Complex Numbers Study Notes

4-2.5 Complex Numbers

Vocabulary

• Definition of i:

  • i is defined as:

  • $i = ext{√}(-1)$

  • $i^2 = -1$

  • The imaginary number i was invented to solve and explain the concept of negative square root numbers.

  • Example:

    • $ext{√}(-9) = ext{√}(-1) imes ext{√}(9) = i imes 3 = 3i$

Square Roots of Negative Numbers

• If r is a positive real number, then:

  • $ext{√}(-r) = i ext{√}(r)$

Pure Imaginary Numbers

• These are numbers of the form bi, where b is a nonzero real number. Examples include:

  • $-2i$

  • $8i ext{√}15$

  • $i$

Simplification Examples

Example 1: Simplify the following expressions:

1. $ext{√}({ ext{√}}(-15)) = ext{√}15 ext{√}(-1) = i ext{√}15$

2. $ext{√}(-50) = ext{√}(50) ext{√}(-1) = 5i ext{√}2$

3. $(−6 − 7i) − (1 + 3i) = -7 - 10i$

4. $(1 - i) / (1 + i) = rac{(1 - i)(1 - i)}{(1 + i)(1 - i)} = rac{(1 - 2i - i^2)}{(1 + 1)} = (1 - 2i + 1) = rac{(2 - 2i)}{2} = 1 - i$

5. $(−6 - 5i)(1 + 3i) = −6 − 18i − 5i − 15i^2 = −6 − 18i − 5i + 15 = (9 - 23i)$

6. $i ext{√}(-98) imes (− ext{√}98) = -1 ext{√}(98) = 1 ext{√}(98) = (−1)(−1) ext{√}(98)$

Example 2: Find the factored forms

1. $x^2 + 1 = (x - i)(x + i)$

2. $−1(9x^2 + 100)$

3. $−(3x - 10x)(3x + 10x) = (−6x)(6x) = −36(−1) = 36$

4. $x^2 + 36 = (x + 6i)(x - 6i)$

Quadratic Equations from Roots

Example 5: Find a quadratic equation for the roots

1. Roots: $a = -6i$ and $6i$.

   • Sum: $-6i + 6i = 0;$

   • $-b = 0$, implying $b = 0$;

   • Product: $(−6i)(6i) = −36(−1) = 36;$ thus the equation is $x^2 + 36 = 0$.

2. Roots: $−2$ and $5$.

   • Sum: $−2 + 5 = 3 <br>ightarrow b = −3;$

   • Product: $(−2)(5) = −10;$ hence the equation is $x^2 + 3x - 10 = 0$.

Finding Sum and Product of Roots

Example 6:

1. Given the equation: $5x^2 + 2x + 1$

   • Sum of roots: $-b/a = - rac{2}{5};$

   • Product of roots: $c/a = rac{1}{5}$.

Exercises

1. Simplify the expressions:

   • $ext{√}(-7) = i ext{√}7$

   • $−81 = 9i$

   • $(12 + 5i) - (2 - i) = 10 + 6i$

   • Calculate: $(8i)(4i)(−9i) = -288 i^2 = 288$

   • $= − ext{√}(-81) = 9i$

   • $(−6i)^{2} = (-6)^{2} imes (i)^{2} = 36(-1) = -36$

2. Further simplification:

   • Calculate $10 + ext{√}(-9) - (2 + ext{√}(-25)) = 10 + 3i - (2 + 5i) = 8 - 2i$

Absolute Value and Complex Numbers

• Plotting Complex Numbers:

  • Example: $A = 3 + 2i$;

  • Absolute Value of a complex number z:
    $|z| = ext{√}(x^2 + y^2)$ where x and y are the real and imaginary parts, respectively.

  • For $A$: $|A| = ext{√}(3^2 + 2^2) = ext{√}(9 + 4) = ext{√}(13)$

Quotients as Complex Numbers

• Example:
  $rac{3 - 2i}{5i} = rac{(3 - 2i)(-i)}{(5i)(-i)} = rac{-3i - 2(-1)}{-5} = rac{2 - 3i}{5}$

• Further simplification may apply to other complex divisions.

Quadratic Equations involving Imaginary Solutions

Quadratic Example 1:$x^2 + 2x + 3 = 0$

• Using the quadratic formula:

  • Coefficients:

    • $a = 1$, $b = 2$, $c = 3$

  • Substitute into the formula:
    $x = rac{-b ext{±} ext{√}(b^2-4ac)}{2a}$

  • Calculation:
    $-2 ext{±} ext{√}((2)^2 - 4(1)(3)) = -2 ext{±} ext{√}(4 - 12) = -2 ext{±} ext{√}(-8) = -2 ext{±} 2i ext{√}2$

  • The roots would thus be $x = -1 ext{± } i ext{√}2$

Quadratic Example 2: $2x^2 - 4x + 7 = 0$

• Coefficients:

  • $a = 2$, $b = -4$, $c = 7$

  • Solve using quadratic formula:
    $x = rac{4 ext{±} ext{√}((-4)^2 - 4(2)(7))}{2(2)}$

  • Calculation leads to
    $= rac{4 ext{±} ext{√}(16 - 56)}{4} = rac{4 ext{±} ext{√}(-40)}{4}$

  • Results in complex roots: $x = 1 ext{±} i ext{√}10$

Inverses and Conjugates of Complex Numbers

1. Additive Inverse:

   • For $A$ (let's say points related to previous examples), compute $-A$.

   • For $B = 3 + 2i$, Additive Inverse is $-B = -3 - 2i$.

2. Complex Conjugate:

   • Represented as $\bar{z} = a - bi$ if $z = a + bi$. For $B$:

   • Conjugate of $B$ is $3 - 2i$.

3. Absolute Value:

   • Calculate absolute values of points similarly to before.

Final Example:

• For the equation $(x + 3)(x - 3i) = 34$

  • Rearranging shows complex solutions involved generating $(x^2 - (3i)^2 = 34)$ leading to complex solutions in terms of radicals.

Concluding Notes

• Understanding complex numbers is essential for advanced mathematics, particularly in fields such as engineering and physics.

• Practice with sums, products, and the quadratic formula helps reinforce these concepts.