Problem 12 - Intermediate Algebra

Question

Determine whether each statement is true or false. a. Every complex number is a real number. b. Every real number is a complex number. c. \(i\) is a real number. d. The square root of a negative number is an imaginary number. e. The product of a complex number and its complex conjugate is always a real number.

Step-by-Step Solution

Answer

a) False b) True c) False d) True e) True

Step 1: Analyzing Statement a

Statement: Every complex number is a real number. Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with \( i^2 = -1 \). Certainly, if \( b = 0 \), the complex number \( a + 0i \) is a real number. However, if \( b eq 0 \), it is not a real number. Thus, the statement is false.

Step 2: Analyzing Statement b

Statement: Every real number is a complex number.A real number can be expressed as \( a + 0i \), which fits the definition of a complex number with \( b = 0 \). Therefore, every real number is also a complex number.Thus, the statement is true.

Step 3: Analyzing Statement c

Statement: \(i\) is a real number.The imaginary unit \( i \) is defined to be the square root of \(-1\), which is not a real number. Therefore, \( i \) is not a real number.Thus, the statement is false.

Step 4: Analyzing Statement d

Statement: The square root of a negative number is an imaginary number.The square root of a negative number can be written as \( \sqrt{-1} \, \sqrt{positive} \), where \( \sqrt{-1} = i \). Thus, the square root of any negative number always includes \( i \), making it an imaginary number.Thus, the statement is true.

Step 5: Analyzing Statement e

Statement: The product of a complex number and its complex conjugate is always a real number.For a complex number \( a + bi \), the complex conjugate is \( a - bi \). Their product is:\[ (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2 \]Since both \( a^2 \) and \( b^2 \) are real numbers, \( a^2 + b^2 \) is also a real number.Thus, the statement is true.

Key Concepts

Imaginary Numbers

Imaginary numbers arise when we encounter the square root of negative numbers. Quite different from real numbers, imaginary numbers help in solving equations that would otherwise have no real solutions. The imaginary unit is denoted as \(i\), where \(i\) is defined by the property:\[i^2 = -1\] This means that \(i\) is the square root of \(-1\). When you see the term \(i\) in a number, it's your cue that the number involves the imaginary component. Imaginary numbers can combine with real numbers to form what we call complex numbers, expressed as \(a + bi\).

• The `a` represents the real part.
• The `bi` is the imaginary part.

An example is \(3 + 2i\): 3 is the real part and \(2i\) is the imaginary part.

Real Numbers

Real numbers are the numbers we're most familiar with in everyday mathematics. They include all the whole numbers, fractions, and decimals that represent a fixed point on the number line. What makes real numbers particularly interesting is that every real number can also be considered a complex number. This is because a real number can be written as \(a + 0i\). Here, \(a\) is the real component, and \(0i\) indicates there is no imaginary component, making it purely real.

• For example, 5 is completely the same as \(5 + 0i\) when expressed as a complex number.
• This shows the seamless nature between real and complex numbers.

It's essential to know that not all complex numbers are real. Only those with an imaginary part of zero qualify as real numbers.

Complex Conjugate

A fascinating and useful concept within the realm of complex numbers is the complex conjugate. If you have a complex number, say \(a + bi\), its complex conjugate will be \(a - bi\). They differ only by the sign of the imaginary part. This simple switch has powerful implications in mathematics and engineering, especially when dealing with equations and simplifying expressions.

• When a complex number is multiplied by its conjugate, the result is always a real number.
• The multiplication can be boiled down to \((a + bi)(a - bi) = a^2 + b^2\), stripping away the imaginary component.

Understanding and using complex conjugates can greatly simplify solutions and bring equations back to the reality we easily comprehend.