Problem 18 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ \frac{(1+i)(1-2 i)}{(2+i)(4-3 i)} $$

Step-by-Step Solution

Answer

The expression is \( \frac{7}{25} - \frac{i}{25} \).

Step 1: Multiply the Numerator

Multiply the complex numbers in the numerator: \((1+i)(1-2i) = 1(1) + 1(-2i) + i(1) + i(-2i)\). First, calculate \(1 \cdot 1 = 1\), \(1 \cdot -2i = -2i\), \(i \cdot 1 = i\), and \(i \cdot -2i = -2i^2\). Since \(i^2 = -1\), we have \(-2i^2 = 2\). So the product is \(1 - 2i + i + 2 = 3 - i\).

Step 2: Multiply the Denominator

Multiply the complex numbers in the denominator: \((2+i)(4-3i) = 2(4) + 2(-3i) + i(4) + i(-3i)\). Calculate \(2 \cdot 4 = 8\), \(2 \cdot -3i = -6i\), \(i \cdot 4 = 4i\), and \(i \cdot -3i = -3i^2\). Using \(i^2 = -1\), we have \(-3i^2 = 3\). Thus, the product is \(8 - 6i + 4i + 3 = 11 - 2i\).

Step 3: Divide the Real and Imaginary Parts

The division is now \(\frac{3-i}{11-2i}\). To simplify, multiply the numerator and denominator by the conjugate of the denominator: \(\frac{(3-i)(11+2i)}{(11-2i)(11+2i)}\). First, calculate the denominator \((11)^2 + (2)^2 = 121 + 4 = 125\). Then, multiply: Numerator: \((3-i)(11+2i) = 3(11) + 3(2i) - i(11) - i(2i)\). This leads to \(33 + 6i - 11i - 2i^2\). With \(-2i^2 = 2\), it simplifies to \(33 - 5i + 2 = 35 - 5i\). Thus, the expression is \(\frac{35 - 5i}{125}\).

Step 4: Simplify the Expression

Simplify each part: For the real part, \(\frac{35}{125} = \frac{7}{25}\). For the imaginary part, \(\frac{-5i}{125} = -\frac{i}{25}\). Thus, the expression in the form \(a + ib\) is \(\frac{7}{25} - \frac{i}{25}\).

Key Concepts

Multiplying Complex Numbers

Complex numbers may include both a real part and an imaginary part, typically written in the form \(a + ib\). The imaginary unit \(i\) has the property that \(i^2 = -1\). When multiplying complex numbers, we use the distributive property, often called FOIL, to expand the expression.

Consider two complex numbers \((a + bi)\) and \((c + di)\). Their product is calculated as follows:

• First, multiply the real parts: \(a \cdot c\).
• Outside, multiply the real part of the first by the imaginary part of the second: \(a \cdot di\).
• Inside, multiply the imaginary part of the first by the real part of the second: \(bi \cdot c\).
• Last, multiply the imaginary parts: \(bi \cdot di\). Remember \(i^2 = -1\), so this becomes \(-bd\).

Collect all the terms: if \(i^2\) appears, substitute \(-1\) to simplify. This method is demonstrated in the problem where \((1+i)(1-2i)\) yields the result \(3-i\) after combining and simplifying terms.

Complex Conjugate

The complex conjugate of a complex number \(a+bi\) is \(a-bi\). Conjugates are particularly useful when dividing complex numbers because they enable us to eliminate the imaginary part in the denominator.

When dividing, such as \(\frac{3-i}{11-2i}\), multiply the numerator and the denominator by the conjugate of the denominator, \(11+2i\). This approach removes the imaginary unit \(i\) from the denominator, which is a crucial step.

• The denominator becomes \((11-2i)(11+2i)\).
• This is calculated as \(11^2 - (2i)^2\), simplifying to \(121 + 4 = 125\).
• The numerator is expanded similarly to regular multiplication, yielding \(35 - 5i\).

Hence, after applying the conjugate, the division becomes a straightforward real number fraction, facilitating further simplification.

Simplifying Algebraic Fractions

Once you have removed the imaginary part from the denominator using the complex conjugate, the resulting expression is an algebraic fraction which may still need further simplification.

The expression from the example results in \(\frac{35-5i}{125}\). To simplify:

• Start by separating the real and imaginary parts: \(\frac{35}{125}\) and \(\frac{-5i}{125}\).
• Reduce these fractions to their simplest terms or recognize them as being divisible by a common factor.
• For example, \(\frac{35}{125}\) reduces to \(\frac{7}{25}\) and \(\frac{-5i}{125}\) simplifies to \(-\frac{i}{25}\).

Thus, the expression in its simplest form is \(\frac{7}{25} - \frac{i}{25}\). This step is crucial, as presenting the complex number in its simplest \(a+ib\) form helps in both interpretation and further computations.