Grade 11 → Algebra → Complex Numbers ↓
Powers and Roots of Complex Numbers
Introduction to complex numbers
A complex number is a mathematical entity that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying the equation i² = -1.
Example of complex number
Consider the complex number 3 + 4i. Here, 3 is the real part, and 4 is the imaginary part.
Powers of complex numbers
Raising complex numbers to an exponent requires repeatedly multiplying the number by itself. A common way to handle this is to use de Moivre's theorem.
De Moivre's theorem
De Moivre's theorem states: For a complex number z = r(cos θ + i sin θ) and any integer n,
z^n = r^n (cos(nθ) + i sin(nθ))where r is the modulus of the complex number and θ is the argument.
Example of complex number power
Let's find (1 + i)^3:
First, convert 1 + i to polar form. Modulus r = √(1² + 1²) = √2 and argument θ = arctan(1/1) = π/4.
Use of de Moivre's theorem:
(√2)^3 [cos(3π/4) + i sin(3π/4)] = 2√2 [cos(3π/4) + i sin(3π/4)]Thus, (1 + i)^3 = -2 + 2i.
Roots of complex numbers
Finding the roots of a complex number is to determine another complex number that, when raised to a certain power, equals the original number. The n-th roots of a complex number can be found using the nth root formula.
Formula for roots of complex numbers
Given a complex number z = r(cos θ + i sin θ), its n-th roots are as follows:
w_k = √[n]{r} (cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n))For k = 0, 1, 2, ..., n-1.
Example of complex roots
Find the square root of 4i:
First, express 4i in polar form. Here, the modulus r = 4 and the argument θ = π/2.
The square roots are given as:
w_k = √[2]{4} (cos((π/2 + 2kπ)/2) + i sin((π/2 + 2kπ)/2))Calculate for k = 0 and k = 1:
- For
k = 0:w_0 = 2 (cos π/4 + i sin π/4) = 2(√2/2 + i√2/2) = √2 + i√2. - For
k = 1:w_1 = 2 (cos 5π/4 + i sin 5π/4) = 2(-√2/2 - i√2/2) = -√2 - i√2.
Thus, the square roots of 4i are √2 + i√2 and -√2 - i√2.
Visualization of powers and roots
Looking at complex numbers and their operations such as powers and roots can help to understand them better. Below are some illustrations depicting these operations.
This is a basic Argand diagram. The blue line represents the real axis and the red line represents the imaginary unit i. Complex multiplications and powers cause rotations and scalings on this plane.
A diagram showing square root alignment, showing how roots arise on the complex plane. The example given here shows the square roots taken.
Further considerations and symmetry
Note that raising a number to higher powers or taking multiple roots causes these values to be symmetrically distributed over the complex plane. This symmetry comes from the uniform distribution of angles
Understanding symmetry
For example, the nth roots of a complex number are spread symmetrically around a circle on the Argand plane. This is due to the angle increment of 2π/n for each successive root.
Symmetrical root example
Consider the roots of unity, such as the cube roots of 1 which are 1, e^(2πi/3), and e^(4πi/3).
These origins are located at equal angles of 120° around a circle centered at the origin with radius 1.
Conclusion
Understanding the operation of powers and roots in the realm of complex numbers requires a basic understanding of polar coordinates, the Argand plane, and the properties of angles and symmetries. Mastering these principles makes practical calculations and applications of complex numbers possible in a variety of mathematical contexts.
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