Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Multiply as indicated. Write each product in standand form. $$i(2+7 i)(2-7 i)$$

Step-by-Step Solution

Answer

The product is in standard form: \\(0 + 53i\\).

Step 1: Multiply the Complex Conjugates

First, we'll multiply the complex conjugates \(2 + 7i\) and \(2 - 7i\). Recall that one of the useful properties of complex conjugates is: \((a+bi)(a-bi) = a^2 + b^2\).\ We have, \[(2 + 7i)(2 - 7i) = 2^2 + 7^2 = 4 + 49 = 53\]

Step 2: Distribute the Imaginary Unit

Next, distribute the \(i\) across the product we calculated in Step 1.\ So we have:\[i(53) = 53i\]

Step 3: Write in Standard Form

The result from Step 2 is already in standard form for a complex number, which is \(a + bi\). In this case, \a = 0\ and \b = 53\, making the expression \(0 + 53i\).

Key Concepts

Complex Conjugates

Complex conjugates are a fundamental concept when dealing with complex numbers. A complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For any complex number \(a + bi\), its conjugate is \(a - bi\).
The beauty of complex conjugates lies in their multiplication property:

• When you multiply a complex number by its conjugate, the product is always a real number, specifically \(a^2 + b^2\).
• This property eliminates imaginary parts, leaving a neat real integer.
• The product of a complex number and its conjugate often helps in simplifying complex equations, especially when dividing complex numbers.

Understanding how conjugates neutralize the imaginary part in a complex number multiplication is key in various mathematical applications.

Standard Form

Writing complex numbers in standard form is important for clarity and consistency. The standard form of a complex number is \(a + bi\), where \(a\) is the real component and \(bi\) is the imaginary component.
This form is favored because it clearly distinguishes between the real and imaginary parts. When expressing a final answer, especially in textbook exercises, adhering to the standard form can ensure that readers immediately recognize the structure of the complex number.

• In standard form, \(a\) can be zero, \(b\) can be zero, or both \(a\) and \(b\) can be non-zero.
• The purpose of standard form is to simplify complex numbers into a universally understandable format.
• Standard form is also crucial for matching solutions in academic settings, such as tests or standardized exams.

Imaginary Unit

The imaginary unit, denoted as \(i\), is foundational to understanding complex numbers. It is defined as the square root of \(-1\), so \(i^2 = -1\). This concept allows mathematicians to expand the number system beyond real numbers to include complex numbers.
Understanding the imaginary unit involves recognizing that it behaves like a variable with unique properties:

• \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), after which the powers of \(i\) repeat in cycles.
• \(i\) makes it possible to work with square roots of negative numbers, which are not defined in the realm of real numbers.
• When multiplying with \(i\), it can alter both the real and imaginary components of a complex number.

In the end, mastery of the concept of the imaginary unit underlies the effective manipulation of complex numbers and transforms abstract mathematical ideas into real-world applications.