1.2 Real and Imaginary Parts — Operations, Conjugate, Modulus

Equality of Complex Numbers

Two complex numbers $z_1=a+bi$ and $z_2=c+di$ are equal if and only if: $a=c\text{and}b=d$ That is, their real parts are equal and their imaginary parts are equal.

Example: If $z_1=(x+2)+(y-1)i=5+3i=z_2$, then $x+2=5\Rightarrow x=3$ and $y-1=3\Rightarrow y=4$.

---

Basic Operations on Complex Numbers

Let $z_1=a+bi$ and $z_2=c+di$.

Effect of multiplying by $i$: $iz=i(a+bi)=ai+b{i}^2=-b+ai$ So $\mathrm{Re}(iz)=-\mathrm{Im}(z)$ and $\mathrm{Im}(iz)=\mathrm{Re}(z)$.

Reciprocal of a complex number: $\frac{1}{a+bi}=\frac{1}{a+bi}\cdot \frac{a-bi}{a-bi}=\frac{a-bi}{a^2+b^2}$ So $\mathrm{Re}\left(\frac{1}{z}\right)=\frac{a}{a^2+b^2}$ and $\mathrm{Im}\left(\frac{1}{z}\right)=\frac{-b}{a^2+b^2}$.

---

Complex Conjugate

The complex conjugate of $z=a+bi$ is defined as: $\bar{z}=a-bi$

Key properties:

• $z+\bar{z}=2a=2\mathrm{Re}(z)$ (always real)
• $z-\bar{z}=2bi=2i\mathrm{Im}(z)$ (always imaginary)
• $z\cdot \bar{z}=a^2+b^2=|z{|}^2$ (always a non-negative real)
• $z$ is purely real $⟺z=\bar{z}$ (i.e., $b=0$)
• $z$ is purely imaginary $⟺z=-\bar{z}$ (i.e., $a=0$)

---

Modulus (Absolute Value) of a Complex Number

The modulus of $z=a+bi$ is defined as: $|z|=\sqrt{a^2+b^2}$

This represents the distance of the point $(a,b)$ from the origin in the Argand plane.

Key properties of modulus:

• $|z{|}^2=z\bar{z}$
• $|z_1{z}_2|=|z_1|\cdot |z_2|$
• $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$, $z_2\ne 0$
• Triangle Inequality: $|z_1+z_2|\le |z_1|+|z_2|$

Example: If $|z_1{z}_2|=10$ and $z_1=2+i$, find $|z_2|$. $|z_1|=\sqrt{4+1}=\sqrt{5},|z_2|=\frac{10}{\sqrt{5}}=2\sqrt{5}$

---

Simultaneous Linear Equations with Complex Coefficients

To solve a system of linear equations where coefficients are complex numbers, apply the equality condition: equate real and imaginary parts separately after simplification.

Method:

1. Write the system with complex unknowns (or separate into real/imaginary unknowns).
2. Use elimination or substitution to reduce to one unknown.
3. Multiply through by conjugates where needed to simplify.
4. Equate real and imaginary parts to extract the solution.

Example: Solve for $z$ and $w$: $z+iw=2+3i$ $iz-w=1-i$

From equation 1: $z=2+3i-iw$. Substitute into equation 2: $i(2+3i-iw)-w=1-i$ $2i+3i^2-i^{2}w-w=1-i$ $2i-3+w-w=1-i$

Solve step by step equating real and imaginary parts to find $z$ and $w$.

---