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Questions and Answers
What is the result of the expression $\frac{z_1}{z_2}$ if $z_1 = 1+i$ and $z_2 = 2-3i$?
What is the result of the expression $\frac{z_1}{z_2}$ if $z_1 = 1+i$ and $z_2 = 2-3i$?
- $\frac{5-7i}{13}$
- $\frac{1+3i}{-5}$ (correct)
- $\frac{5+7i}{13}$
- $\frac{1-3i}{-5}$ (correct)
Which property of complex numbers is illustrated by the equation $\frac{z_1}{z_2} = \frac{(z_1)}{z_2}$?
Which property of complex numbers is illustrated by the equation $\frac{z_1}{z_2} = \frac{(z_1)}{z_2}$?
- Subtraction of Complex Numbers
- Addition of Complex Numbers (correct)
- Division of Complex Numbers
- Multiplication of Complex Numbers
In the verification section, which equation relates to the modulus of a complex number |z|?
In the verification section, which equation relates to the modulus of a complex number |z|?
- |z| = \sqrt{z^2 + \overline{z}^2}
- z = \overline{z} + (real)
- z = (z) if z is real (correct)
- z = z + \overline{z}
Which of the following equations shows subtraction of complex numbers correctly?
Which of the following equations shows subtraction of complex numbers correctly?
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What does the property $\frac{z_1+z_2}{z_2} = \frac{z_1+z_2}{z_2}$ demonstrate?
What does the property $\frac{z_1+z_2}{z_2} = \frac{z_1+z_2}{z_2}$ demonstrate?
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Study Notes
Complex Numbers
- Complex numbers can be added, subtracted, and multiplied.
- The modulus of a complex number, |z|, is a measure of its magnitude.
- For a real number, z = (z), which means the number itself is equal to its real part.
- The following properties are true for complex numbers:
- ( \frac{z_1}{z_2} = \frac{(z_1)}{z_2} )
- ( \frac{z_1}{z_2} = \frac{z_1 \cdot z_2}{z_2} )
- ( \frac{z_1-z_2}{z_2} = \frac{z_1-z_2}{z_2} )
- ( \frac{z_1+z_2}{z_2} = \frac{z_1+z_2}{z_2} )
- These properties can be verified using examples.
- Complex numbers are essential tools in mathematics and other fields.
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