Problem 73 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{7+3 i}{5+3 i}=\frac{7}{5} $$

Step-by-Step Solution

Answer

The statement is false. The correct statement should be \(\frac{7+3 i}{5+3 i} = \frac{22}{17} - \frac{3}{17} i\).

Step 1: Simplification of Complex Fraction

The first task is to simplify \(\frac{7+3 i}{5+3 i}\). To perform division between complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(5+3i\) is \(5 - 3i\). Hence, rewrite it as \(\frac{(7+3 i)(5-3i)}{(5+3 i)(5-3i)}\).

Step 2: Multiply the Numerator

The next step is to multiply the numerator(7+3i) with the conjugate of the denominator(5-3i).\nFollowing the distributive property of multiplication over addition, we get:\n\((7+3 i)(5-3i)\) => \(35 -21i+15i-9i^2\).\nSubstitute \(i^2\) with -1(as \(i^2 = -1\)) to get:\n\(35 -21i+15i+9\) => \(44 -6i\).

Step 3: Multiply the Denominator

Similarly, multiplying the denominator (5+3i) with its conjugate (5-3i) gives:\n\((5+3 i)(5-3i)\) => \(25 - 15i +15i - 9i^2\).\nSubstitute \(i^2\) with -1 to get:\n\(25 - 15i +15i + 9\) => \(34\).

Step 4: Final Simplification

Now simplify the fraction after obtaining the simplified numerator and denominator to get the final simplified complex number:\n\(\frac{44 -6i}{34}\). Divide the real part and the imaginary part of the numerator by the denominator to get \(\frac{44}{34} - \frac{6}{34} i \) => \(\frac{22}{17} - \frac{3}{17} i\)

Step 5: Comparing the results

Upon comparing the results, it can clearly be seen that \(\frac{7+3 i}{5+3 i} \neq \frac{7}{5}\). Therefore, the given statement is false.

Step 6: Correcting the statement

In order to make the statement true, replace the second half of the equation with the simplified complex number, thus \(\frac{7+3 i}{5+3 i} = \frac{22}{17} - \frac{3}{17} i\) is a true statement.

Key Concepts

Conjugate of Complex Numbers

A complex number is typically expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The conjugate of a complex number swaps the sign of the imaginary part. So, if you have a complex number \(a + bi\), its conjugate is \(a - bi\).
This concept is incredibly useful in mathematics, especially when you are dealing with fractions that include complex numbers.
When you multiply a complex number by its conjugate, it eliminates the imaginary part, resulting in a real number.
This is why the conjugate is often used as a tool to simplify complex fractions. As seen in the exercise where the complex number was \(5+3i\), its conjugate is \(5-3i\). Multiplying by this conjugate is a key step in simplifying the original expression.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property and the concept of \(i^2 = -1\).
If you need to multiply two complex numbers, \((a + bi)\) and \((c + di)\), the process involves multiple steps:

• First, distribute each term in the first complex number across the second, resulting in \(ac + adi + bci + bdi^2\).
• Next, remember that \(i^2=-1\), so the term \(bdi^2\) becomes \(-bd\).
• The final result is \((ac - bd) + (ad + bc)i\).

In the exercise, multiplying \(7+3i\) by \(5-3i\) correctly follows this approach, leading to the simplified result \(44-6i\).
This step is crucial for reducing fractions with complex denominators.

Division of Complex Numbers

Dividing complex numbers is a bit more involved than simple multiplication because you want to avoid leaving a complex number in the denominator.
The trick is to multiply both the numerator and the denominator by the conjugate of the denominator.
This effectively 'rationalizes' the denominator, leaving a real number which is easier to handle.
For instance, when dividing \(\frac{7+3i}{5+3i}\), the denominator becomes \(34\) after multiplication by its conjugate \((5-3i)\).

• The numerator becomes \(44 - 6i\).

Finally, you divide each part of the complex numerator by the real number denominator.
This results in the simplified expression \(\frac{22}{17} - \frac{3}{17}i\).
Using the conjugate to handle division ensures clarity in your final expressions.

Simplification of Fractions

Simplifying fractions, especially those involving complex numbers, clears up expressions and makes them more manageable.
First, you want to get rid of any complex parts in the denominator by using its conjugate, which turns the denominator into a real number.

• This can simplify the entire expression, leaving you with a cleaner, more straightforward result.
• In our case, from \(\frac{44-6i}{34}\), we reached \(\frac{22}{17} - \frac{3}{17}i\).

Next, simplify further by breaking down the fraction into its real and imaginary parts, divided separately by the real denominator.
This separation makes it clearer and easier to compare or manipulate further, proving the original statement false and providing the correct reduced form.