Problem 12 - Elementary and Intermediate Algebra

Question

Fill in the blanks. 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: Understanding Complex vs. Real Numbers

To determine the truth of these statements, recall the definitions: A real number is a point on the real number line, while a complex number is of the form \(a + bi\) (where \(i = \sqrt{-1}\) and \(a, b\) are real numbers). It follows that not every complex number is real, specifically when \(b eq 0\). Thus, statement a. is false.

Step 2: Complex to Real Relationship

Every real number can be expressed as a complex number by setting \(b = 0\), which makes the number \(a + 0i\) (a purely real complex number). Therefore, every real number is indeed a complex number, making statement b. true.

Step 3: Reviewing 'i' in the Real Number Context

The imaginary unit \(i\) represents \(\sqrt{-1}\) and is not located on the real number line. Thus, \(i\) is not a real number, making statement c. false.

Step 4: Analyzing Square Roots of Negative Numbers

The square root of any negative number cannot yield a real number, and is defined using the imaginary unit \(i\). So, the square root of a negative number is an imaginary number, making statement d. true.

Step 5: Product of a Complex Number and Its Conjugate

For a complex number \(z = a + bi\), its conjugate \(\overline{z}\) is \(a - bi\). Multiplying these gives \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\), which is always a sum of squares and therefore, a real number. Hence, statement e. is true.

Key Concepts

Real Numbers

Real numbers are what most people first encounter in mathematics. They include all the numbers you can find on the number line.
These numbers consist of both positive and negative whole numbers, fractions, and decimals without imaginary components. Real numbers are simple and form the foundation of everyday arithmetic.

• Examples of real numbers include \( -3, 0, 7, 3.5, \text{and } \pi \).
• Real numbers can be easily added, subtracted, multiplied, and divided, given that the division does not involve zero.

Important to note is that every real number can also be considered a complex number. This is because a complex number has the form \( a + bi \) where, for real numbers, \( b = 0\). Thus, the expression becomes merely \( a + 0i = a \), a real number. This means while not all complex numbers are purely real (i.e., when \( b \) is not zero), all real numbers qualify as complex numbers.

Imaginary Numbers

Imaginary numbers may seem mysterious, but they help solve mathematical problems that real numbers cannot. The foundation of imaginary numbers is the unit \( i \), which represents the square root of \( -1 \).

• An imaginary number takes the form \( bi \), where \( b \) is a real number.
• Numbers like \( 3i, -5i, \text{and } 6.7i \) are purely imaginary.

Combining imaginary numbers with real numbers creates complex numbers. It's essential to remember that imaginary numbers cannot be placed on a traditional number line because they do not have a real "location" like how physical objects can be ordered or measured. An interesting aspect of imaginary numbers is that while they remain prominent in certain calculations, solutions involving real-world situations eventually return to real numbers, often through the product involving their complex conjugates.

Complex Conjugate

A complex conjugate is like a mirror image of a complex number with respect to the real number axis. If you have a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).

• The only difference between \( z \) and \( \overline{z} \) is the sign before the \( i \).
• A real number is a special case of a complex number where the imaginary part is zero, meaning the complex conjugate is the same as the number itself.

One of their important properties is that the product of a complex number and its conjugate always results in a non-negative real number. This is shown through the formula:\[(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\]Because both \( a^2 \) and \( b^2 \) are always non-negative, their sum will always be a real and non-negative number, anchoring the complex concept back into the real number spectrum.