1. Find the absolute value of the complex number 2 + 3i.
Solution:
The absolute value of a complex number is the distance between the origin and the point representing the complex number on the complex plane. We can use the Pythagorean theorem to find the distance:
|2 + 3i| = sqrt(2^2 + 3^2) = sqrt(13)
Therefore, the absolute value of the complex number 2 + 3i is sqrt(13).
2. Find the square of the complex number 1 + i.
Solution:
To square a complex number, we can use the formula:
(a + bi)^2 = a^2 + 2abi - b^2
where a and b are the real and imaginary parts of the complex number.
In this case, a = 1 and b = 1, so we have:
(1 + i)^2 = 1^2 + 2(1)(i) - 1^2 = 2i
Therefore, the square of the complex number 1 + i is 2i.
3. Find all complex solutions to the equation z^3 = 1.
Solution:
To solve this equation, we can write 1 as a complex number using polar coordinates:
1 = 1(cos(0) + i sin(0))
where 0 is any multiple of 2π.
Then, the cube roots of 1 are given by:
z = 1^(1/3) = cos(0/3) + i sin(0/3)
z = 1^(1/3) = cos(2π/3) + i sin(2π/3)
z = 1^(1/3) = cos(4π/3) + i sin(4π/3)
Therefore, the three complex solutions to the equation z^3 = 1 are:
z = 1, z = -1/2 + i sqrt(3)/2, and z = -1/2 - i sqrt(3)/2.
4. Find all complex solutions to the equation z^4 = -16.
Solution:
To solve this equation, we can write -16 as a complex number using polar coordinates:
-16 = 16(cos(π) + i sin(π))
Then, the fourth roots of -16 are given by:
z = (-16)^(1/4) = 2(cos(π/4) + i sin(π/4))
z = (-16)^(1/4) = 2(cos(3π/4) + i sin(3π/4))
z = (-16)^(1/4) = 2(cos(5π/4) + i sin(5π/4))
z = (-16)^(1/4) = 2(cos(7π/4) + i sin(7π/4))
Therefore, the four complex solutions to the equation z^4 = -16 are:
z = 2(cos(π/4) + i sin(π/4)),
z = 2(cos(3π/4) + i sin(3π/4)),
z = 2(cos(5π/4) + i sin(5π/4)), and
z = 2(cos(7π/4) + i sin(7π/4)).
Note that we can also write these solutions in rectangular form if desired.
5. Find all complex numbers z such that |z - 3 - 4i| = 5.
Solution:
The equation |z - 3 - 4i| = 5 represents the set of complex numbers that are a distance of 5 units from the point (3,4) in the complex plane. Geometrically, this represents a circle with radius 5 centered at (3,4).
To find all complex numbers z that satisfy this equation, we can write z = x + yi and substitute into the equation, then use the distance formula for two points to obtain:
sqrt((x-3)^2 + (y-4)^2) = 5
Squaring both sides gives:
(x-3)^2 + (y-4)^2 = 25
Expanding and simplifying:
x^2 - 6x + y^2 - 8y + 24 = 0
Completing the square for both x and y:
(x - 3)^2 + (y - 4)^2 = 1
This is the equation of a circle with radius 1 centered at (3,4). Therefore, the set of complex numbers that satisfy |z - 3 - 4i| = 5 is the set of complex numbers that lie on the circle with radius 5 centered at (3,4) in the complex plane.
In rectangular form, the equation of this circle is:
(x - 3)^2 + (y - 4)^2 = 25
or
x^2 + y^2 - 6x - 8y + 24 = 0
which is the same equation we obtained earlier by completing the square.
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