## An Interactive Introduction to Complex Numbers

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### Complex Conjugates

#### Example 2.1:

Open the Applet Basic Calculations and enter $$z_1 = 1 + 2 \cdot i$$ and $$z_2 = 1.5 \cdot e^{150^\circ i}$$. Then click on $$\bar z_1$$ and $$\bar z_2$$, respectively.

$$\bar z_1 = 1 - 2 \cdot i$$ is called the "complex conjugate of $$z_1 = 1 + 2 \cdot i$$ " and $$\bar z_2 = 1.5 \cdot e^{210^\circ i} = 1.5 \cdot e^{-150^\circ i}$$ is the complex conjugate of $$z_2 = 1.5 \cdot e^{150^\circ i}$$.

#### Rule 2.1 a

The complex conjugate of $$z = a + b \cdot i$$ in cartesian form is $$\bar z = a - b \cdot i$$ with $${\mathop{\rm Re}\nolimits} \left( {\bar z} \right) = {\mathop{\rm Re}\nolimits} \left( z \right)$$ and $${\mathop{\rm Im}\nolimits} \left( {\bar z} \right) = -{\mathop{\rm Im}\nolimits} \left( z \right)$$.

#### Rule 2.1 b

The complex conjugate of $$z = r \cdot e^{\varphi i}$$ in exponential form is $$\bar z = r \cdot e^{ - \varphi i}$$ with $$\left| z \right| = \left| {\bar z} \right|$$ and $$\arg \left( {\bar z} \right) = - \arg \left( z \right)$$.

#### Rule 2.1 c

$$\bar {\bar z} = z$$

Graphically, the complex conjugate $$\bar z$$ mirrors $$z$$ at the real axis and vice versa.

#### Example 2.2:

Open the Applet Basic Calculations and set $$z_1 = 2 + i$$ and $$z_2 = 1 + 2 \cdot i$$. Then click on $$z_1 + z_2$$. Note that both the real parts and the imaginary parts are added up.

#### Rule 2.2

The sum of two complex numbers $$z_1 =a_1 + b_1 \cdot i$$ and $$z_2 =a_2 + b_2 \cdot i$$ in cartesian form is $$z_1 + z_2 = a_1 + b_1 \cdot i + a_2 + b_2 \cdot i = a_1 + a_2 + \left( {b_1 + b_2 } \right) \cdot i$$ with $${\mathop{\rm Re}\nolimits} \left( {z_1 + z_2 } \right) = a_1 + a_2$$ and $${\mathop{\rm Im}\nolimits} \left( {z_1 + z_2 } \right) = b_1 + b_2$$.

There is no general rule for the addition of complex numbers in exponential form.

### Subtraction

#### Example 2.3:

Open the Applet Basic Calculations and set $$z_1 = 3 + 6 \cdot i$$ and $$z_2 = 4 + 2 \cdot i$$. Then click on $$z_1 - z_2$$. Note that both the real parts and the imaginary parts are subtracted.

#### Rule 2.3

The difference between two complex numbers $$z_1 =a_1 + b_1 \cdot i$$ and $$z_2 =a_2 + b_2 \cdot i$$ in cartesian form is $$z_1 - z_2 = a_1 + b_1 \cdot i - a_2 - b_2 \cdot i = a_1 - a_2 + \left( {b_1 - b_2 } \right) \cdot i$$ with $${\mathop{\rm Re}\nolimits} \left( {z_1 - z_2 } \right) = a_1 - a_2$$ and $${\mathop{\rm Im}\nolimits} \left( {z_1 - z_2 } \right) = b_1 - b_2$$.

There is no general rule for the subtraction of complex numbers in exponential form.

### Multiplication

#### Example 2.4 a:

Open the Applet Basic Calculations and set $$z_1 = 1 + 2 \cdot i$$ and $$z_2 = 3 + 4 \cdot i$$. Then click on $$z_1 \cdot z_2$$. Note the multiplication in exponential form first. Obviously, the absolute values are multiplied, whereas the exponents are added up.

#### Rule 2.4 a

The product of two complex numbers $$z_1 = r_1 \cdot e^{\varphi_1 \cdot i }$$ and $$z_2 = r_2 \cdot e^{\varphi_2 \cdot i }$$ in exponential form is $$z_1 \cdot z_2 = r_1 \cdot e^{\varphi_1 \cdot i } \cdot r_2 \cdot e^{\varphi_2 \cdot i } = r_1 \cdot r_2 \cdot e^{\varphi _1 \cdot i + \varphi _2 \cdot i } = r_1 \cdot r_2 \cdot e^{\left( {\varphi _1 + \varphi _2 } \right)\cdot i}$$ with $$\left| {z_1 \cdot z_2 } \right| = r_1 \cdot r_2$$ and $$\arg \left( {z_1 \cdot z_2 } \right) = \varphi _1 + \varphi _2$$.

#### Example 2.4 b:

How would you multiply $$z_1 = 1 + 2 \cdot i$$ and $$z_2 = 3 + 4 \cdot i$$ in cartesian form intuitively? Probably by calculating
$$z_1 \cdot z_2 = \left( {1 + 2 \cdot i} \right) \cdot \left( {3 + 4 \cdot i} \right)$$ $$= 1 \cdot 3 + 1 \cdot 4 \cdot i + 2 \cdot 3 \cdot i + 2 \cdot i \cdot 4 \cdot i =$$ $$1 \cdot 3 + \left( {1 \cdot 4 + 2 \cdot 3} \right) \cdot i + 2 \cdot 4 \cdot i^2$$ $$= 1 \cdot 3 - 2 \cdot 4 + \left( {1 \cdot 4 + 2 \cdot 3} \right) \cdot i$$ $$= - 5 + 10 \cdot i$$.
The applet in example 2.4 a shows that the result is correct.

#### Rule 2.4 b

The product of two complex numbers $$z_1 =a_1 + b_1 \cdot i$$ and $$z_2 =a_2 + b_2 \cdot i$$ in cartesian form is
$$z_1 \cdot z_2 = \left( {a_1 + b_1 \cdot i} \right) \cdot \left( {a_2 + b_2 \cdot i} \right) = a_1 \cdot a_2 + b_1 \cdot a_2 \cdot i + a_1 \cdot b_2 \cdot i + b_1 \cdot b_2 \cdot i^2$$ $$= a_1 \cdot a_2 - b_1 \cdot b_2 + \left( {a_1 \cdot b_2 + a_2 \cdot b_1 } \right) \cdot i$$ with $${\mathop{\rm Re}\nolimits} \left( {z_1 \cdot z_2 } \right) = a_1 \cdot a_2 - b_1 \cdot b_2$$ and $${\mathop{\rm Im}\nolimits} \left( {z_1 \cdot z_2 } \right) = a_1 \cdot b_2 + a_2 \cdot b_1$$.

#### Rule 2.4 c

$$z \cdot \bar z = \left( {a + b \cdot i} \right) \cdot \left( {a - b \cdot i} \right) = a^2 - \left( {b \cdot i} \right)^2 = a^2 - b^2 \cdot \left( { - 1} \right) = a^2 + b^2 = \left| z \right|^2$$
$$z \cdot \bar z = r \cdot e^{\varphi \cdot i} \cdot r \cdot e^{ - \varphi \cdot i} = r^2 \cdot e^{\left( {\varphi - \varphi } \right) \cdot i} = r^2 \cdot e^{0 \cdot i} = r^2 = \left| z \right|^2$$

### Division

#### Example 2.5 a:

Open the Applet Basic Calculations and set $$z_1 = 1 + 2 \cdot i$$ and $$z_2 = 3 + 4 \cdot i$$. Then click on $$z_1 / z_2$$. Note the division in exponential form first. Obviously, the absolute values are divided, whereas the exponents are subtracted.

#### Rule 2.5 a

The quotient of two complex numbers $$z_1 = r_1 \cdot e^{\varphi _1 \cdot i}$$ and $$z_2 = r_2 \cdot e^{\varphi _1 \cdot i}$$ in exponential form is
$$\frac{{z_1 }}{{z_2 }} = \frac{{r_1 \cdot e^{\varphi _1 \cdot i} }}{{r_2 \cdot e^{\varphi _2 \cdot i} }} = \frac{{r_1 }}{{r_2 }} \cdot e^{\varphi _1 \cdot i - \varphi _2 \cdot i} = \frac{{r_1 }}{{r_2 }} \cdot e^{\left( {\varphi _1 - \varphi _2 } \right) \cdot i}$$ with $$\left| {\frac{{z_1 }}{{z_2 }}} \right| = \frac{{r_1 }}{{r_2 }}$$ and $$\arg \left( {\frac{{z_1 }}{{z_2 }}} \right) = \varphi _1 - \varphi _2$$.

#### Example 2.5 b:

Although not quite obvious one can also calculate the quotient of $$z_1 = 1 + 2 \cdot i$$ and $$z_2 = 3 + 4 \cdot i$$ in cartesian form. To get a result, expand the quotient with $$\bar z_2 = 3 - 4 \cdot i$$ and consider rule 2.4 c.
Then $$\frac{{z_1 }}{{z_2 }} = \frac{{1 + 2 \cdot i}}{{3 + 4 \cdot i}} = \frac{{1 + 2 \cdot i}}{{3 + 4 \cdot i}} \cdot \frac{{3 - 4 \cdot i}}{{3 - 4 \cdot i}} = \frac{{3 - 4 \cdot i + 6 \cdot i - 8 \cdot i^2 }}{{9 - 16 \cdot i^2 }} = \frac{{11 + 2 \cdot i}}{{25}} = 0.44 + 0.08 \cdot i$$

#### Rule 2.5 b

The quotient of two complex numbers $$z_1 = a_1 + b_1 \cdot i$$ and $$z_2 = a_2 + b_2 \cdot i$$ in cartesian form is
$$\frac{{z_1 }}{{z_2 }} = \frac{{a_1 + b_1 \cdot i}}{{a_2 + b_2 \cdot i}} = \frac{{\left( {a_1 + b_1 \cdot i} \right) \cdot \left( {a_2 - b_2 \cdot i} \right)}}{{\left( {a_2 + b_2 \cdot i} \right) \cdot \left( {a_2 - b_2 \cdot i} \right)}}$$ $$= \frac{{a_1 \cdot a_2 - a_1 \cdot b_2 \cdot i + a_2 \cdot b_1 \cdot i - b_1 \cdot b_2 \cdot i^2 }}{{a_2^2 - \left( {b_2 \cdot i} \right)^2 }}$$ $$= \frac{{a_1 \cdot a_2 + b_1 \cdot b_2 + \left( {a_2 \cdot b_1 - a_1 \cdot b_2 } \right) \cdot i}}{{a_2^2 + b_2^2 }}$$ $$= \frac{{a_1 \cdot a_2 + b_1 \cdot b_2 }}{{\left| {z_2 } \right|^2 }} + \frac{{a_2 \cdot b_1 - a_1 \cdot b_2 }}{{\left| {z_2 } \right|^2 }} \cdot i\quad a_2 \ne 0 \vee b_2 \ne 0$$
with $${\mathop{\rm Re}\nolimits} \left( {\frac{{z_1 }}{{z_2 }}} \right) = \frac{{a_1 \cdot a_2 + b_1 \cdot b_2 }}{{a_2^2 + b_2^2 }} = \frac{{a_1 \cdot a_2 + b_1 \cdot b_2 }}{{\left| {z_2 } \right|^2 }}$$ and $${\mathop{\rm Im}\nolimits} \left( {\frac{{z_1 }}{{z_2 }}} \right) = \frac{{a_2 \cdot b_1 - a_1 \cdot b_2 }}{{a_2^2 + b_2^2 }} = \frac{{a_2 \cdot b_1 - a_1 \cdot b_2 }}{{\left| {z_2 } \right|^2 }}$$.

#### Rule 2.5 c

$$\frac{{z_1 }}{{z_2 }} = \frac{{z_1 \cdot \bar z_2 }}{{z_2 \cdot \bar z_2 }} = \frac{{z_1 \cdot \bar z_2 }}{{\left| {z_2 } \right|^2 }}$$

#### Exercise 2.1:

Determine the complex conjugates of $$z_1 = 1 - 4 \cdot i$$ and $$z_2 = 0.5 \cdot e^{117^\circ i}$$. Use the Applet Basic Calculations to check your answers.

#### Exercise 2.2:

We have $$z_1 = 1 - 4 \cdot i$$ and $$z_2 =- 1 + 3 \cdot i$$. Determine $$z_1 + z_2$$, $$z_1 - z_2$$, $$z_1 \cdot z_2$$ and $$z_1 / z_2$$. Use the Applet Basic Calculations to check your answers.

#### Exercise 2.3:

We have $$z_1 = 1.5 \cdot e^{60^\circ i}$$ and $$z_2 = e^{320^\circ i}$$. Determine $$z_1 + z_2$$, $$z_1 - z_2$$, $$z_1 \cdot z_2$$ and $$z_1 / z_2$$. If necessary, turn the numbers into cartesian form. Use the Applet Basic Calculations to check your answers.

Dr. Jens Siebel, Last update: 08/15/2010