Problem 119 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i\) a. \((2 i)^{2}\) b. \((2+i)^{2}\)

Step-by-Step Solution

Answer

a. \(-4 + 0i\), b. \(3 + 4i\)

Step 1: Square the Imaginary Unit

Given the expression \((2i)^2\), we will use the property that \(i^2 = -1\). Thus,\[(2i)^2 = (2)^2 imes (i)^2 = 4 imes i^2\]Substitute \(i^2 = -1\) to find the result:\[4 imes -1 = -4\]Thus, \((2i)^2 = -4\). In \(a + bi\) form, this is \(-4 + 0i\).

Step 2: Square the Binomial with Complex Terms

The given expression is \((2 + i)^2\). Use the binomial theorem for squaring:\[(2 + i)^2 = 2^2 + 2 imes 2 imes i + i^2\]Calculate each term separately:- \(2^2 = 4\)- \(2 imes 2 imes i = 4i\)- \(i^2 = -1\)Substitute these into the expanded form:\[(2 + i)^2 = 4 + 4i + (-1)\ = 3 + 4i\]Thus, the result in \(a + bi\) form is \(3 + 4i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \(i\), is a fundamental concept in the field of complex numbers. It's defined as the square root of negative one, \(i = \sqrt{-1}\). This definition allows us to extend the real number system into the complex realm, providing solutions to equations that don't have real number solutions.
Here are a few key points about the imaginary unit:

• When you square the imaginary unit, you get \(i^2 = -1\). This is the most important property of \(i\).
• The powers of \(i\) repeat in a cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This cyclical behavior repeats for higher powers.
• In complex numbers, which are expressed as \(a + bi\), the term \(bi\) represents the imaginary part.

When you perform operations with complex numbers, this property of \(i^2 = -1\) is leveraged to simplify expressions, as we saw in the exercise where \( (2i)^2 \) simplifies to \(-4\).

Binomial Theorem

The binomial theorem is a crucial tool when working with powers of expressions and plays an essential role in algebra and calculus. In the context of complex numbers, it simplifies the process of raising binomials to powers, such as squaring a binomial like \((2 + i)^2\).
The binomial theorem states:

• When expanding \((a + b)^n\), you apply the formula \((a + b)^n = \sum_{k=0}^{n} {\binom{n}{k} a^{n-k} b^k}\), where \({\binom{n}{k}}\) is a binomial coefficient.
• For small values of \(n\), such as 2 in our example, it's simple enough to use the identity: \((a + b)^2 = a^2 + 2ab + b^2\).
• This formula allows you to find each term individually, thus simplifying the calculation significantly.

By using this principle, we efficiently calculated \((2+i)^2\) in the solution, resulting in \(3 + 4i\) by expanding and simplifying each component term.

Complex Conjugate

A complex conjugate might not have been explicitly required in the provided operations, yet it remains an essential concept tied to complex numbers. Each complex number \(a + bi\) has a complex conjugate, which is \(a - bi\). These two numbers have identical real parts but opposite imaginary parts.
Key features of complex conjugates:

• The product of a complex number and its conjugate is always a real number: \((a + bi)(a - bi) = a^2 + b^2\).
• Conjugates are used to divide complex numbers by eliminating the imaginary unit in the denominator.
• In mathematics and engineering, they're instrumental in simplifying expressions and finding orthogonal projections.

Understanding complex conjugates is critical for operations involving division or simplifying ratios of complex numbers. Though not part of the exercise, appreciating this concept aids in grasping the full scope of complex arithmetic.