Problem 103 - Algebra for College Students

Question

Can the sum of two nonreal complex numbers be a real number? Defend your answer.

Step-by-Step Solution

Answer

Yes, if their imaginary parts cancel each other out.

Step 1: Understand the components of a complex number

A complex number is usually in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Two nonreal complex numbers must both have non-zero imaginary parts (i.e., \(b eq 0\)).

Step 2: Consider the sum of two nonreal complex numbers

Let's consider two nonreal complex numbers: \( (a_1 + b_1i) \) and \( (a_2 + b_2i) \). When we add them together, the sum becomes \((a_1 + a_2) + (b_1 + b_2)i\).

Step 3: Determine when this sum is a real number

The resulting sum \((a_1 + a_2) + (b_1 + b_2)i\) becomes a real number if and only if the imaginary part is zero, which means \(b_1 + b_2 = 0\). This means that the imaginary parts of the two numbers must cancel each other out.

Step 4: Conclusion

If the imaginary parts \(b_1\) and \(b_2\) are such that they sum to zero, the sum of two nonreal complex numbers can indeed be a real number. Thus, it is possible for the sum to be real under specific conditions where \(b_1 = -b_2\).

Key Concepts

Imaginary Part

In complex numbers, the imaginary part is the component that includes the letter \(i\), the symbol for the imaginary unit. This unit, \(i\), represents the mathematical concept of the square root of -1. In a typical complex number notation \(a + bi\), \(b\) is the coefficient of \(i\), and is referred to as the imaginary part. It's important to note that the imaginary part is essentially what differentiates complex numbers from real numbers.Finding the imaginary part is straightforward. Just look at the term accompanying the \(i\). For example:

• In the complex number \(3 + 4i\), the imaginary part is 4.
• In the number \(-2 + 5i\), the imaginary part is 5.

It’s crucial to remember that the imaginary part can be positive, negative, or even zero. When the imaginary part is zero, the complex number behaves like a real number.

Real Part

The real part of a complex number is just what it sounds like—the part that’s real and not multiplied by the imaginary unit \(i\). In the expression \(a + bi\), the real part is \(a\).Let's take a look at some examples to strengthen our understanding:

• For the complex number \(7 + 3i\), the real part is 7.
• In the complex number \(-4 + 6i\), the real part is -4.

By being able to identify the real part, you can easily work with complex numbers in both addition and subtraction. It helps to think of the real part as counting quantities that are "normal", or on the familiar real number line, whereas the imaginary part stretches into a new dimension, known as the imaginary axis.

Complex Number Addition

Adding complex numbers is relatively simple, and it involves combining the respective real and imaginary parts separately. If you have two complex numbers, \(a_1 + b_1i\) and \(a_2 + b_2i\), their sum is given by:\[(a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i\]The process can be broken down as:

• Add the real parts \(a_1\) and \(a_2\) to get the new real part.
• Add the imaginary parts \(b_1\) and \(b_2\) to get the new imaginary part.

An interesting property of complex number addition is that the sum may remain a complex number, or it could become a purely real number if the imaginary parts cancel each other out. For instance:- Adding \(3 + 4i\) and \(5 - 4i\) yields \((3 + 5) + (4 - 4)i = 8 + 0i = 8\), a real number.Understanding complex number addition can help you solve equations and problems efficiently where complex numbers are involved, especially when checking for conditions like provided in the initial exercise.