Problem 92 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((-4-3 i)(4+2 i)\)

Step-by-Step Solution

Answer

The simplified expression is \(-10 - 20i\).

Step 1: Distribute Each Term in the First Expression

The given expression is \((-4-3i)(4+2i)\). Start by distributing the first term \(-4\) over the terms of the second expression \((4 + 2i)\): \(-4 \times 4 = -16\) and \(-4 \times 2i = -8i\).

Step 2: Distribute the Imaginary Term

Now, distribute the \(-3i\) over the terms of the second expression \((4 + 2i)\): \((-3i) \times 4 = -12i\)and\((-3i) \times (2i) = -6i^2\).Remember that \(i^2 = -1\),so \(-6i^2 = 6\).

Step 3: Combine Like Terms

Combine the real and imaginary parts from the results of the distribution.Real part:\(-16 + 6 = -10\).Imaginary part:\(-8i - 12i = -20i\).

Step 4: Write in Standard Form

The simplified expression is the sum of the real part and the imaginary part: \(-10 - 20i\).Writing in the standard form \(a + bi\),we have: \(-10 + (-20i)\).

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics is akin to tidying up a room—it involves combining like terms and reducing the complexity of expressions to make them easier to understand or evaluate. When working with complex numbers, this process is essential. Complex numbers have both real and imaginary components, typically denoted as \(a+bi\). To simplify, we distribute each term across others, then combine all similar terms.

• Distribution Process: Each term in one of the parentheses is multiplied by every term in the other. This method is commonly referred to as the FOIL method (First, Outer, Inner, Last) when extending to binomials.
• Combining Like Terms: After distribution, terms are combined based on their components—real parts go with real parts, imaginary with imaginary.
• Simplification of \(i^2\): A key mistake is forgetting that \(i^2 = -1\). Remembering this is crucial when simplifying terms like \(-6i^2\), which becomes \(6\) since \(-6 imes (-1) = 6\).

Simplification not only aids understanding but also mastering complex numbers and their operations.

Standard Form

The standard form of a complex number is a neat way to write complex numbers, and it is represented as \(a + bi\). This standardization is crucial because it allows mathematicians to easily understand and manipulate complex numbers and avoids any confusion. Let's break this down:

• Real Component \(a\): This is the first term and is a real number. It's what makes the complex number grounded in the real number system.
• Imaginary Component \(bi\): The second term, where \(b\) is a real number, and \(i\) is the imaginary unit \(\sqrt{-1}\). This part defines the number's position in the imaginary realm.

Whenever tasked with expressing a result in standard form, ensure the last step involves arranging the number in the format \(a + bi\). This consistent method helps in comparing and operating on complex numbers with ease.

Imaginary Numbers

Imaginary numbers might sound like something out of a sci-fi novel, but in mathematics, they are very real tools used to solve equations that can't be solved using real numbers alone. Imaginary numbers are based on the imaginary unit \(i\), which is defined as \(\sqrt{-1}\).

• Imaginary Unit \(i\): This is a fundamental concept, where \(i^2 = -1\). It denotes an extension of the real number system, allowing for solutions beyond real numbers.
• Why Use Imaginary Numbers? They enable the taking of square roots of negative numbers, essential in various fields like engineering and physics, where real numbers fall short.
• Applications: Beyond pure mathematics, imaginary numbers have practical applications. They are crucial for solving electrical problems, signal processing, and even quantum physics.

Understanding imaginary numbers opens the doors to advanced mathematical concepts and their real-world applications, bridging gaps where real numbers alone would fail.