Solving the Complex Equation z^4 -1 and Its Applications
Solving the Complex Equation z^4 -1 and Its Applications
Complex numbers, denoted as z, are a fundamental concept in mathematics. They extend the real number system and offer a more complete framework for solving equations such as z4 -1. This article delves into the trigonometric form of complex numbers, DeMoivre's Theorem, and the process of finding the fourth roots of a complex number. By understanding these concepts, we can solve the given equation and explore its broader implications.
Understanding Complex Numbers and DeMoivre's Theorem
The equation z4 -1 can be solved within the realm of complex numbers. To find the solutions, we express -1 in its polar form, which is a useful representation in complex number theory. The polar form of a complex number is given by:
r cos(θ) i sin(θ)
For -1, the modulus r is 1, and the argument θ is π (or 180°). Thus, we can write:
-1 1 (cosπ i sinπ)
Applying DeMoivre's Theorem
DeMoivre's Theorem is a powerful tool for solving powers of complex numbers in polar form. The theorem states that for a complex number in the form r(cosθ i sinθ), the n-th power of this number is:
(r(cosθ i sinθ))n rn(cos(nθ) i sin(nθ))
In the case of solving z4 -1, we can rewrite the equation in polar form and then apply DeMoivre's Theorem to find the fourth roots. The equation becomes:
z4 1 (cosπ i sinπ)
To find the values of z, we take the fourth root of both sides:
zk r1/4(cos(θ 2kπ/4) i sin(θ 2kπ/4))
Calculating the Fourth Roots
For our equation, r 1, θ π, and n 4. Therefore, we have:
zk 11/4(cos(π 2kπ/4) i sin(π 2kπ/4))
Calculating for k 0, 1, 2, 3:
For k 0:z0 cos(π/4) i sin(π/4) (1/√2) i (1/√2)
For k 1:z1 cos(3π/4) i sin(3π/4) -1/√2 i (1/√2)
For k 2:z2 cos(5π/4) i sin(5π/4) -1/√2 - i (1/√2)
For k 3:z3 cos(7π/4) i sin(7π/4) 1/√2 - i (1/√2)
Therefore, the four solutions to the equation z4 -1 are:
z0 (1/√2) i (1/√2) z1 -1/√2 i (1/√2) z2 -1/√2 - i (1/√2) z3 1/√2 - i (1/√2)Further Insights
The solutions to the equation z4 -1 are evenly distributed on the unit circle in the complex plane, rotated by π/4 radians. This distribution can be represented as:
{eibpi/4 | b 0, 1, 2, 3, 4, 5, 6, 7}
Thus, the roots can also be expressed as:
{eπi/4, e3πi/4, e5πi/4, e7πi/4}Using the trigonometric form, we find:
eπi/4 cos(π/4) i sin(π/4) (1/√2) i (1/√2)
e3πi/4 cos(3π/4) i sin(3π/4) -1/√2 i (1/√2)
e5πi/4 cos(5π/4) i sin(5π/4) -1/√2 - i (1/√2)
e7πi/4 cos(7π/4) i sin(7π/4) (1/√2) - i (1/√2)
Conclusion
The solutions to the equation z4 -1 demonstrate the power and flexibility of the complex number system. By utilizing the polar form and DeMoivre's Theorem, we can find the roots of complex equations, providing valuable insights into the structure of the complex plane.