Problem 7 - Algebra for College Students

Question

Label each statement true or false. The sum of two complex numbers is sometimes a real number.

Step-by-Step Solution

Answer

True.

Step 1: Understand Complex Numbers

Complex numbers are numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).

Step 2: Understand When Sum is Real

The sum of two complex numbers \((a + bi) + (c + di) = (a+c) + (b+d)i\) is real if the imaginary part \((b+d)i\) equals zero, which occurs when \(b + d = 0\).

Step 3: Consider Special Cases

Consider two complex numbers like \(3 + 4i\) and \(-3 - 4i\). Their sum is \((3 + 4i) + (-3 - 4i) = 0 + 0i\), which is a real number, showing that the imaginary part cancels to zero.

Step 4: Generalize the Condition

From the special case above, we can generalize that the sum of two complex numbers is real when their imaginary parts are additive inverses (i.e., sum to zero). This condition can sometimes be met, depending on the choice of numbers.

Step 5: Final Assessment of Statement

Based on the condition found, the statement "The sum of two complex numbers is sometimes a real number" is true because there exist cases where the imaginary parts cancel each other, resulting in a real number.

Key Concepts

Imaginary Unit

When dealing with complex numbers, the concept of the imaginary unit is pivotal. The imaginary unit is represented by the symbol \(i\), and it is defined by the property \(i^2 = -1\). This definition comes from the need to extend the real number system to solve equations like \(x^2 + 1 = 0\), which have no solutions within the real numbers.

With this, the imaginary unit allows us to include solutions to polynomials and other equations that require more than just real numbers. So, whenever you see \(i\), remember it's not a typical number but a crucial component of complex numbers. It provides a bridge from real numbers to complex numbers.

Real Numbers

Real numbers are numbers that can be found on the number line. They include both positive and negative integers, fractions, and irrational numbers. Real numbers are everywhere in everyday life, whether we're measuring distances, calculating areas, or balancing checkbooks.

• Real numbers have no imaginary part (when considering the complex number form \(a + bi\), here \(b = 0\)).
• They can be whole numbers, fractions, square roots, or mathematical constants like \(\pi\) and \(e\).

In the realm of complex numbers, real numbers form the "real part" and stand alone on the number line without any imaginary component.

Complex Addition

Complex addition involves adding two complex numbers. Consider two complex numbers \(a + bi\) and \(c + di\). To add these numbers, simply perform the addition separately for their real and imaginary parts:

• Add the real components: \(a + c\).
• Add the imaginary components: \(bi + di = (b + d)i\).

The result is \((a+c) + (b+d)i\).

A special case arises when the imaginary parts cancel each other out. This can occur if \(b + d = 0\). In such cases, the result is purely real, demonstrating that the sum of two complex numbers can sometimes be a real number.

Additive Inverses

Additive inverses are numbers that, when added together, yield zero. This concept extends into the complex number domain. For any complex number \(b + di\), its additive inverse is \(-b - di\). Adding these will result in their sum being zero:

• Consider the imaginary component of a complex number \(bi\); its inverse is \(-bi\).
• When summed \(bi + (-bi)\), the imaginary part cancels out, resulting in zero.

This principle is key to understanding when the sum of two complex numbers can be a real number since it depends on their imaginary parts being additive inverses. When the imaginary parts effectively neutralize each other, the remaining sum is purely real.