Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ i(3+4 i)^{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-24 - 7i\).

Step 1: Expand the Square

We start by expanding \[(3 + 4i)^2 = (3 + 4i)(3 + 4i).\] Applying the distributive property (FOIL method), we get:\[= 3 \times 3 + 3 \times 4i + 4i \times 3 + 4i \times 4i = 9 + 12i + 12i + 16i^2. \]Since \(i^2 = -1\), we substitute this in:\[= 9 + 24i - 16. \]Simplifying gives:\[-7 + 24i.\]

Step 2: Multiply by i

Now, multiply the expanded form \(-7 + 24i\) by \(i\):\[i(-7 + 24i) = i(-7) + i(24i). \]This can be rewritten as:\[-7i + 24i^2.\]Again using \(i^2 = -1\), \[24i^2 = 24(-1) = -24.\]Thus the multiplication becomes:\[-24 - 7i.\]

Step 3: Identify Real and Imaginary Parts

The expression is now re-written in the standard form of a complex number, where \(-24\) is \(a\) (the real part), and \(-7\) is \(b\) (the imaginary part of the number in terms of \(i\)).Thus, \[-24 - 7i\] is already in the form \(a + bi\).

Key Concepts

Distributive Property

The distributive property is a fundamental aspect of mathematics, and it plays a crucial role in simplifying expressions, especially when dealing with polynomials and complex numbers. It states that for any three numbers, the multiplication distributes over addition. For example, if you have three numbers, say, \(a\), \(b\), and \(c\), the distributive property shows that \(a(b + c) = ab + ac\). This principle makes it a lot easier to manage expressions that involve both numbers and variables.

In the exercise we are discussing, this property was explicitly used to expand \((3 + 4i)(3 + 4i)\). This involved performing multiplication on each term in the bracket, essentially treating each term separately and then summing them all up.

• First, calculate \(3 \times 3\) to get \(9\).
• Next, find \(3 \times 4i\), which gives \(12i\).
• Then, \(4i \times 3\) also results in \(12i\).
• Finally, \(4i \times 4i\) becomes \(16i^2\). Remember, since \(i^2 = -1\), replace \(16i^2\) with \(-16\).

With these steps, the complete expansion simplifies to \(9 + 24i - 16\), which further simplifies to \(-7 + 24i\) after combining like terms. This consistent use of the distributive property ensures that complex expressions can be systematically broken down and understood.

Imaginary Numbers

Imaginary numbers might seem confusing at first glance, but they are an invaluable tool in mathematics, especially in engineering and physics. The most basic imaginary number is \(i\), defined as the square root of \(-1\). This means \(i^2\) is equal to \(-1\), an essential rule when working with imaginary numbers.

In the given exercise, imaginary numbers come into play prominently. When expanding \((3 + 4i)^2\), you involve terms like \(4i \times 4i\), leading to \(16i^2\). Since we know \(i^2 = -1\), this simplifies to \(16(-1)\), or \(-16\).

The use of "i" in multiplication further demonstrates how imaginary numbers transition between real numbers and what might initially seem to be abstract concepts. After the initial expansion in the exercise, the continuing calculation used \(i(-7 + 24i)\). Multiplying each term by \(i\), we get \(-7i + 24i^2\), which simplifies to \(-7i - 24\) using the property of \(i^2\).

An understanding of imaginary numbers helps one to appreciate the beauty of mathematics in providing solutions where no real numbers could suffice.

Standard Form of Complex Numbers

Complex numbers introduce a new dimension to mathematics beyond just real numbers. These numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, while \(b\) multiplied by \(i\) represents the imaginary part.

One of the goals when working with complex numbers is often to simplify them into this standard form. Doing so makes it easier to interpret and use in various mathematical applications, such as in electrical engineering or wave physics.

In the exercise example, after carrying out operations and simplifications, the expression \(-24 - 7i\) emerges. Here, \(-24\) is the real part \(a\), and \(-7\) is the coefficient of \(i\), representing the imaginary part \(b\). Thus, the final expression neatly fits into the form \(a + bi\), showcasing the usefulness of the standard form in organizing complex numbers.

Understanding this structure enables one to perform further mathematical operations such as addition, subtraction, and multiplication of complex numbers with ease. This clarity is what makes complex numbers powerful and practical in both theoretical and applied contexts.