Problem 64 - Algebra 2

Question

Simplify. $$ \frac{2+3 i}{1+2 i} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{8}{5} - \frac{1}{5}i \).

Step 1: Identify the Complex Expression

We need to simplify the expression \( \frac{2 + 3i}{1 + 2i} \). This is a division of complex numbers. Our goal is to express this as a single complex number.

Step 2: Find the Conjugate

The first step in simplifying is to eliminate the imaginary part in the denominator. To do this, multiply the numerator and the denominator by the conjugate of the denominator, \( 1 - 2i \).

Step 3: Multiply the Numerator and Denominator

Calculate the multiplication:- Numerator: \( (2 + 3i)(1 - 2i) = 2 \times 1 + 2 \times (-2i) + 3i \times 1 + 3i \times (-2i) \)Using \( i^2 = -1 \), simplify to \( 2 - 4i + 3i - 6(i^2) = 2 - i + 6 = 8 - i \).- Denominator: \( (1 + 2i)(1 - 2i) = 1 \times 1 - 2i \times 2i = 1 + 4 = 5 \).

Step 4: Write the Simplified Complex Number

Divide the new numerator by the new denominator to express the complex number in standard form: \( \frac{8 - i}{5} = \frac{8}{5} - \frac{i}{5} \). Therefore, the simplified expression is \( \frac{8}{5} - \frac{1}{5}i \).

Key Concepts

Conjugate of a Complex Number

Every complex number has a twin called its conjugate. This helps in simplifying expressions involving complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\). These are almost identical, except for their imaginary components having opposite signs.
To eliminate the imaginary part from a denominator, multiply by the conjugate. This process converts the denominator into a real number. Such simplification is essential in dividing complex numbers.

Understanding Imaginary Numbers

Imaginary numbers might sound like they are make-believe, but they are very real and useful in mathematics. An imaginary number is one in which a square of the number is negative. The basic unit of imaginary numbers is \(i\), where \(i^2 = -1\).
When you see an imaginary number such as \(3i\) in calculus or engineering, it represents \(3 \times i\). It's crucial to remember that the real part and the imaginary part of a complex number play distinct roles. In our exercise, both \(2 + 3i\) and \(1 + 2i\) have real and imaginary parts.

Standard Form of Complex Numbers

When it comes to complex numbers, the standard form is as simple as writing them out clearly. It is expressed as \(a + bi\), where:

• \(a\) is the real part
• \(b\) is the imaginary part

For proper understanding and operations involving complex numbers, always convert them to their standard form. After simplification, as in our solution, the complex number \(\frac{8}{5} - \frac{1}{5}i\) is in standard form, with the real part \(\frac{8}{5}\) and imaginary part \(\frac{-1}{5}i\).

Simplifying Complex Fractions

Simplifying complex fractions can seem tricky, but with a methodical approach, it becomes manageable. The primary technique involves multiplying both numerator and denominator by the conjugate of the denominator. This eliminates the imaginary component of the denominator.
In our problem, we multiplied by \(1 - 2i\) leading to a real number denominator.
After multiplication, remember to simplify using \(i^2 = -1\) when necessary. Finally, express your result in the standard form, ensuring both real and imaginary components are clear. This approach is key to mastering complex arithmetic.