1.2 Algebra of Complex Numbers

• Complex numbers are of the form a + bi
• What is the real part of a complex number $a +$$bi$?


  $a$
• The imaginary part of a complex number is denoted $Im(z) =$$b$.
• The imaginary unit $i$ has the property i^2 = -1
• What are the real and imaginary parts of $z =$$3 +$$4i$?


  Re(z) = 3, Im(z) = 4</latex>
• What are the real and imaginary parts of $z =$$5 - 2i$?


  $Re(z) =$$5, Im(z) =$$- 2$
• In the complex number z = -3 + i</latex>, <latex>Re(z) = -3
• What are the real and imaginary parts of $z =$$4 +$$7i$?


  $Re(z) =$$4, Im(z) =$$7$
• In the complex number $4 +$$7i$, the imaginary part is $7$.
• What does the real part of a complex number represent?

  The real number without $i$
• The imaginary unit $i$ is defined as \sqrt{ - 1
• To add two complex numbers, add their real parts and imaginary parts separately.
• If $z_{1} =$$a +$$bi$ and $z_{2} =$$c +$$di$, what is z_{1} + z_{2}</latex>?


  $(a + c) +$$(b + d)i$
• Add $z_{1} =$$3 +$$2i$ and $z_{2} =$$1 - 4i$.


  $4 - 2i$
• The imaginary part of the sum of two complex numbers is the sum of their individual imaginary parts.
• If $z_{1} =$$a +$$bi$ and $z_{2} =$$c +$$di$, what is $z_{1} +$$z_{2}$?


  $(a + c) +$$(b + d)i$
• To add two complex numbers, add their real parts together and their imaginary parts together separately
• To subtract complex numbers, you subtract the real parts and the imaginary parts separately.
• Complex numbers are written in the form a + bi
• The imaginary unit i</latex> has the property $i^{2} =$$- 1$.
• What is the notation for the real part of a complex number $z$?


  $Re(z) =$$a$
• The imaginary part of a complex number is the coefficient of i
• The real part of $z =$$- 3 +$$i$ is $- 3$.
• What is the formula for adding two complex numbers $z_{1} =$$a +$$bi$ and $z_{2} =$$c +$$di$?


  $z_{1} +$$z_{2} =$$(a + c) +$$(b + d)i$
• Steps for subtracting two complex numbers $z_{1} =$$a +$$bi$ and $z_{2} =$$c +$$di$


  1️⃣ Subtract the real parts: $Re(z_{1}) - Re(z_{2}) =$$a - c$

  2️⃣ Subtract the imaginary parts: $Im(z_{1}) - Im(z_{2}) =$$b - d$

  3️⃣ Combine the results to form the new complex number: $(a - c) +$$(b - d)i$
• What method is used to multiply two complex numbers and simplify the result?

  FOIL
• When multiplying complex numbers, you must use the property $i^{2} =$$- 1$ to simplify the result.
• What is the general formula for multiplying two complex numbers $z_{1} =$$a +$$bi$ and $z_{2} =$$c +$$di$?


  $(ac - bd) +$$(ad + bc)i$
• To multiply two complex numbers $z_{1} =$$a +$$bi$ and z_{2} = c + di</latex>, use the FOIL method to distribute each term and simplify using $i^{2} =$$- 1$.FOIL
• Multiplying $z_{1} =$$3 +$$2i$ and $z_{2} =$$1 - 4i$ results in $11 - 10i$.
• What is the complex conjugate of a complex number $z =$$a +$$bi$?


  $\bar{z} =$$a - bi$
• The complex conjugate of a complex number $z =$$a +$$bi$ is obtained by changing the sign of the imaginary part.
• Match each complex number with its complex conjugate:

  $3 +$$4i$ ↔️ $3 - 4i$

  $- 2 - 5i$ ↔️ $- 2 +$$5i$

  $6i$ ↔️ $- 6i$

  $7$ ↔️ $7$
• In a complex number $z =$$a +$$bi$, the imaginary part is denoted as $Im(z) =$$b$.
• In the complex number $z =$$3 +$$4i$, the real part is 3.
• What are the real and imaginary parts of the complex number $z =$$5 - 2i$?


  $Re(z) =$$5, Im(z) =$$- 2$
• To add two complex numbers, their real parts and imaginary parts are added separately.
• What is the sum of $z_{1} =$$3 +$$2i$ and $z_{2} =$$1 - 4i$?


  $4 - 2i$
• To subtract complex numbers, subtract their real parts and their imaginary parts separately.
• What is the result of subtracting $z_{2} =$$1 - 4i$ from $z_{1} =$$3 +$$2i$?


  $2 +$$6i$