A complex number can be represented by an expression of the form \(a+bi\) where \(a\) and \(b\) are real numbers and \(i\) is a symbol with the property that \(i^{2}=-1\).

The real part of the complex number \(a+bi\) is the real number \(a\) and the imaginary part is the real number \(b\).

Example : The real part of \(3+5i\) is the real number \(3\) and the imaginary part is the real number \(5\).

Equality

Let a, b, c and d be real numbers.

\(a+bi=c+di\)	\(\longleftrightarrow\)	\(a=c,\ \ \ b=d\)
\(a+bi=0\)	\(\longleftrightarrow\)	\(a=0, \ \ \ b=0\)

4 Laws

Let a, b, c and d be real numbers.

Sum	\((a+bi)+(c+di)=(a+c)+(b+d)i\)
Difference	\((a+bi)-(c+di)=(a-c)+(b-d)i\)
Product	\((a+bi)\cdot(c+di)=(ac-bd)+(ad+bc)i\)
Division	\(\frac{a+bi}{c+di}=\frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i\)　 where \(c+di\not= 0\)

Example : \(\frac{2+3i}{1+i}=\frac{(2+3i)(1-i)}{(1+i)(1-i)}=\frac{5+i}{2}\)

Conjugate

For the complex number \(z=a+bi\), we define its complex conjugate to be \(\overline{z}=a-bi\). The geometric interpretation of the complex conjugate is that \(\overline{z}\) is the reflection of \(z\) in the real axis.

Example : If \(z=2+3i\), then \(\overline{z}=2-3i\)

Sum	\(\overline{z+\omega}=\overline{z}+\overline{\omega}\)
Difference	\(\overline{z-\omega}=\overline{z}-\overline{\omega}\)
Product	\(\overline{z\cdot \omega}=\overline{z}\cdot \overline{\omega}\)
Division	\(\overline{\frac{ z }{\omega}}=\frac{\overline{z}}{\overline{\omega}}\)

Real / Complex Parts

Let \(z=a+bi\) be a complex number. Then the real part of \(z=a+bi\) is the real number \(a\) and the imaginary part is the real number \(b\).

Real Part of \(z\)	\(a=\frac{z+\overline{z}}{2}\)
Complex Part of \(z\)	\(b=\frac{z-\overline{z}}{2i}\)

Example : If \(z=2+3i\), then the real part of \(z\) is \(2=\frac{z+\overline{z}}{2}=\frac{(2+3i)+(2-3i)}{2}\).

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If the complex number \(z=a+bi\) is a real number or a pure complex number, they satisfy the following conditions:

\(z\) is a real number.	\(\longleftrightarrow\)	\(\overline{z}=z\)
\(z\) is a pure complex number.	\(\longleftrightarrow\)	\(\overline{z}=-z\not= 0\)

Example : \(z=3i\) is a pure complex number. Thus, we have \(\overline{z}=-3i=-z\)

Square root of negative numbers

Since \(i^{2}=-1\), one of square roots of \(-1\) is \(i\). However, we also have \((-i)^{2}=i^{2}=-1\) and so \(-i\) is also a square root of \(-1\). We say that \(i\) is the principal square root of \(-1\) and write \(\sqrt{-1}=i\).

In general, if \(a>0\), we write \(\sqrt{-a}=\sqrt{a} i \).

Example: \(\sqrt{-3}=\sqrt{3}i, \ \ \ \sqrt{-25}=5i \)

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With this convention, formula for the roots of the quadratic equation \(ax^{2}+bx+c\) are valid even when \(b^{2}-4ac<0\).

Example: Find the roots of the equation \(x^{2}+3x+3=0\).

By the quadratic formula, we have

$$x=\frac{-3\pm\sqrt{(-3)^{2}-4\cdot 1\cdot 3}}{2}=\frac{-3\pm\sqrt{-3}}{2}=\frac{-3\pm\sqrt{3}i}{2}$$

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