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A complex number is a number that can be expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). Here, \( a \) is called the real part of \( z \), denoted by \( \Re(z) = a \), and \( b \) is called the imaginary part, denoted by \( \Im(z) = b \).
Complex numbers extend the real number system and are fundamental in many areas of mathematics, physics, and engineering.
Given \( z = a + bi \), the real part is \( a \), and the imaginary part is \( b \). Both \( a \) and \( b \) are real numbers. For example, if \( z = 3 - 4i \), then \( \Re(z) = 3 \) and \( \Im(z) = -4 \).
To add or subtract two complex numbers, add or subtract their corresponding real and imaginary parts:
\[(a + bi) + (c + di) = (a + c) + (b + d)i\]Multiply using distributive property and use \( i^2 = -1 \):
\[(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\]To divide \( \frac{a + bi}{c + di} \), multiply numerator and denominator by the conjugate of the denominator:
\[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\]The conjugate of \( z = a + bi \) is denoted by \( \overline{z} \) and defined as:
\[\overline{z} = a - bi\]The conjugate reflects the complex number about the real axis in the complex plane.
The modulus of \( z = a + bi \) is its distance from the origin in the complex plane:
\[|z| = \sqrt{a^2 + b^2}\]The argument \( \theta = \arg(z) \) is the angle made with the positive real axis, measured counterclockwise:
\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]Complex numbers can be represented as points or vectors in the complex plane (also called the Argand plane), where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Here, the vector from the origin to the point \( (3,4) \) represents the complex number \( 3 + 4i \). The length of this vector is the modulus \( |z| = 5 \), and the angle \( \theta \) is the argument.
Using modulus and argument, a complex number can be expressed as:
\[z = r(\cos \theta + i \sin \theta)\]where \( r = |z| \) and \( \theta = \arg(z) \).This form is useful for multiplication, division, and finding powers and roots of complex numbers.
Problem: Find \( (3 + 2i) + (1 - 5i) \).
Solution:
Add real parts: \( 3 + 1 = 4 \)
Add imaginary parts: \( 2i + (-5i) = -3i \)
\[ (3 + 2i) + (1 - 5i) = 4 - 3i \]Problem: Find the product \( (2 + 3i)(4 - i) \).
Solution:
\[ (2 + 3i)(4 - i) = 2 \times 4 + 2 \times (-i) + 3i \times 4 + 3i \times (-i) \] \[ = 8 - 2i + 12i - 3i^2 = 8 + 10i - 3(-1) = 8 + 10i + 3 = 11 + 10i \]Problem: Find the conjugate and modulus of \( z = -1 + 4i \).
Solution:
Conjugate: \( \overline{z} = -1 - 4i \)
Modulus: \( |z| = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \)
Problem: Compute \( \frac{3 + 2i}{1 - i} \).
Solution:
Multiply numerator and denominator by conjugate of denominator: \[ \frac{3 + 2i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} \] Numerator: \[ 3 \times 1 + 3 \times i + 2i \times 1 + 2i \times i = 3 + 3i + 2i + 2i^2 = 3 + 5i + 2(-1) = 3 + 5i - 2 = 1 + 5i \] Denominator: \[ 1 - i^2 = 1 - (-1) = 1 + 1 = 2 \] So, \[ \frac{3 + 2i}{1 - i} = \frac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i \]Problem: Find the argument of \( z = -\sqrt{3} + i \).
Solution:
\[ \theta = \tan^{-1} \left(\frac{1}{-\sqrt{3}}\right) = \tan^{-1}(-\frac{1}{\sqrt{3}}) = -30^\circ \] Since \( a = -\sqrt{3} < 0 \) and \( b = 1 > 0 \), \( z \) lies in the second quadrant. So, \[ \arg(z) = 180^\circ - 30^\circ = 150^\circ \] Or in radians, \[ \arg(z) = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]| Concept | Formula | Description |
|---|---|---|
| Complex Number | \( z = a + bi \) | Standard form, where \( a, b \in \mathbb{R} \) and \( i^2 = -1 \) |
| Real Part | \( \Re(z) = a \) | Real component of \( z \) |
| Imaginary Part | \( \Im(z) = b \) | Imaginary component of \( z \) |
| Conjugate | \( \overline{z} = a - bi \) | Reflection of \( z \) about the real axis |
| Modulus | \( |z| = \sqrt{a^2 + b^2} \) | Distance of \( z \) from origin in complex plane |
| Argument | \( \arg(z) = \tan^{-1} \left(\frac{b}{a}\right) \) | Angle made by \( z \) with positive real axis |
| Addition | \( (a + bi) + (c + di) = (a + c) + (b + d)i \) | Add real and imaginary parts separately |
| Multiplication | \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \) | Use distributive property and \( i^2 = -1 \) |
| Division | \( \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \) | Multiply numerator and denominator by conjugate of denominator |
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