If the mathematical symbols do not show up properly within a few seconds, open the pdf-version of this chapter.
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 aThe 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 bThe 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.
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 2.2The 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.
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 2.3The 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.
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 aThe 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 \(\arg \left( {z_1 \cdot z_2 } \right) = \varphi _1 + \varphi _2\). |
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 bThe 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 |
Rule 2.4 c\(z \cdot \bar z = \left( {a + b \cdot i} \right) \cdot \left( {a - b \cdot i} \right) = a^2 - \left( {b \cdot i} \right)^2 = a^2 - b^2 \cdot \left( { - 1} \right) = a^2 + b^2 = \left| z \right|^2\) |
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 aThe 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 |
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 bThe 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\(\frac{{z_1 }}{{z_2 }} = \frac{{z_1 \cdot \bar z_2 }}{{z_2 \cdot \bar z_2 }} = \frac{{z_1 \cdot \bar z_2 }}{{\left| {z_2 } \right|^2 }}\) |
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.
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.
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