- #1
Hijaz Aslam
- 66
- 1
I am bit confused with the Eueler representation of Complex Numbers.
For instance, we say that [tex]e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1[/tex].
The derivation of [tex]e^{i\theta}=cos(\theta)+isin(\theta)[/tex] is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using taylor series. But what is the mathematical meaning of ##e^{i\pi}##? How can a constant (##e##) be raised to an 'entity' like ##i=\sqrt{-1}##?
This problem started to concern me when I tried the following out.
A theorem states that : [tex]|z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1\bar{z_2})=|z_1|^2+|z_2|^2+2|z_1||z_2|cos(\theta_1-\theta_2)[/tex]
But I tried solving this out using the Euler number like: [tex]|z_1+z_2|^2=|(z_1+z_2)^2|=|(r_1e^{i\theta_1}+r_2e^{i\theta_2})^2|= |r_1e^{i\theta_1}|^2+|r_2e^{i\theta_2}|^2+|2r_1r_2e^{i(\theta_1+\theta_2)}|=r_1+r_2+2r_1r_2=|z_1|^2+|z_2|^2+2|z_1||z_2|[/tex]
I know that am seriously wrong somewhere. Can I follow out the "complex" algebra of 'complex numbers' by using Euler's form in simple algebra?
For instance, we say that [tex]e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1[/tex].
The derivation of [tex]e^{i\theta}=cos(\theta)+isin(\theta)[/tex] is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using taylor series. But what is the mathematical meaning of ##e^{i\pi}##? How can a constant (##e##) be raised to an 'entity' like ##i=\sqrt{-1}##?
This problem started to concern me when I tried the following out.
A theorem states that : [tex]|z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1\bar{z_2})=|z_1|^2+|z_2|^2+2|z_1||z_2|cos(\theta_1-\theta_2)[/tex]
But I tried solving this out using the Euler number like: [tex]|z_1+z_2|^2=|(z_1+z_2)^2|=|(r_1e^{i\theta_1}+r_2e^{i\theta_2})^2|= |r_1e^{i\theta_1}|^2+|r_2e^{i\theta_2}|^2+|2r_1r_2e^{i(\theta_1+\theta_2)}|=r_1+r_2+2r_1r_2=|z_1|^2+|z_2|^2+2|z_1||z_2|[/tex]
I know that am seriously wrong somewhere. Can I follow out the "complex" algebra of 'complex numbers' by using Euler's form in simple algebra?
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