2.1 Complex Numbers

• Real numbers can be plotted on a number line, whereas imaginary numbers cannot
• How do you add or subtract complex numbers?

  Combine real and imaginary parts
• What is the imaginary unit $i$ equal to?


  \sqrt{ - 1}</latex>
• What is the general form of a complex number?

  $a +$$bi$
• What do you combine when adding or subtracting complex numbers?

  Real and imaginary parts
• Complex numbers are of the form $a +$$bi$, where $a$ and $b$ are real numbers.


  True
• To add or subtract complex numbers, combine their respective real and imaginary
• The sum of $(3 + 2i)$ and $(1 - i)$ is 4 + i
• An imaginary number is any number of the form bi, where $b$ is a real number.
• In a complex number $z =$$a +$$bi$, the real part is a.
• The sum of $(3 + 2i)$ and (1 - i)</latex> is 4 + i.
• The formula for multiplying two complex numbers is $z_{1} \times z_{2} =$$(a_{1}a_{2} - b_{1}b_{2}) +$$(a_{1}b_{2} +$$a_{2}b_{1})i$.conjugate
• What is the conjugate of the complex number z = a_{2} + b_{2}i</latex>?

  $a_{2} - b_{2}i$
• What is the formula for dividing two complex numbers $z_{1} =$$a_{1} +$$b_{1}i$ and $z_{2} =$$a_{2} +$$b_{2}i$?


  $\frac{z_{1}}{z_{2}} =$ \frac{(a_{1} + b_{1}i)(a_{2} - b_{2}i)}{a_{2}^{2} + b_{2}^{2}}
• What is the conjugate of the complex number $z =$$a +$$bi$?


  $z^ *$$=$$a - bi$
• The real part of a complex number and its conjugate are the same.


  True
• What are the complex roots of the quadratic equation $x^{2} +$$2x +$$5 =$$0$?


  $- 1 \pm 2i$
• When squared, imaginary numbers yield a negative result


  True
• Imaginary numbers are essential for constructing complex numbers.
• What is the imaginary part of a complex number $z =$$a +$$bi$?


  $bi$
• How do you subtract complex numbers?

  Combine real and imaginary parts separately
• Subtracting $(1 - i)$ from $(3 + 2i)$ results in $2 +$$3i$


  True
• When adding complex numbers $z_{1} =$$a_{1} +$$b_{1}i$ and z_{2} = a_{2} + b_{2}i</latex>, the imaginary part of the sum is $(b_{1} +$$b_{2})i$.


  True
• Subtracting $(1 - i)$ from $(3 + 2i)$ results in $2 +$$3i$.


  True
• To divide complex numbers, you multiply both the numerator and denominator by the conjugate of the denominator.


  True
• What is the result of multiplying (3 + 2i)</latex> by $(1 - i)$?


  5 - i
• Why is the conjugate used in dividing complex numbers?

  To eliminate the imaginary part in the denominator
• What is the definition of the imaginary unit $i$?


  $i =$$\sqrt{ - 1}$
• What is the real part of a complex number $z =$$a +$$bi$?


  $a$
• What is the result of adding (3 + 2i)</latex> and $(1 - i)$?


  $4 +$$i$
• The difference between $(3 + 2i)$ and $(1 - i)$ is $2 +$$3i$.


  True
• Imaginary numbers can be plotted on a number line.

  False
• The real part of a complex number can be zero.


  True
• When subtracting complex numbers, you add the imaginary parts.


  True
• The conjugate of $a +$$bi$ is $a - bi$.


  True
• To divide complex numbers, we use the concept of the conjugate
• The multiplication of two complex numbers involves the distributive property
• The conjugate of a complex number is obtained by changing the sign of its imaginary part.
• What is the result of dividing (3 + 2i)</latex> by $(1 - i)$?


  $\frac{1}{2} +$$\frac{5}{2}i$
• What is the definition of the imaginary unit $i$?


  $i =$$\sqrt{ - 1}$