1. Introduction to Complex Numbers
A complex number is a number of the form:
\[ z = a + ib \]where:
- \( a \) = real part
- \( b \) = imaginary part
- \( i = \sqrt{-1} \)
The set of all complex numbers is denoted by:
\[ \mathbb{C} = \{ a + ib \mid a, b \in \mathbb{R} \} \] —2. Field of Complex Numbers
The set \( \mathbb{C} \) forms a field because:
- It is closed under addition and multiplication.
- Additive and multiplicative identities exist (0 and 1).
- Every non-zero complex number has a multiplicative inverse.
- Associative, commutative, and distributive laws hold.
If \( z = a + ib \), then its conjugate is:
\[ \bar{z} = a – ib \]The modulus of \( z \) is:
\[ |z| = \sqrt{a^2 + b^2} \]The multiplicative inverse of \( z \neq 0 \) is:
\[ z^{-1} = \frac{\bar{z}}{|z|^2} \]
Because every non-zero element has an inverse,
\( \mathbb{C} \) is a field.
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3. Polar Form of a Complex Number
A complex number can also be written in polar (trigonometric) form:
\[ z = r(\cos\theta + i\sin\theta) \]where:
\[ r = |z|, \quad \theta = \arg(z) \] —4. De Moivre’s Theorem
If
\[ z = r(\cos\theta + i\sin\theta) \]then for any integer \( n \),
\[ z^n = r^n (\cos n\theta + i\sin n\theta) \]
De Moivre’s Theorem:
Raise the modulus to the power \( n \), and multiply the argument by \( n \).
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Raise the modulus to the power \( n \), and multiply the argument by \( n \).
5. Example
Find \( ( \cos\theta + i\sin\theta )^3 \).
\[ (\cos\theta + i\sin\theta)^3 = \cos 3\theta + i\sin 3\theta \] —6. Applications of De Moivre’s Theorem
(a) Finding Powers of Complex Numbers
\[ (1 + i)^4 \]First convert to polar form, then apply De Moivre’s theorem.
—(b) Finding Roots of Complex Numbers
The nth roots of unity are:
\[ z_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right), \quad k = 0,1,2,\dots,n-1 \] —(c) Expanding Trigonometric Expressions
Using De Moivre’s theorem:
\[ \cos 3\theta = 4\cos^3\theta – 3\cos\theta \] \[ \sin 3\theta = 3\sin\theta – 4\sin^3\theta \] —7. Final Summary
- The set \( \mathbb{C} \) forms a field.
- Complex numbers can be written in algebraic and polar forms.
- De Moivre’s theorem helps compute powers and roots easily.
- It has important applications in trigonometry and higher mathematics.
The complex number system extends real numbers and provides powerful tools for advanced mathematics.