Problem 18 - Intermediate Algebra

Question

Add or subtract as indicated. $$ (-4-12 i)+(-3+16 i) $$

Step-by-Step Solution

Answer

The result is \(-7 + 4i\).

Step 1: Identify Real and Imaginary Parts

The given expression is \[(-4-12i) + (-3+16i).\]Identify the real parts and the imaginary parts from both complex numbers. For the first complex number \(-4-12i\), the real part is \(-4\) and the imaginary part is \(-12i\). For the second complex number \(-3+16i\), the real part is \(-3\) and the imaginary part is \(+16i\).

Step 2: Add the Real Parts

Add the real parts of the complex numbers identified in Step 1:\[-4 + (-3) = -4 - 3 = -7.\]

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers identified in Step 1:\[-12i + 16i = 4i.\]

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to form the final complex number:\[-7 + 4i.\]

Key Concepts

Adding Complex Numbers

Adding complex numbers might seem tricky at first, but it's straightforward once you understand the core process. A complex number is typically in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. To add complex numbers, you simply add their respective real parts and imaginary parts separately.

• Step 1: Identify the real and imaginary parts of both numbers.
• Step 2: Add the real parts together.
• Step 3: Add the imaginary parts together.
• Step 4: Combine these sums to get the final complex number.

For example, adding \((-4-12i) + (-3+16i)\) involves first adding the real parts: \(-4 + (-3) = -7\), and then adding the imaginary parts \(-12i + 16i = 4i\). Combining these gives the result \(-7 + 4i\).
It's just like combining like terms in algebra!

Real and Imaginary Parts

Every complex number has two components: a real part and an imaginary part. The real part is similar to what you'd encounter in traditional arithmetic, whereas the imaginary part involves the imaginary unit \(i\), where \(i^2 = -1\).

• Real Part: Appears without an \(i\) and represents a real number.
• Imaginary Part: Appears with an \(i\) and represents an imaginary number.

In our example \(-4 - 12i\), the real part is \(-4\) and the imaginary part is \(-12i\). Similarly, in \(-3 + 16i\), the real part is \(-3\) and the imaginary part is \(+16i\).
When dealing with complex numbers, always keep track of these parts, as they guide all operations, from addition to subtraction and beyond.

Complex Number Operations

Complex number operations expand your mathematical toolbox beyond real numbers, allowing for arithmetic with numbers that include real and imaginary parts. These operations include addition, subtraction, multiplication, and division, each with its unique steps.

• Addition & Subtraction: Involves combining or separating real and imaginary parts.
• Multiplication: Uses the distributive property and \(i^2 = -1\) to simplify products.
• Division: Involves multiplying by the conjugate to rationalize the denominator.

Following our sample exercise, by focusing on addition, you gain insight into the foundational steps needed to operate with complex numbers. Understanding the structure of these numbers ensures you're ready to tackle more advanced calculations efficiently.