une puissance, et déterminera également la forme polaire, le conjugué, le module et l'inverse du nombre complexe.
Solution
Your input: simplify and calculate different forms of $$$i$$$
The expression is already simplified.
Polar form
For a complex number $$$a+bi$$$, polar form is given by $$$r(\cos(\theta)+i \sin(\theta))$$$, where $$$r=\sqrt{a^2+b^2}$$$ and $$$\theta=\operatorname{atan}\left(\frac{b}{a}\right)$$$
We have that $$$a=0$$$ and $$$b=1$$$
Thus, $$$r=\sqrt{\left(0\right)^2+\left(1\right)^2}=1$$$
Also, $$$\theta=\operatorname{atan}\left(\frac{1}{0}\right)=\frac{\pi}{2}$$$
Therefore, $$$i=\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}$$$
Inverse
The inverse of $$$i$$$ is $$$\frac{1}{i}$$$
Multiply and divide by $$$i$$$ (keep in mind that $$$i^2=-1$$$):
$$${\color{red}{\left(\frac{1}{i}\right)}}={\color{red}{\left(- i\right)}}$$$
Hence, $$$\frac{1}{i}=- i$$$
Conjugate
The conjugate of $$$a + i b$$$ is $$$a - i b$$$: the conjugate of $$$i$$$ is $$$- i$$$
Modulus
The modulus of $$$a + i b$$$ is $$$\sqrt{a^{2} + b^{2}}$$$: the modulus of $$$i$$$ is $$$1$$$
Answer
$$$i=i=1.0 i$$$
The polar form of $$$i$$$ is $$$\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}$$$
The inverse of $$$i$$$ is $$$\frac{1}{i}=- i=- 1.0 i$$$
The conjugate of $$$i$$$ is $$$- i=- 1.0 i$$$
The modulus of $$$i$$$ is $$$1$$$