Question

OPEN-ENDED Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related?

Short answer

Any two imaginary numbers in the form of \(bi\) and \(-bi\) (conjugates) have a sum and product equal to real numbers. The sum of these numbers is 0, and their product is \(b^2\).

Step-by-step solution

Identify the imaginary numbers

Let's denote two imaginary numbers as \(bi\) and \(-bi\), where \(b\) is any real number and \(i\) is the imaginary unit.

Find the sum and product of the imaginary numbers

Sum of the two imaginary numbers \(bi\) and \(-bi\) is \(bi - bi\) which results in \(0\), a real number. The product of the two imaginary numbers is \((bi) * (-bi) = -b^2i^2\). We know that \(i^2 = -1\), so substituting here would give us a real number.

Substitution

Substitute \(i^2\) with \(-1\) in the product, we get \(b^2\), which is a real number.

Conclusion

Therefore, the sum and product of the given imaginary numbers \(bi\) and \(-bi\) are real numbers. The imaginary numbers are related in such a way that they are conjugates of each other.

Key concepts

Understanding Complex Numbers

In the world of mathematics, the realm of numbers extends beyond what we can typically quantify in the physical world. This brings us to the concept of complex numbers. A complex number is expressed in the form of a + bi, where a is the real part, b is the imaginary part, and i is the square root of -1, which is an imaginary unit. Here's the fascinating part: even though i represents something beyond traditional comprehension – the square root of a negative number – when combined with real numbers, you get an entirely new set of numbers with both magnitude and direction.

For instance, the number 3 + 4i has a real component of 3 and an imaginary part of 4i. Complex numbers are crucial in various fields, including engineering and physics, because they can represent oscillations, waves, and many other phenomena that can't be described using only real numbers.

The Role of Conjugate Pairs

When diving into the properties of complex numbers, one stumbles upon the concept of conjugate pairs. A conjugate pair consists of two complex numbers that have the same real component but opposite imaginary components; essentially, they are mirror images of each other across the real axis on the complex plane. Given a complex number a + bi, its conjugate is a - bi.

The significance of conjugates becomes clear when considering operations such as addition and multiplication. When a complex number is added to its conjugate, the result is purely a real number, as the imaginary parts cancel each other out. This is exactly what was demonstrated in the textbook exercise with bi and -bi. Likewise, when a complex number is multiplied by its conjugate, the result is also a real number. This property of conjugate pairs is not only important in simplifying complex expressions but is also vital in many applications, such as finding the magnitude of a complex number and solving polynomial equations.

Real Numbers as a Foundation

While complex numbers and their conjugates constitute a vast and intriguing topic, they are built upon the bedrock of real numbers. Real numbers include all the numbers on the number line, encompassing rational numbers like fractions and integers, as well as irrational numbers like the square root of 2 or the number pi. For the lack of a better term, they are the numbers we encounter in everyday life. Simple operations such as addition, subtraction, multiplication, and division – as long as we're not dividing by zero – all stay within the realm of real numbers. They form a complete and ordered field, and every point on the number line corresponds to a real number.

In the context of the exercise, the sum and product of the imaginary numbers resulted in real numbers, showcasing the seamless interaction between complex numbers and real numbers. The real numbers serve as the horizontal axis in the complex plane, anchoring complex numbers to a dimension we understand and can easily work with.