Introduction to Complex Numbers

Imaginary unit and complex number operations

Introduction to Complex Numbers

The Imaginary Unit

The imaginary unit ii is defined as: i=āˆ’1i = \sqrt{-1}

Therefore: i2=āˆ’1i^2 = -1

Complex Numbers

A complex number has the form: a+bia + bi

where:

  • aa = real part
  • bb = imaginary part
  • ii = imaginary unit

Example: 3+4i3 + 4i

Powers of ii

Pattern repeats every 4:

  • i1=ii^1 = i
  • i2=āˆ’1i^2 = -1
  • i3=i2ā‹…i=āˆ’ii^3 = i^2 \cdot i = -i
  • i4=i2ā‹…i2=1i^4 = i^2 \cdot i^2 = 1
  • i5=ii^5 = i (pattern repeats)

Adding and Subtracting

Combine like terms (real with real, imaginary with imaginary):

(3+4i)+(2āˆ’i)=(3+2)+(4iāˆ’i)=5+3i(3 + 4i) + (2 - i) = (3 + 2) + (4i - i) = 5 + 3i

Multiplying

Use FOIL and remember i2=āˆ’1i^2 = -1:

(2+3i)(1+4i)(2 + 3i)(1 + 4i) =2+8i+3i+12i2= 2 + 8i + 3i + 12i^2 =2+11i+12(āˆ’1)= 2 + 11i + 12(-1) =2+11iāˆ’12= 2 + 11i - 12 =āˆ’10+11i= -10 + 11i

Complex Conjugates

The conjugate of a+bia + bi is aāˆ’bia - bi.

Property: (a+bi)(aāˆ’bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2 (always real!)

šŸ“š Practice Problems

1Problem 1easy

ā“ Question:

Simplify: āˆš(-16)

šŸ’” Show Solution

Step 1: Recall that i = āˆš(-1): The imaginary unit i is defined as āˆš(-1)

Step 2: Factor out -1: āˆš(-16) = āˆš(16 Ɨ (-1)) = āˆš16 Ɨ āˆš(-1)

Step 3: Simplify: = 4i

Answer: 4i

2Problem 2easy

ā“ Question:

Simplify: āˆ’16\sqrt{-16}

šŸ’” Show Solution

Factor out āˆ’1-1: āˆ’16=āˆ’1ā‹…16\sqrt{-16} = \sqrt{-1 \cdot 16}

=āˆ’1ā‹…16= \sqrt{-1} \cdot \sqrt{16}

=iā‹…4= i \cdot 4

=4i= 4i

Answer: 4i4i

3Problem 3easy

ā“ Question:

Add: (3 + 2i) + (5 - 4i)

šŸ’” Show Solution

Step 1: Group real and imaginary parts: (3 + 2i) + (5 - 4i) = (3 + 5) + (2i - 4i)

Step 2: Combine like terms: Real parts: 3 + 5 = 8 Imaginary parts: 2i - 4i = -2i

Step 3: Write in standard form: 8 - 2i

Answer: 8 - 2i

4Problem 4medium

ā“ Question:

Add: (5āˆ’2i)+(āˆ’3+7i)(5 - 2i) + (-3 + 7i)

šŸ’” Show Solution

Combine real parts and imaginary parts separately:

Real parts: 5+(āˆ’3)=25 + (-3) = 2 Imaginary parts: āˆ’2i+7i=5i-2i + 7i = 5i

Answer: 2+5i2 + 5i

5Problem 5medium

ā“ Question:

Multiply: (2 + 3i)(4 - i)

šŸ’” Show Solution

Step 1: Use FOIL method: First: 2 Ɨ 4 = 8 Outer: 2 Ɨ (-i) = -2i Inner: 3i Ɨ 4 = 12i Last: 3i Ɨ (-i) = -3iĀ²

Step 2: Combine: 8 - 2i + 12i - 3iĀ²

Step 3: Remember that iĀ² = -1: -3iĀ² = -3(-1) = 3

Step 4: Combine all terms: 8 - 2i + 12i + 3 = (8 + 3) + (-2i + 12i) = 11 + 10i

Answer: 11 + 10i

6Problem 6medium

ā“ Question:

Divide: (6 + 8i)/(1 - i)

šŸ’” Show Solution

Step 1: Multiply by conjugate of denominator: The conjugate of (1 - i) is (1 + i)

Step 2: Multiply numerator and denominator: (6 + 8i)/(1 - i) Ɨ (1 + i)/(1 + i)

Step 3: Expand numerator: (6 + 8i)(1 + i) = 6 + 6i + 8i + 8iĀ² = 6 + 14i + 8(-1) = 6 + 14i - 8 = -2 + 14i

Step 4: Expand denominator: (1 - i)(1 + i) = 1 + i - i - iĀ² = 1 - (-1) = 1 + 1 = 2

Step 5: Divide: (-2 + 14i)/2 = -2/2 + 14i/2 = -1 + 7i

Answer: -1 + 7i

7Problem 7hard

ā“ Question:

Multiply: (3āˆ’2i)(3+2i)(3 - 2i)(3 + 2i)

šŸ’” Show Solution

Notice these are conjugates! Use the formula (a+bi)(aāˆ’bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2

Or use FOIL: (3āˆ’2i)(3+2i)(3 - 2i)(3 + 2i) =9+6iāˆ’6iāˆ’4i2= 9 + 6i - 6i - 4i^2 =9āˆ’4(āˆ’1)= 9 - 4(-1) =9+4= 9 + 4 =13= 13

Answer: 1313 (a real number!)

8Problem 8hard

ā“ Question:

Find all solutions to xĀ² + 4 = 0 in the complex number system.

šŸ’” Show Solution

Step 1: Solve for xĀ²: xĀ² = -4

Step 2: Take square root of both sides: x = Ā±āˆš(-4)

Step 3: Simplify āˆš(-4): āˆš(-4) = āˆš(4 Ɨ (-1)) = āˆš4 Ɨ āˆš(-1) = 2i

Step 4: Write both solutions: x = 2i or x = -2i

Step 5: Verify x = 2i: (2i)Ā² + 4 = 4iĀ² + 4 = 4(-1) + 4 = -4 + 4 = 0 āœ“

Step 6: Verify x = -2i: (-2i)Ā² + 4 = 4iĀ² + 4 = 4(-1) + 4 = -4 + 4 = 0 āœ“

Answer: x = Ā±2i

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