Problem 27 - Intermediate Algebra

Question

$$ (4-7 i)+(2+3 i) $$

Step-by-Step Solution

Answer

The result is \(6 - 4i\).

Step 1: Identify the Numbers

Identify the complex numbers in the expression: \((4 - 7i)\) and \((2 + 3i)\). Each number is in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Add the Real Parts

Add the real parts of the complex numbers: \(4\) and \(2\). This gives:\[4 + 2 = 6\]

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers: \(-7i\) and \(+3i\). This gives: \[-7i + 3i = -4i\]

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to form the resulting complex number: \(6 - 4i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers might seem daunting, but it's actually quite simple. First, let's recap what a complex number is. A complex number consists of two parts: a real part and an imaginary part. These numbers are typically written in the form: \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).

To add two complex numbers, such as \((4 - 7i)\) and \((2 + 3i)\), you follow a straightforward process:

• Add the real parts together: \(4 + 2 = 6\).
• Add the imaginary parts: \(-7i + 3i = -4i\).

Finally, combine these results to get the final sum, which is \(6 - 4i\). This demonstrates how separately handling the real and imaginary changes simplifies the addition process.

Real and Imaginary Parts

Understanding real and imaginary parts is crucial when working with complex numbers. Each complex number has both, and it’s essential to know how to identify and manipulate them effectively.

The **real part** of a complex number is the component that appears without the \(i\) notation. For example, in the number \(4 - 7i\), the real part is \(4\).
The **imaginary part** is the component attached to the \(i\) notation. In \(4 - 7i\), the imaginary part is \(-7i\). Note that it includes the imaginary unit \(i\) and ensures you correctly interpret the sign in front of it.

When performing operations like addition on complex numbers, handling these components separately ensures accuracy. By summing like parts, i.e., real with real and imaginary with imaginary, you avoid mixing two different quantities that represent different properties within the complex number.

Complex Number Arithmetic

Arithmetic involving complex numbers follows similar rules to those with real numbers, but always respects the properties of the imaginary unit \(i\). The basic operations include addition, subtraction, multiplication, and division involving real and imaginary parts. Here's a brief overview of performing complex number arithmetic:

• **Addition and Subtraction**: As seen before, add or subtract corresponding real parts and imaginary parts separately.
• **Multiplication**: Use the distributive property, just like when expanding binomials, remembering that \(i^2 = -1\). For example, \((a + bi)(c + di) = ac + adi + bci + bdi^2\) simplifies to \((ac - bd) + (ad + bc)i\).
• **Division**: For division, it involves multiplying the numerator and the denominator by the conjugate of the denominator to eliminate \(i\) from the denominator, then simplifying.

Understanding these operations helps unveil the power of complex numbers in mathematics, allowing for more solutions to exist in various equations beyond just real numbers.