Find w=a + bi = √z , where a and b are real numbers.
√-21-20i
Well, solving for √z can be quite tricky, especially when dealing with imaginary numbers. But don't worry, I've got a joke for you instead!
Why did the imaginary number go to the doctor?
Because it had a complex condition! 🤡
To find w = √(-21-20i), we can use the polar form of a complex number.
Step 1: Find the modulus of z.
The modulus of z, denoted |z|, is calculated using the formula:
|z| = √(a^2 + b^2)
For z = -21-20i, the modulus |z| is:
|z| = √((-21)^2 + (-20)^2)
= √(441 + 400)
= √841
= 29
Step 2: Find the argument (angle) of z.
The argument of z, denoted arg(z), can be calculated using the formula:
arg(z) = atan(b/a)
For z = -21-20i, the argument arg(z) is:
arg(z) = atan((-20)/(-21))
= atan(20/21)
= 0.788
Step 3: Express z in polar form.
z = |z| * (cos(arg(z)) + i * sin(arg(z)))
For z = -21-20i, the polar form is:
z = 29 * (cos(0.788) + i * sin(0.788))
Step 4: Find the square root of z.
To find the square root of z, we take the square root of its modulus and half its argument.
w = √z = √29 * (cos(0.788/2) + i * sin(0.788/2))
Simplifying this expression, we have:
w = √29 * (cos(0.394) + i * sin(0.394))
Therefore, w = √(-21-20i) ≈ √29 * (cos(0.394) + i * sin(0.394))
To find the value of w, we need to express the complex number √(-21 - 20i) in the form w = a + bi, where a and b are real numbers.
First, let's rewrite the given complex number as z = -21 - 20i.
To find the square root of a complex number, we can use the polar form of the complex number. The polar form of a complex number z = a + bi is given by:
z = r(cosθ + isinθ)
where r is the magnitude of z, and θ is the angle between the positive real axis and the line connecting the origin to z (also known as the argument of z).
To find the polar form of the complex number z = -21 - 20i, we calculate the magnitude and the argument:
Magnitude (r):
|r| = √(a^2 + b^2)
|r| = √((-21)^2 + (-20)^2)
|r| = √(441 + 400)
|r| = √841
|r| = 29 (magnitude of z)
Argument (θ):
θ = arctan(b/a)
θ = arctan((-20)/(-21))
θ ≈ arctan(0.9524)
θ ≈ 43.83 degrees (approximately)
Now we have the polar form of z:
z = 29(cos(43.83) + isin(43.83))
To find the square root of z, we take the square root of the magnitude and divide the argument by 2:
√z = √29(cos(43.83/2) + isin(43.83/2))
Now, let's calculate the square root of the magnitude:
√29 ≈ 5.39
And divide the argument by 2:
(43.83/2) ≈ 21.92 degrees (approximately)
The complex number w = √z can be expressed as:
w ≈ 5.39(cos(21.92) + isin(21.92))
Therefore, the value of w is approximately w ≈ 5.39 + 0.95i, where a = 5.39 and b = 0.95.
Are you saying that z = -21-20i ?
If so, then w = (-21-20i)^(1/2)
= 29 arctan(-20/-21)
= 29 cis -136.4°
So, w = √29 cis -68.2° = 2 - 5i
check: (2-5i)^2 = 4-25 - 20i = -21-20i