Solved Examples of Complex Number Addition, Subtraction, and Multiplication

📚 Quick Study Guide

• 🔢 Complex numbers are of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).
• ➕ To add complex numbers, add the real parts and the imaginary parts separately: $(a + bi) + (c + di) = (a + c) + (b + d)i$.
• ➖ To subtract complex numbers, subtract the real parts and the imaginary parts separately: $(a + bi) - (c + di) = (a - c) + (b - d)i$.
• ✖️ To multiply complex numbers, use the distributive property (FOIL method) and remember that $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.
• 💫 The conjugate of a complex number $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate results in a real number: $(a + bi)(a - bi) = a^2 + b^2$.

Practice Quiz

1. What is the result of $(3 + 2i) + (1 - i)$?
   1. 4 + i
   2. 2 + 3i
   3. 4 - i
   4. 2 - 3i
2. What is the result of $(5 - 3i) - (2 + i)$?
   1. 3 - 4i
   2. 7 - 2i
   3. 3 + 4i
   4. 7 + 2i
3. What is the result of $(2 + i)(3 - 2i)$?
   1. 8 - i
   2. 4 + 7i
   3. 8 + i
   4. 4 - 7i
4. Simplify: $(1 + i)^2$
   1. 2i
   2. 2
   3. 0
   4. -2i
5. What is the result of $(-2 + 4i) + (5 - i)$?
   1. 3 + 3i
   2. -7 + 5i
   3. 3 - 3i
   4. -7 - 5i
6. What is the result of $(4 - 2i) - (1 - 5i)$?
   1. 3 + 3i
   2. 5 - 7i
   3. 3 - 3i
   4. 5 + 7i
7. What is the result of $(1 - i)(2 + 3i)$?
   1. 5 + i
   2. -1 + 5i
   3. 5 - i
   4. -1 - 5i