Problem 17 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(8 - i\).

Step 1: Expand the Numerator

Multiply the two complex numbers in the numerator. Use the distributive property (also known as the FOIL method for binomials). The expression becomes:\[(3-i)(2+3i) = 3*2 + 3*3i - i*2 - i*3i\]Simplify each term to get:\[6 + 9i - 2i - 3i^2\]Since \(i^2 = -1\), replace \(-3i^2\) with \(3\). This simplifies to:\[6 + 7i + 3 = 9 + 7i\].

Step 2: Simplify the Denominator

Multiply both the numerator and the denominator by the conjugate of the denominator to remove the imaginary number in the denominator. The conjugate of \(1+i\) is \(1-i\). Multiply:\[(1+i)(1-i) = 1^2 - (i)^2 = 1 + 1 = 2\]Thus, the denominator simplifies to 2.

Step 3: Rationalize the Expression

Now, multiply the simplified numerator by the conjugate of the denominator:Fraction becomes:\[\frac{(9+7i)(1-i)}{2} = \frac{9(1-i) + 7i(1-i)}{2}\]Expanding gives:\[\frac{9 - 9i + 7i - 7i^2}{2}\]Using \(i^2 = -1\), \(-7i^2\) becomes \(7\). Simplify to:\[\frac{9 + 7 + (7i - 9i)}{2} = \frac{16 - 2i}{2}\].

Step 4: Simplify the Expression

Finally, divide both real and imaginary components by 2:\[\frac{16}{2} - \frac{2i}{2} = 8 - i\].Thus, the expression in the form \(a + ib\) is \(8 - i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics. It allows us to multiply a single term across a sum or a difference. In essence, it ensures that we can distribute one value into another set of values without missing any elements along the way. Take for instance the expression

• \((a + b)c\) which becomes \(ac + bc\)

This property is especially handy when dealing with polynomials and complex numbers.
When applied to complex numbers like \((3-i)(2+3i)\), each term in the first bracket is multiplied by each term in the second bracket:

• First, \(3 \times 2 = 6\)
• Then, \(3 \times 3i = 9i\)
• Next, \(-i \times 2 = -2i\)
• Lastly, \(-i \times 3i = -3i^2\)

Completing the multiplication, and using the fact that \(i^2 = -1\), we further simplify the expression.

FOIL Method

The FOIL method is an acronym that stands for First, Outer, Inner, Last. It is a specific application of the distributive property used for multiplying two binomials. The method involves four simple steps:1. **First:** Multiply the first terms in each binomial.2. **Outer:** Multiply the outer terms in the multiplication expression.3. **Inner:** Multiply the inner terms.4. **Last:** Multiply the last terms in each binomial.To apply FOIL to our exercise, consider the multiplication

• \((3-i)(2+3i)\)

By following the FOIL steps:

• **First:** \(3 \times 2 = 6\)
• **Outer:** \(3 \times 3i = 9i\)
• **Inner:** \(-i \times 2 = -2i\)
• **Last:** \(-i \times 3i = -3i^2\)

Summing these, and knowing that \(i^2 = -1\), simplifies the expression to \(9 + 7i\).

Complex Conjugate

A complex conjugate of a complex number is found by changing the sign of the imaginary part. So, if we have a complex number \(a + bi\), its complex conjugate is \(a - bi\). Using the conjugate is crucial in simplifying expressions, especially when it comes to rationalizing denominators.
In this exercise, we need to remove the imaginary denominator. The denominator is \(1+i\), and its conjugate is \(1-i\).
By multiplying the numerator and the denominator by this conjugate:

• \((1+i)(1-i) = 1 - i^2 = 2\)

This results in a purely real number in the denominator, hence effectively eliminating the imaginary component from it.

Rationalizing Denominator

Rationalizing the denominator is a technique used to eliminate imaginary or irrational numbers from the denominator. The aim is to transform it into a rational number. With complex numbers, this often involves using the complex conjugate.
For instance, when working with the fraction

• \(\frac{(9+7i)(1-i)}{1+i}\)

we multiply both the numerator and the conjugate of the denominator, \(1-i\) in this case. The result:

• \(1+i\) multiplied by \(1-i\) gives the real number 2.

The numerator, after simplification becomes \(16-2i\) and the new rational denominator is 2:

• Finally, dividing through gives \(8-i\) in the standard form \(a+ib\).

Rationalizing helps in expressing complex numbers in a more workable form, aiding deeper analysis or calculation.