Problem 107 - Precalculus with Limits

Question

True or False? In Exercises 106 and \(107,\) determine whether the statement is true or false. Justify your answer. The product of two complex numbers is zero only when the modulus of one (or both) of the complex numbers is zero.

Step-by-Step Solution

Answer

The statement is true. The product of two complex numbers is zero only when the modulus of one (or both) of the complex numbers is zero.

Step 1: Understanding complex numbers

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The imaginary unit or unit imaginary number (i) is a mathematical constant which is equal to the square root of -1. For any real number, the square of this number is a negative real number, which is not a real number.

Step 2: Modulus of Complex numbers

The modulus of a complex number, usually denoted |z|, is the absolute value or magnitude of the complex number. It is given by the formula \(|z| = \sqrt{a^2 + b^2}\) where 'a' is the real part and 'b' is the imaginary part of the complex number.

Step 3: Product of two complex Numbers

The product of two complex numbers, say \(z1=a+bi\) and \(z2=c+di\) where a, b, c, and d are real numbers, is defined as \(z1*z2=(ac-bd) + (ad+bc)i\). If the product of two complex numbers is zero, based on the definition, both the real part and the imaginary part of the product need to be zero.

Step 4: Check the given Statement

For the product complex number to be zero, either or both complex numbers should be zero. This aligns with the given statement, so we verify that the statement is true.

Key Concepts

Modulus of Complex Numbers

The modulus of a complex number provides a way to measure its size. Think of it as the distance of the complex number from the origin in the complex plane. When we talk about a complex number, like \(a + bi\), the modulus is represented as \(|z|\). This is calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

Here, \(a\) is the real part and \(b\) is the imaginary part. This formula is essentially the Pythagorean theorem applied to the complex plane.
This concept is crucial because the modulus tells us about the 'size' or 'amplitude' of a complex number, regardless of its direction. If the modulus is zero, the complex number is zero, which also implies both \(a\) and \(b\) are zero.
Understanding the modulus is key when analyzing the statement that the product of two complex numbers is zero only when one or both of them have a zero modulus.

Product of Complex Numbers

Understanding the product of complex numbers involves a bit of algebra. When you multiply two complex numbers, there's a specific formula to follow. Let's consider two complex numbers, \(z_1 = a + bi\) and \(z_2 = c + di\).

• Their product is \(z_1 \times z_2 = (ac - bd) + (ad + bc)i\)

Here, the real part \((ac - bd)\) and the imaginary part \((ad + bc)i\) both need to be zero for the product to be zero.
For this to happen, either one (or both) of the complex numbers must be zero, leading to the conclusion that their moduli should be zero as well. This adheres to the original statement that the product of two complex numbers can be zero only when the modulus of one or both is zero. This highlights the importance of both the modulus and understanding the product, as they are interlinked in determining whether their product is zero.

Imaginary Unit

The imaginary unit, often denoted as \(i\), is fundamental to complex numbers. It is defined as \(i^2 = -1\). This may seem strange at first because we know from basic math that squaring any real number gives a positive result. However, the creation of \(i\) allows us to extend beyond the real number system.
When paired with real numbers, \(i\) helps in constructing complex numbers, such as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. It's important to note that the imaginary unit is essential to operating with complex numbers, especially in multiplication.
Because \(i\) represents a square root of \(-1\), it permits the combination of real and "imaginary" components into one number, which can then be used to solve equations that aren't solvable with just real numbers. This imaginary unit is crucial in understanding products and moduli of complex numbers, as it forms the backbone of complex number operations.