Problem 72 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((3+7 i)(3-7 i)\) is an imaginary number.

Step-by-Step Solution

Answer

\((3+7 i)(3-7 i)\) is a real number. The statement is false. To make a true statement it should be: \((3+7 i)(3-7 i)\) is a real number.

Step 1: Identify the Expressions

The given expressions \((3+7 i)\) and \((3-7 i)\) are conjugate complex numbers. The multiplication rule of complex numbers states that the product of a complex number and its conjugate equals the square of the modulus of that complex number.

Step 2: Perform the Multiplication

Multiply the given complex numbers using foil method where (First Outer Inner Last). \((3+7 i)(3-7 i)\) equals to \((3*3)+(3*-7i)+(7i*3)+(7i*-7i)\) which simplifies to \(9-21i+21i-49\).

Step 3: Simplify the Result

Combine the real and imaginary parts. Upon simplification the imaginary parts cancel out and \(9-49\) gives \(-40\).

Key Concepts

Conjugate Complex Numbers

Conjugate complex numbers are pairs of complex numbers that have the same real component but opposite imaginary parts. Given a complex number, say, \(a + bi\), the conjugate is \(a - bi\). These pairs are important in various mathematical computations and have unique properties.
For example:

• The conjugate of \(3 + 7i\) is \(3 - 7i\).
• When multiplied together, conjugates always yield a real number.

This property makes conjugates particularly useful in simplifying expressions and rationalizing denominators in complex fractions.

Multiplication of Complex Numbers

When multiplying complex numbers, we often use the FOIL method, which stands for First, Outer, Inner, Last. This method helps us keep track of each part of the multiplication. Consider the expression \((a + bi)(c + di)\). The expanded form becomes:
\(ac + adi + bci + bdi^2\).
The key step is to remember that \(i^2 = -1\), which simplifies the expression.

• Apply the FOIL method to \((3+7i)(3-7i)\).
• Result: \(9 - 21i + 21i - 49i^2\).

After simplification, the imaginary parts \(-21i + 21i\) cancel out, and we replace \(i^2\) with \(-1\), resulting in a real number \(-40\). This example illustrates how conjugates help to eliminate imaginary parts.

Imaginary Numbers

Imaginary numbers are a class of numbers which, when squared, yield a negative result. They are represented as a multiple of \(i\), where \(i\) is the imaginary unit and \(i^2 = -1\).
These numbers play a vital role in complex number arithmetic and have a wide range of applications in fields such as engineering and physics.

• Imaginary numbers originate from the need to find the square roots of negative numbers.
• They complete the number system, extending it beyond real numbers.

An essential fact to remember is that while some operations with complex numbers might initially involve imaginary numbers, they can often simplify to real numbers, as seen in the multiplication of conjugates.

Modulus of a Complex Number

The modulus of a complex number, often denoted as \(|a + bi|\), measures its "size" or magnitude. It is calculated as the square root of the sum of the squares of its real and imaginary parts: \(\sqrt{a^2 + b^2}\).
This concept mirrors taking the magnitude of a vector in a two-dimensional plane.

• For the complex number \(3 + 7i\), the modulus is \(\sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}\).
• When multiplying a complex number by its conjugate, the result is the square of the modulus.

Understanding the modulus is crucial, especially when dealing with polar forms of complex numbers and exploring their geometric interpretations.