Problem 16 - College Algebra

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(2+7 i)(2-7 i)$$

Step-by-Step Solution

Answer

The product of the two complex numbers is \(53\).

Step 1: Distribute the product

Begin by distributing the product just like you would with a binomial. Multiply each term in the first parentheses by each term in the second one.\n You get: \n\( (2)(2) + (2)(-7i) + (7i)(2) + (7i)(-7i) \).

Step 2: Simplify the expression

Next, simplify these terms: \n\( 4 - 14i + 14i - 49i^2 \). Remember, \(i^2 = -1 \). Replacing \(i^2\) with \(-1\), you get: \n\( 4 - 49(-1) \).

Step 3: Calculate the Result

Finally, the last step boils down to a simple arithmetic operation. You compute: \n\(4 + 49 = 53\).

Key Concepts

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which satisfies \(i^2 = -1\). This form is particularly useful because it clearly separates the real part of the complex number, \(a\), from the imaginary part, \(b\). When multiplying complex numbers, it's important to use this standard form to systematically apply multiplication rules.

For example, when given the product \((2+7i)(2-7i)\), each complex number is already in standard form. The real part of each complex number is 2, and the imaginary parts are \(7i\) and \(-7i\) respectively. Multiplication of these follows the same principles as multiplying polynomials, and in the end, the result is simplified back into the standard form, which, for the given problem, is a real number \(53\). This reinforces that complex numbers can multiply to yield real numbers.

Distributive Property

When multiplying complex numbers, the distributive property allows a systematic approach to find the product. This property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. Applying this to complex numbers, each term in one binomial is multiplied by every term in the other binomial.

In the exercise \((2+7i)(2-7i)\), the distributive property is applied twice. First, distribute the real number 2 in the first complex number across the second complex number: \((2)(2) + (2)(-7i)\). Secondly, distribute the imaginary number \(7i\) across the second complex number: \((7i)(2) + (7i)(-7i)\). These steps create four terms that combine both like and unlike terms. Simplification of these results leads to the final standard form. The distributive property ensures that we account for all possible products in the multiplication process.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers, defined by the property that \(i^2 = -1\). This definition is what sets apart complex numbers from real numbers, as no real number's square gives a negative result. The imaginary unit allows us to represent the square roots of negative numbers and perform arithmetic that would otherwise be undefined within the realm of real numbers.

In our example, one of the terms from the distributive property is \((-49i^2)\), which incorporates the imaginary unit. Since \(i^2\) equals -1, this term simplifies to \(+49\). This shows how the rules governing the imaginary unit interact with the multiplication of complex numbers. When multiplying complex numbers, it's important to remember that \(i^2\) should always be replaced with -1 as part of the simplification process, which can lead to real number results even in operations that initially include imaginary parts.