Problem 20 - Intermediate Algebra

Question

Add or subtract as indicated. $$ (12-9 i)-(14-6 i) $$

Step-by-Step Solution

Answer

\(-2 - 3i\)

Step 1: Identify the Expression

You are given the expression \((12-9i)-(14-6i)\). Your task is to subtract the two complex numbers within the parentheses.

Step 2: Separate the Real and Imaginary Parts

For each complex number, separate the real and imaginary components. The first complex number is \(12-9i\), where 12 is the real part and \(-9i\) is the imaginary part. The second complex number is \(14-6i\), where 14 is the real part and \(-6i\) is the imaginary part.

Step 3: Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number: \(12 - 14 = -2\).

Step 4: Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: \(-9i - (-6i) = -9i + 6i = -3i\).

Step 5: Combine the Results

Combine the results from the previous two steps to get the final complex number. The real part is \(-2\) and the imaginary part is \(-3i\), resulting in the complex number \(-2 -3i\).

Key Concepts

Imaginary Numbers

Imaginary numbers may sound a bit abstract at first, but they become much more approachable when you understand their purpose. In mathematics, an imaginary number is a number that, when squared, gives a negative result. This is something real numbers cannot do. The imaginary unit is denoted as \( i \), and by definition, \( i^2 = -1 \).

• Imaginary numbers are crucial in fields such as engineering, physics, and signal processing.
• They help in modeling phenomena such as electrical currents and wave patterns.

Imaginary numbers usually work in tandem with real numbers to create complex numbers. Here, you’ll often see terms like \( bi \), where \( b \) is a real number and \( i \) signifies its imaginary component. So, in our exercise, \(-9i\) and \(-6i\) are the imaginary parts of the complex numbers we're dealing with.

Real Numbers

Real numbers are the numbers that we typically think of and use in everyday life. They include all the rational numbers like integers, fractions, and decimals, as well as irrational numbers that cannot be expressed as fractions, like \( \pi \) and \( \sqrt{2} \).

• Real numbers form a foundational set in mathematics and are considered linear or one-dimensional.
• They can be visualized along a continuous number line.

In the context of our exercise, the real parts are "12" and "14" from the complex numbers \( 12-9i \) and \( 14-6i \), respectively. These are the parts we subtract directly when handling complex numbers.

Subtraction of Complex Numbers

Subtracting complex numbers might seem tricky at first, but it follows the same basic principles as regular subtraction. A complex number consists of a real part and an imaginary part.

• When you subtract complex numbers, you handle the real and imaginary parts separately.
• You subtract the real parts of the numbers and the imaginary parts independently.

In our exercise \((12-9i)-(14-6i)\): - Subtract the real parts: \(12 - 14 = -2\).- Subtract the imaginary parts: \(-9i - (-6i) = -9i + 6i = -3i\). Finally, we combine these results to get \(-2 - 3i\). This outcome shows how real and imaginary components work together to form a complex result, making it a blend of subtraction and combination.