Kalkylator för komplexa tal
Utför operationer med komplexa tal steg för steg
Kalkylatorn försöker förenkla vilket komplext uttryck som helst och visar stegen. Den utför addition, subtraktion, multiplikation, division och upphöjning till potens samt bestämmer den polära formen, komplexkonjugatet, modulen och inversen för det komplexa talet.
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 $$$1$$$