express the roots of unity in standard form a+bi.
1.) cube roots of unity
2.) fourth roots of unity
3.) sixth roots of unity
4.) square roots of unity
The following link will explain in detail the roots of unity.
If you have answers to be checked, feel free to post them.
I don't see the link. I don't think this website allows you to put up links
Sorry, it was my omission.
http://en.wikipedia.org/wiki/Root_of_unity
To express the roots of unity in the standard form a+bi, we need to find the complex numbers that satisfy the equation z^n = 1, where n represents the root number.
1. Cube Roots of Unity:
To find the cube roots of unity, we need to solve the equation z^3 = 1. We can rewrite 1 as 1 + 0i.
Let z = a + bi, where a and b are real numbers.
Substituting z into the equation, we get:
(a + bi)^3 = 1
Expanding and simplifying, we have:
a^3 + 3a^2bi + 3ab^2i^2 + b^3i^3 = 1
a^3 + 3a^2bi - 3ab^2 - b^3i = 1
Since complex numbers are equal if and only if their corresponding real and imaginary parts are equal, we can equate the real and imaginary parts separately:
a^3 - 3ab^2 = 1 (Real part)
3a^2b - b^3 = 0 (Imaginary part)
Solving these two equations gives us the values of a and b, which represent the real and imaginary parts of the cube roots of unity in the form a+bi.
2. Fourth Roots of Unity:
For the fourth roots of unity, we need to solve the equation z^4 = 1.
Following a similar process as above, let z = a + bi:
(a + bi)^4 = 1
Expanding and simplifying:
a^4 + 4a^3bi + 6a^2b^2i^2 + 4ab^3i^3 + b^4i^4 = 1
a^4 + 4a^3bi - 6a^2b^2 - 4ab^3i + b^4 = 1
Equating the real and imaginary parts, we get two equations:
a^4 - 6a^2b^2 + b^4 = 1 (Real part)
4a^3b - 4ab^3 = 0 (Imaginary part)
Solving these equations will give us the values of a and b for the fourth roots of unity, expressed in the standard form a+bi.
3. Sixth Roots of Unity:
To find the sixth roots of unity, we solve the equation z^6 = 1. Proceeding in the same manner as above, we can find the values of a and b for the sixth roots of unity.
4. Square Roots of Unity:
The square roots of unity are a special case since every complex number has two square roots. In this case, we need to solve the equation z^2 = 1.
Let z = a + bi:
(a + bi)^2 = 1
Expanding and simplifying:
a^2 + 2abi + b^2i^2 = 1
a^2 + 2abi - b^2 = 1
Equating the real and imaginary parts, we get two equations:
a^2 - b^2 = 1 (Real part)
2ab = 0 (Imaginary part)
From the equation 2ab = 0, we have two possibilities:
1) a = 0 and b ≠ 0, which gives us one square root of unity: b+0i, where b is a non-zero real number.
2) a ≠ 0 and b = 0, which gives us another square root of unity: a+0i, where a is a non-zero real number.
Hence, for square roots of unity, there are two numbers: a+0i and 0+bi, where a and b are non-zero real numbers.