Problem 39 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \frac{26+39 i}{2-3 i} $$

Step-by-Step Solution

Answer

The result is \(-5 + 12i\).

Step 1: Identify the need for a conjugate

In order to simplify the expression \( \frac{26+39i}{2-3i} \), we need to eliminate the imaginary number in the denominator. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2-3i\) is \(2+3i\).

Step 2: Multiply the numerator with the conjugate

Multiply \((26+39i)\) by \((2+3i)\):\[(26+39i)(2+3i) = 26 \cdot 2 + 26 \cdot 3i + 39i \cdot 2 + 39i \cdot 3i\]Simplify the expression:\[= 52 + 78i + 78i + 117i^2\]Since \(i^2 = -1\), substitute:\[= 52 + 156i - 117\]Combine the real parts:\[= -65 + 156i\]

Step 3: Multiply the denominator with the conjugate

Now multiply \((2-3i)\) by its conjugate \((2+3i)\):\[(2-3i)(2+3i) = 2 \cdot 2 + 2 \cdot 3i - 3i \cdot 2 - 3i \cdot 3i\]Simplify the expression:\[= 4 + 6i - 6i - 9i^2\]Since \(i^2 = -1\), substitute:\[= 4 + 9\]Combine:\[= 13\]

Step 4: Divide by the denominator

Now, take the result from Step 2 and divide it by the result from Step 3:\[\frac{-65 + 156i}{13}\]Simplify both terms of the fraction:\[= -\frac{65}{13} + \frac{156}{13}i\]Calculate each part:\(-\frac{65}{13} = -5\) and \(\frac{156}{13} = 12\)Thus the expression simplifies to:\(-5 + 12i\).

Key Concepts

Conjugate

To handle complex numbers effectively, understanding the concept of a conjugate is essential. In mathematical terms, the conjugate of a complex number helps in operations like division.
A complex number is in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary coefficient multiplied by \(i\), the imaginary unit.

• The conjugate of \(a + bi\) is \(a - bi\), where the sign of the imaginary part is flipped.
• For example, the conjugate of \(2 - 3i\) is \(2 + 3i\).

Using the conjugate is especially useful in simplifying expressions with complex numbers having a complex denominator. By multiplying the numerator and the denominator by the conjugate of the denominator, it helps to eliminate the imaginary part in the denominator, thereby making further calculations simpler.

Imaginary Numbers

Imaginary numbers are a form of numbers that extend the concept of one-dimensional number lines used for real numbers to two dimensions. The basis of imaginary numbers is the imaginary unit \(i\).
The imaginary unit \(i\) has a special property:

• \(i^2 = -1\)
• Therefore, \(i\) is defined as \( \sqrt{-1} \).

Complex numbers consist of real and imaginary components, forming expressions like \(a + bi\). These numbers are vital in fields like engineering, physics, and computer science, where they help solve equations that do not have real solutions only. In complex arithmetic, imaginary numbers are manipulated in tandem with real numbers but have distinct rules defined mainly by the behavior of \(i\). When multiplying or dividing complex numbers, recognizing and implementing these rules ensures accurate results.

Simplifying Expressions

Simplifying expressions, particularly those involving complex numbers, is about reducing them to the simplest form to make computations easier. This involves combining like terms and using operations on complex numbers, taking advantage of properties like the conjugate.
To simplify a complex expression in division:

• First, identify the conjugate of the denominator.
• Multiply both the numerator and the denominator by this conjugate.
• In multiplication, distribute each term carefully, keeping track of the imaginary unit \(i\) and using \(i^2 = -1\).
• After obtaining the product, simplify by combining like terms—real numbers and imaginary numbers separately.
• Finally, divide each term of the resulting expression by the simplified denominator.

These steps help transform a complex expression into a more straightforward form \(a + bi\), where calculations are more direct and intuitive.