Notes on Complex Numbers: Addition, Subtraction, Multiplication, Division by Conjugate, and Powers of i

Complex Numbers: Overview, Operations, Division by Conjugate, and Powers of i

• Definition and notation

  • A complex number is written as $z = a + bi$ where:
  • $a \in \mathbb{R}$ is the real part
  • $b \in \mathbb{R}$ is the imaginary part
  • $i$ is the imaginary unit with $i^2 = -1$
  • Basic idea: separate the real and imaginary components when performing operations

• Addition and subtraction

  • To add or subtract complex numbers, add/subtract real parts and imaginary parts separately:
  • $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • $(a + bi) - (c + di) = (a - c) + (b - d)i$
  • Example (conceptual, not from transcript):
  • $(3 + 4i) + (2 - i) = (3 + 2) + (4 - 1)i = 5 + 3i$

• Multiplication (FOIL)

  • Multiply as you would polynomials, distributing terms:
  • $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Intuition: real parts multiply, imaginary parts produce cross terms, and i^2 terms contribute to the real component via $i^2 = -1$
  • Example:
  • $(3 + 2i)(-1 + 4i) = (3)(-1) + (3)(4i) + (2i)(-1) + (2i)(4i)$
  • $= -3 + 12i - 2i + 8i^2 = -3 + 10i + 8(-1) = -11 + 10i$
  • Note: in the transcript, there was discussion of using FOIL and recognizing $i^2 = -1$, which turns negative terms into real contributions
  • Helpful identity: if you know one factor and its conjugate, you can simplify certain products using the same idea as division (see below)

• Division by a complex number (conjugate method)

  • To divide by a complex number in the denominator, multiply numerator and denominator by the conjugate of the denominator
  • Conjugate: for $a + bi$, the conjugate is $a - bi$ (flip the sign of the imaginary part)
  • Rationale: using $(a + bi)(a - bi) = a^2 + b^2$ eliminates the imaginary part in the denominator
  • Procedure:
  • Given $\dfrac{N}{D}$ with $D = a + bi$, compute
    • $\dfrac{N}{D} = \dfrac{N}{a+bi} \cdot \dfrac{a - bi}{a - bi} = \dfrac{N(a - bi)}{(a+bi)(a-bi)}$
    • Denominator becomes real: $a^2 + b^2$
    • Numerator expands to a complex number; you can then separate real and imaginary parts
  • Example from transcript (division):
  • Divide $3 - 2i$ by $-7 + 3i$
  • Multiply numerator and denominator by the conjugate of the denominator $-7 - 3i$:
    • Denominator: $(-7 + 3i)(-7 - 3i) = (-7)^2 + (3)^2 = 49 + 9 = 58$
    • Numerator:
    • $(3 - 2i)(-7 - 3i) = 3(-7) + 3(-3i) + (-2i)(-7) + (-2i)(-3i)$
    • $= -21 - 9i + 14i + 6i^2$
    • $= -21 + 5i + 6(-1) = -27 + 5i$
  • Result:
    • $\dfrac{3 - 2i}{-7 + 3i} = \dfrac{-27 + 5i}{58} = -\dfrac{27}{58} + \dfrac{5}{58}i$
  • Takeaway: when dividing by a complex number, the conjugate trick eliminates the imaginary part in the denominator and yields a real denominator

• Powers of i (cyclic behavior)

  • Powers of the imaginary unit follow a 4-term cycle:
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
  • Then it repeats: $i^5 = i, i^6 = -1, <br />\ni^7 = -i, i^8 = 1, \dots$
  • There are only four possible results for powers of $i$: $i, -1, -i, 1$
  • Quick rule: if the exponent is divisible by 4, the result is $1$
  • General shortcut: for any integer $n \ge 0$, write $n = 4k + r$ with $r \in {0,1,2,3}$, then
  • $i^n = i^r = \begin{cases} 1 &amp; \text{if } r=0 \ i &amp; \text{if } r=1 \ -1 &amp; \text{if } r=2 \ -i &amp; \text{if } r=3 \end{cases}$
  • Example ideas mentioned: determine quickly whether four divides a given exponent by checking remainder modulo 4
  • Connection to real-world use: powers of i underpin Euler's formula and signal processing, but here we focus on the cyclic pattern relevant to solving equations

• Key observations and practical implications (from transcript context)

  • Addition/subtraction of complex numbers is straightforward: operate on real parts together and imaginary parts together
  • Multiplication via FOIL is direct; watch out for i^2 terms and simplify using $i^2 = -1$
  • Division is the most involved operation among the three basic operations due to the need for the conjugate to rationalize the denominator
  • The conjugate technique is a standard tool from algebra that extends naturally to complex numbers
  • The powers of i exhibit a simple, predictable cycle that can be exploited to simplify expressions like $i^n$ for large exponents

• Quick reference table (for memorization)

  • Complex number form: $z = a + bi$ where $a,b \in \mathbb{R}$ and $i^2 = -1$
  • Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$
  • Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Conjugate of $a + bi$: $a - bi$
  • Division by conjugate: $\dfrac{N}{a + bi} = \dfrac{N(a - bi)}{a^2 + b^2}$
  • Powers of i:
  • $i^{4k} = 1$, $i^{4k+1} = i$, $i^{4k+2} = -1$, $i^{4k+3} = -i$

• Connections to foundational concepts and examples mentioned in the transcript

  • The concept of Real and Imaginary parts aligns with the idea of decomposing a complex number into two orthogonal components
  • The FOIL approach mirrors polynomial multiplication learned earlier, extended to include imaginary terms
  • The conjugate method for division mirrors the technique of rationalizing denominators from algebra, adapted to complex numbers
  • The cyclic nature of i’s powers reflects the underlying structure of complex arithmetic and connects to broader topics such as roots of unity in higher mathematics

• Quick practice prompts (to reinforce the notes)

  • Compute the product:
  • $(5 + 2i)(3 - 4i)$
  • Divide:
  • $\dfrac{3 - i}{-2 + 5i}$ using the conjugate of the denominator
  • Determine the value:
  • $i^{17}$ and explain using the modulo-4 rule

• Practical takeaway

  • For quick work with complex numbers, memorize the multiplication formula, the conjugate division method, and the i-cycle. These tools let you handle addition, subtraction, multiplication, and division efficiently, and they provide a foundation for more advanced topics in complex analysis and engineering applications.