z¹=2-2i z²=6-5i
To analyze the complex numbers \( z^1 = 2 - 2i \) and \( z^2 = 6 - 5i \), you might be interested in various operations such as addition, subtraction, multiplication, division, or finding the modulus and argument of each.
Addition: \[ z^1 + z^2 = (2 - 2i) + (6 - 5i) = (2 + 6) + (-2 - 5)i = 8 - 7i \]
Subtraction: \[ z^1 - z^2 = (2 - 2i) - (6 - 5i) = (2 - 6) + (-2 + 5)i = -4 + 3i \]
Multiplication: \[ z^1 \cdot z^2 = (2 - 2i)(6 - 5i) \] Using the distributive property: \[ = 2 \cdot 6 + 2 \cdot (-5i) - 2i \cdot 6 - 2i \cdot (-5i) \] \[ = 12 - 10i - 12i + 10i^2 \] Recall that \( i^2 = -1 \): \[ = 12 - 10i - 12i - 10 = 2 - 22i \]
Division: \[ \frac{z^1}{z^2} = \frac{2 - 2i}{6 - 5i} \] To divide, multiply the numerator and denominator by the conjugate of the denominator: \[ = \frac{(2 - 2i)(6 + 5i)}{(6 - 5i)(6 + 5i)} \] Denominator: \[ = 6^2 + 5^2 = 36 + 25 = 61 \] Numerator: \[ = (2 \cdot 6 + 2 \cdot 5i - 2i \cdot 6 - 2i \cdot 5i) \] \[ = 12 + 10i - 12i + 10 = 22 - 2i \] So, \[ \frac{z^1}{z^2} = \frac{22 - 2i}{61} = \frac{22}{61} - \frac{2}{61}i \]
Modulus: \[ |z^1| = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \] \[ |z^2| = \sqrt{6^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} \]
Argument: \[ \text{arg}(z^1) = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) \] This is in the fourth quadrant, where the angle is \( -\frac{\pi}{4} \) (or \( \frac{7\pi}{4} \) in standard position). \[ \text{arg}(z^2) = \tan^{-1}\left(\frac{-5}{6}\right) \] The angle is in the fourth quadrant.
If you have a specific operation or a different aspect of the complex numbers you would like to explore, please let me know!
