Problem 45 - Intermediate Algebra

Question

Write each quotient in the form \(a+b i .\) See Example 5. $$ \frac{3+5 i}{1+i} $$

Step-by-Step Solution

Answer

The quotient is \(4+i\).

Step 1: Identify the Problem Type

This exercise requires us to express a complex division in the standard form \( a + b i \).

Step 2: Multiply by Conjugate

To eliminate the imaginary part in the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\). So the fraction becomes: \[ \frac{(3+5i)(1-i)}{(1+i)(1-i)} \]

Step 3: Simplify the Denominator

Multiply the conjugates in the denominator using the formula \((a+b)(a-b) = a^2 - b^2\). Thus, \((1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2\). Now the new denominator is 2.

Step 4: Expand the Numerator

Multiply the numerators: \((3+5i)(1-i) = 3(1) + 3(-i) + 5i(1) - 5i^2\). Simplify to get: \[ 3 - 3i + 5i + 5\] which reduces to \[8 + 2i\] (since \(-5i^2 = 5\)).

Step 5: Divide Each Term by the Denominator

Now divide each term in the simplified numerator by 2: \(\frac{8}{2} + \frac{2i}{2}\) which becomes \(4 + i\).

Step 6: Express the Result

After completing the division, the quotient in the form \(a+bi\) is \(4+i\).

Key Concepts

Complex Division

Performing division with complex numbers might initially sound daunting, but it becomes quite manageable when broken down into clear steps. In complex division, our primary aim is to rewrite the division of one complex number by another into the standard form \(a+bi\), where \(a\) and \(b\) are real numbers.

To divide complex numbers, we tackle the problem by multiplying both the numerator and the denominator by the conjugate of the denominator. This nifty trick transforms the denominator into a real number, effectively eliminating the imaginary part from it. It's akin to rationalizing denominators when dealing with square roots, and it set the stage for easy simplification of the complex expression into a neat real and imaginary combination.

Conjugate of a Complex Number

The conjugate of a complex number is crucial when simplifying complex division. Given a complex number \(a+bi\), its conjugate is \(a-bi\).

The beauty of using conjugates lies in their multiplication, which effectively removes the imaginary part from the denominator:

• When you multiply \((a+bi)(a-bi)\), you apply the difference of squares formula: \(a^2 - b^2\).
• In the realm of complex numbers, knowing that \(i^2 = -1\) helps us to replace \(-b^2\) with \(+b^2\). Hence, the product of a number and its conjugate is a real number, \(a^2 + b^2\).

This process ensures the denominator becomes a simple real number, paving the way for easier division and simplification of the original complex expression.

Simplifying Complex Expressions

Simplification is the final touch that transforms an initially intimidating complex expression into a simple, more understandable form. After multiplying by the conjugate, both the numerator and the denominator are altered.

The numerator's complex arithmetic requires you to distribute and simplify the resulting terms. For example, multiplying \((3+5i)(1-i)\) involves calculating:

• \(3 \times 1 = 3\)
• \(3 \times -i = -3i\)
• \(5i \times 1 = 5i\)
• \(5i \times -i = -5(i^2) = 5\)

After calculating, terms are combined and simplified: \(3 - 3i + 5i + 5 = 8 + 2i\).

Finally, divide the real and imaginary parts by the real denominator achieved through the use of conjugates. For our exercise, this means dividing 8 and 2i by 2, leaving you with \(4+i\) in the form \(a + bi\). Thus, the initial complex expression has been converted into a straightforward complex number.