Problem 78 - Elementary and Intermediate Algebra

Question

Divide. Write all answers in the form \(a+b i\) See Example 8. $$ \frac{5-3 i}{4+2 i} $$

Step-by-Step Solution

Answer

The answer is \(0.7 - 1.1i\).

Step 1: Identify the Conjugate

To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number \(a+bi\) is \(a-bi\). Here, the conjugate of \(4+2i\) is \(4-2i\).

Step 2: Multiply Numerator and Denominator

Multiply both the numerator \(5-3i\) and the denominator \(4+2i\) by the conjugate of the denominator \(4-2i\). This gives you:\[\frac{(5-3i)(4-2i)}{(4+2i)(4-2i)}\]

Step 3: Expand the Products

Expand the numerator and the denominator. For the numerator: \[(5-3i)(4-2i) = 5 \times 4 + 5 \times (-2i) - 3i \times 4 - 3i \times (-2i) = 20 - 10i - 12i + 6i^2\]\[= 20 - 22i + 6(-1) = 20 - 22i - 6\]\[= 14 - 22i\]For the denominator:\[(4+2i)(4-2i) = 4^2 - (2i)^2 = 16 - (-4) = 16 + 4 = 20\]

Step 4: Simplify the Expression

Now simplify the expression by dividing both the real and imaginary parts of the expanded numerator by the denominator:\[\frac{14 - 22i}{20} = \frac{14}{20} - \frac{22i}{20} = 0.7 - 1.1i\]So in the form \(a + bi\), it is \(0.7 - 1.1i\).

Key Concepts

Complex Conjugate

In the world of complex numbers, the concept of a complex conjugate is quite important—especially when it comes to division. A complex conjugate involves changing the sign of the imaginary part of a complex number.
For a complex number in the form of \(a + bi\), its conjugate is \(a - bi\).
This method is crucial for simplifying the division of complex numbers, and helps in eliminating the imaginary unit from the denominator.

• Imagine you have \(4+2i\), the conjugate will be \(4-2i\).
• This switch helps in achieving a real number when the conjugate is multiplied with the original complex number.

The multiplication of a complex number by its conjugate always results in a real number. This property makes it possible to handle complex division more easily, as we see when multiplying both the numerator and the denominator by the conjugate of the denominator.

Complex Number Multiplication

Understanding how to multiply complex numbers allows you to combine and manipulate them in a wide range of calculations. In essence, multiplying complex numbers involves using the distributive property, sometimes referred to as FOIL (First, Outer, Inner, Last) method.
When you multiply two complex numbers, remember to treat them as expressions with two parts: the real and the imaginary.

• For instance, when multiplying \((5-3i)\) and \((4-2i)\), start by distributing each term in the first complex number with each term in the second.
  • First: Multiply the real parts \(5 \times 4 = 20\)
  • Outer: Multiply the outer terms \(5 \times -2i = -10i\)
  • Inner: Multiply the inner terms \(-3i \times 4 = -12i\)
  • Last: Multiply the imaginary parts, keeping in mind that \(i^2 = -1\) \(-3i \times -2i = 6\times i^2 = -6\)

Combine all these terms, and you'll achieve the multiplication result: \(20 - 22i - 6\), which simplifies to \(14 - 22i\). This illustrates how careful tracking can lead to an accurate result in complex number multiplication.

Simplifying Complex Fractions

Simplifying complex fractions becomes effortless once you grasp that the division of complex numbers depends largely on getting rid of the imaginary component in the denominator. This is made possible by employing the conjugate, as discussed.
By multiplying the numerator and the denominator of a given fraction by the conjugate of the denominator, the division transforms into a simpler real number division task.

• For instance, when you have the fraction \(\frac{14 - 22i}{20}\), break it down.
  • First, divide the real part by the denominator: \(\frac{14}{20} = 0.7\).
  • Next, separately divide the imaginary part: \(\frac{-22i}{20} = -1.1i\).

This process transforms even the most daunting-looking complex fraction into a comprehensible form, such as \(0.7 - 1.1i\). By understanding how to simplify fractions in this way, you gain valuable skills useful in both calculus and algebra. This strategy simplifies and clarifies complex numbers, leading to more effective problem solving.