Problem 24 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ (2+3 i)\left(\frac{2-i}{1+2 i}\right)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2 + 3i\).

Step 1: Simplify the Expression Inside Parentheses

Start by squaring the complex number \( \frac{2-i}{1+2i} \). Expand it as \( \left(\frac{2-i}{1+2i}\right)^2 = \frac{(2-i)^2}{(1+2i)^2} \). Compute \((2-i)^2\) and \((1+2i)^2\).

Step 2: Expand and Simplify the Numerator and Denominator

Calculate \((2-i)^2 = 4 - 4i + i^2 = 3 - 4i\), since \(i^2 = -1\). For \((1+2i)^2\), compute it as \(1 + 4i + 4i^2 = -3 + 4i\).

Step 3: Rationalize the Denominator

Rationalize \(\frac{3-4i}{-3+4i}\) by multiplying the numerator and the denominator by the complex conjugate \(-3-4i\), resulting in \(\frac{(3-4i)(-3-4i)}{(-3+4i)(-3-4i)}\).

Step 4: Compute the Products

Calculate the product \((3-4i)(-3-4i) = 9 + 12i - 12i + 16 = 25\), and \((-3 + 4i)(-3 - 4i) = 9 + 16 = 25\). So the rationalized form is \(\frac{25}{25} = 1\).

Step 5: Multiply with the Original Complex Number

Now multiply \((2+3i)\) with 1, resulting in \(2 + 3i\).

Step 6: Conclusion

Since multiplying by 1 didn't change \(2 + 3i\), the expression simplifies to \(2 + 3i\).

Key Concepts

Rationalizing Denominators

Rationalizing the denominator is a crucial process when dealing with complex numbers. It is similar to removing the square root from the denominator in real-number fractions, making the expression easier to work with.
You typically multiply both the numerator and denominator by the complex conjugate of the denominator. This step eliminates the imaginary unit \(i\) in the denominator, leaving a real number.
For instance: When you have a fraction like \(\frac{3-4i}{-3+4i}\), multiplying both the numerator and denominator by the complex conjugate \(-3-4i\) will remove \(i\) from the denominator:

• The numerator becomes \((3-4i)(-3-4i) = 25\).
• The denominator is calculated as \((-3+4i)(-3-4i) = 25\).

After this, the denominator is a real number, making computation and algebraic manipulation straightforward. This approach ensures expressions are easily interpretable.

Complex Conjugates

Understanding complex conjugates is essential for working with complex numbers, especially when rationalizing denominators.
The complex conjugate of a complex number \(a + bi\) is \(a - bi\).
It’s basically the same number, but the sign of the imaginary part is reversed. Complex conjugates play a vital role in simplifying complex expressions.
Why are they important? Multiplying a complex number by its conjugate always results in a real number. This property is what makes them useful when rationalizing denominators. In our exercise, using complex conjugates helped to eliminate the imaginary part from the denominator:

• The initial denominator was \(-3+4i\). Its conjugate is \(-3-4i\).
• When multiplied, \((-3+4i)(-3-4i)\) results in the real number 25.

This ability to transform complex expressions to real numbers is central to mastering complex calculations. It highlights the elegance in the mathematics behind complex conjugates.

Multiplication of Complex Numbers

Multiplying complex numbers follows the distributive property, much like multiplying binomials. Each part of the first complex number is multiplied by each part of the second complex number.
To multiply \((a+bi)\) by \((c+di)\), you apply the formula:
\((a+bi)(c+di) = ac + adi + bci + bdi^2\).
Remember: \(i^2 = -1\), which transforms imaginary terms into real ones.
Let's look closer:

• For \((2-i)^2\): Evaluate as \(4 - 4i + i^2 = 3-4i\).
• For \((-3+4i)(-3-4i)\): Result is \(9 + 12i - 12i + 16 = 25\).

The imaginary terms often cancel each other out, simplifying down to a real number or another simple complex number. This step in complex multiplication ensures you're confident handling complex expressions much like real numbers, simplifying calculations inevitably.