Problem 11 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ (2+3 i)^{2} $$

Step-by-Step Solution

Answer

The expression \((2+3i)^2\) simplifies to \(-5 + 12i\).

Step 1: Expand the Expression

To start solving \[(2 + 3i)^{2}\]we use the formula \[(a + b)^{2} = a^{2} + 2ab + b^{2}\]where \(a = 2\) and \(b = 3i\).

Step 2: Calculate Each part

First, compute \[a^{2} = 2^{2} = 4\]Next, compute \[2ab = 2(2)(3i) = 12i\]Lastly, compute \[b^{2} = (3i)^{2} = 9i^{2}\] Remembering that \(i^{2} = -1\), so \[b^{2} = 9(-1) = -9\].

Step 3: Combine and Simplify the Results

Now combine the parts:\[4 + 12i - 9\]Combine the real parts and imaginary parts separately:\[(4 - 9) + 12i = -5 + 12i\]

Key Concepts

Imaginary Unit

In the realm of complex numbers, we encounter the unique and fascinating concept of the imaginary unit, denoted as \(i\). The imaginary unit is defined such that \(i^2 = -1\). This might seem a bit puzzling at first since no real number squared equals -1. It's a tool mathematicians use to expand the real number system, allowing for the solution of equations that would otherwise have no real solutions.

When dealing with complex numbers, the imaginary unit helps represent numbers in the form \(a + ib\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part and \(ib\) is the imaginary part. The concept might appear abstract initially, but it's essential in solving and simplifying complex arithmetic problems, including exercises like \((2+3i)^2\).

Understanding \(i\) is straightforward but critical as it introduces a new dimension to the number world. In practical terms, it lets us define an entirely new class of numbers that help in a myriad of mathematical applications. The crucial takeaway is remembering that \(i^2 = -1\), as it guides us through many problems involving complex numbers.

Complex Arithmetic

Arithmetic involving complex numbers takes us one step beyond the familiar operations with real numbers. Complex numbers allow addition, subtraction, multiplication, and division just like real numbers, each operation requiring careful consideration of both the real and imaginary parts.

When multiplying complex numbers or expressions such as \((2 + 3i)^2\), it is important to expand the expression methodically. We use the distributive property and rules familiar from algebra but apply them with the consideration of the imaginary unit \(i\). For example, when expanding \((2 + 3i)^2\), we carefully calculate each product: the square of the real part, the product of the parts twice, and the square of the imaginary part, remembering that \(i^2 = -1\).

• Calculate \(a^2\) for the real part.
• Multiply \(2ab\), realizing this results in an imaginary component.
• Remember that \(b^2\) contributes a real component \((-b)^2 \times (-1)\).

Combining these results involves adding and simplifying real and imaginary parts separately. This precision illustrates the elegant complexity of working with complex numbers.

Binomial Expansion

Binomial expansion is a critical tool in algebra that simplifies the process of expanding expressions raised to powers. It is especially useful with complex numbers, where the traditional arithmetic approaches require more careful attention to the properties of the numbers involved.

The binomial theorem provides a straightforward formula for expanding expressions of the form \((a+b)^n\), allowing us to compute powers of binomials systematically. In the case of complex numbers like \((2+3i)^2\), binomial expansion provides a methodical path to solving the problem.

• Apply the formula \((a+b)^2 = a^2 + 2ab + b^2\) for this particular case.
• Recognize \(a = 2\) and \(b = 3i\).
• Perform calculations for each term and simplify using \(i^2 = -1\).

By applying the binomial expansion, we avoid the clutter that can come with manual distribution, ensuring accuracy and clarity. The results from this approach give us clean, precise results, allowing us to confidently express the outcome in the standard complex form \(a + ib\). This process streamlines computations involving powers and products of complex numbers.