# Find the principal argument and exponential form of z=i/(1+i) If z=x+iy is a complex number

Complex numbers
Find the principal argument and exponential form of $$\displaystyle{z}=\frac{{i}}{{{1}+{i}}}$$ If z=x+iy is a complex number

2020-11-02

Rewrite the above complex number in the usual form z=x+iy as follows.
$$\displaystyle{z}=\frac{{i}}{{{1}+{i}}}=\frac{{i}}{{{1}+{i}}}\cdot\frac{{{1}-{i}}}{{{1}-{i}}}$$ [multiply and divide by the conjugate 1−i of 1+i]
$$\displaystyle=\frac{{{i}{\left({1}-{i}\right)}}}{{{\left({1}+{i}\right)}{\left({1}-{i}\right)}}}=\frac{{{i}-{i}^{{2}}}}{{{1}^{{2}}-{i}^{{2}}}}={\left({i}+{1}\right)}{\left({1}+{1}\right)}=\frac{{{i}+{1}}}{{2}}=\frac{{1}}{{2}}+\frac{{1}}{{2}}{i}$$
Compare the complex number $$\displaystyle{z}=\frac{{1}}{{2}}+\frac{{1}}{{2}}{i}$$ with z=x+iy and obtain $$\displaystyle{x}=\frac{{1}}{{2}}$$ and $$\displaystyle{y}=\frac{{1}}{{2}}.$$
The argument of a complex number z=x+iy is given by $$\theta =\tan^{-1}(\frac{y}{x})$$

Here, for $$z=\frac{1}{2}+\frac{1}{2i}$$

$$\displaystyle\theta-{{\tan}^{ -{{1}}}{\left(\frac{{\frac{1}{{2}}}}{{\frac{1}{{2}}}}\right)}}={{\tan}^{ -{{1}}}{\left({1}\right)}}=\frac{\pi}{{4}}$$

Since both $$\displaystyle{x}=\frac{1}{{2}}$$ and $$\displaystyle{y}=\frac{1}{{2}}$$ are positive, the complex number $$\displaystyle{z}=\frac{1}{{2}}+\frac{1}{{2}}{i}$$ lie in the 1st quadrant.

Hence, its principal argument is the same as $$\displaystyle\theta=\frac{\pi}{{4}}$$

The exponential form of a complex number z=x+iy with principal argument theta is given by $$\displaystyle{z}={r}{e}^{{{i}\theta}}$$ where $$\displaystyle{r}=\sqrt{{{x}^{2}+{y}^{2}}}$$

For $$\displaystyle{z}=\frac{1}{{2}}+\frac{1}{{2}}{i}$$ the value $$\displaystyle{r}=\sqrt{{{x}^{2}+{y}^{2}}}=\sqrt{{{\left(\frac{1}{{2}}\right)}^{2}+{\left(\frac{1}{{2}}\right)}^{2}}}=\sqrt{{\frac{1}{{2}}}}=\frac{1}{\sqrt{{2}}}$$

Hence, the exponential form of $$\displaystyle{z}=\frac{1}{{2}}+\frac{1}{{2}}{i}$$ is $$\displaystyle{z}=\frac{1}{{2}}+\frac{1}{{2}}{i}$$ is $$\displaystyle{z}=\frac{1}{\sqrt{{2}}}{e}^{{{i}{\left(\frac{\pi}{{4}}\right)}}}$$

Thus, the principal argument of $$\displaystyle{z}=\frac{i}{{{1}+{i}}}$$ is $$\displaystyle{0}=\frac{\pi}{{4}}$$ and the exponential form is $$\displaystyle{z}\frac{1}{\sqrt{{2}}}{e}^{{{i}{\left(\frac{\pi}{{4}}\right)}}}$$