To divide complex numbers, you eliminate the complex part from the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator.
Dividing complex numbers is similar to rationalizing the denominator when working with radicals. The goal is to transform the division expression $\frac{a + bi}{c + di}$ into the standard form $x + yi$, where $x$ and $y$ are real numbers.
The Process of Dividing Complex Numbers
The key to dividing complex numbers lies in using the complex conjugate. For a complex number $c + di$, its complex conjugate is $c - di$. When a complex number is multiplied by its conjugate, the result is always a real number. This is because $(c + di)(c - di) = c^2 - (di)^2 = c^2 - d^2i^2$. Since $i^2 = -1$, this simplifies to $c^2 - d^2(-1) = c^2 + d^2$. The sum of squares ($c^2 + d^2$) is always a non-negative real number.
Here are the Steps for Dividing Complex Numbers:
- Identify the Complex Conjugate: For a division $\frac{a + bi}{c + di}$, identify the denominator, which is $c + di$. Its complex conjugate is $c - di$.
- Multiply by the Conjugate: Multiply both the numerator $(a + bi)$ and the denominator $(c + di)$ by the complex conjugate of the denominator $(c - di)$. This is equivalent to multiplying the fraction by $\frac{c - di}{c - di}$, which is equal to 1 and doesn't change the value of the expression:
$$ \frac{a + bi}{c + di} = \frac{a + bi}{c + di} \times \frac{c - di}{c - di} $$
This step aligns directly with the reference: "Multiply the conjugate with the numerator and the denominator of the complex fraction". - Simplify the Denominator: Expand the denominator $(c + di)(c - di)$. Use the algebraic identity $(x+y)(x-y) = x^2 - y^2$.
$$ (c + di)(c - di) = c^2 - (di)^2 = c^2 - d^2i^2 $$
Now, substitute $i^2 = -1$.
$$ c^2 - d^2(-1) = c^2 + d^2 $$
The denominator becomes a real number. This step aligns directly with the reference: "Apply the algebraic identity (a+b)(a-b)=a2 - b2 in the denominator and substitute i2 = -1". - Simplify the Numerator: Expand the numerator $(a + bi)(c - di)$ using the distributive property (often called FOIL):
$$ (a + bi)(c - di) = ac - adi + bci - bdi^2 $$
Substitute $i^2 = -1$:
$$ ac - adi + bci - bd(-1) = ac - adi + bci + bd $$
Group the real terms and the imaginary terms:
$$ (ac + bd) + (bc - ad)i $$
This step aligns directly with the reference: "Apply the distributive property in the numerator and simplify". - Write in Standard Form: Combine the simplified numerator and denominator and express the result in the standard form $x + yi$:
$$ \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i $$
Here, $x = \frac{ac + bd}{c^2 + d^2}$ and $y = \frac{bc - ad}{c^2 + d^2}$.
Example: Dividing Two Complex Numbers
Let's divide the complex number $3 + 4i$ by $1 - 2i$.
$$ \frac{3 + 4i}{1 - 2i} $$
Here, $a=3$, $b=4$, $c=1$, and $d=-2$.
| Step | Action | Calculation |
|---|---|---|
| 1. Identify Conjugate | The denominator is $1 - 2i$. Its conjugate is $1 + 2i$. | Conjugate: $1 + 2i$ |
| 2. Multiply by Conjugate | Multiply numerator and denominator by $1 + 2i$. | $$ \frac{3 + 4i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} $$ |
| 3. Simplify Denominator | Multiply $(1 - 2i)(1 + 2i)$. Use $(c+di)(c-di) = c^2 + d^2$ with $c=1, d=-2$. | $$ (1)^2 + (-2)^2 = 1^2 + 2^2 = 1 + 4 = 5 $$ |
| 4. Simplify Numerator | Multiply $(3 + 4i)(1 + 2i)$ using distributive property. | $$ (3)(1) + (3)(2i) + (4i)(1) + (4i)(2i) $$ $$ = 3 + 6i + 4i + 8i^2 $$ $$ = 3 + 10i + 8(-1) $$ $$ = 3 + 10i - 8 $$ $$ = (3 - 8) + 10i $$ $$ = -5 + 10i $$ |
| 5. Write in Standard Form | Combine numerator and denominator and separate into real and imaginary parts. | $$ \frac{-5 + 10i}{5} $$ $$ = \frac{-5}{5} + \frac{10i}{5} $$ $$ = -1 + 2i $$ |
Thus, the result of dividing $3 + 4i$ by $1 - 2i$ is $-1 + 2i$.
Understanding how to use the complex conjugate is fundamental for performing division and simplifying complex fractional expressions. This method ensures the denominator becomes a real number, allowing the result to be easily written in the standard $a + bi$ form.
