Problem 19 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(z_{1}\) and \(z_{2}\) are any two complex numbers, then which of the following is not true. (a) \(\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\) (b) \(\left|z_{1}-z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\) (c) \(\left|z_{1}-z_{2}\right| \geq \| z_{1}|-| z_{2}||\) (d) \(\left|z_{1}\right|-\left|z_{2}\right| \geq\left|z_{1}-z_{2}\right|\)

Step-by-Step Solution

Answer

The correct answer is option (d), as it is not true.

Step 1: Analyze Option (a)

Option (a) states \(\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\). This is known as the Triangle Inequality for complex numbers, which is always true. Therefore, option (a) is true.

Step 2: Analyze Option (b)

Option (b) states \(\left|z_{1}-z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\). This is also a form of the Triangle Inequality applied to complex numbers, and it is always true. Thus, option (b) is true.

Step 3: Analyze Option (c)

Option (c) states \(\left|z_{1}-z_{2}\right| \geq \| z_{1}|-| z_{2}||\). This represents the Reverse Triangle Inequality, which is a fundamental property of norms in a vector space, and is always true. So, option (c) is true.

Step 4: Analyze Option (d)

Option (d) states \(\left|z_{1}\right| - \left|z_{2}\right| \geq \left|z_{1} - z_{2}\right|\). This suggests that the difference in magnitudes of two complex numbers is greater than or equal to the magnitude of their difference, which contradicts the Triangle Inequality, making it not true. Therefore, option (d) is not true.

Key Concepts

Understanding the Triangle Inequality

The triangle inequality is a fundamental concept in the study of complex numbers, and it plays a crucial role in many mathematical proofs and applications. It is expressed as: \[ |z_1 + z_2| \leq |z_1| + |z_2| \] This inequality shows that the magnitude (or absolute value) of the sum of two complex numbers is always less than or equal to the sum of their magnitudes. Here's a simple way to think about it: imagine two sides of a triangle with lengths corresponding to \(|z_1|\) and \(|z_2|\) respectively. According to the triangle inequality, the length of the side formed by adding these vectors, \(|z_1 + z_2|\), will always be less or in line with the sum \(|z_1| + |z_2|\).

• This is an inequality not just applicable to complex numbers, but also to vectors in general.
• It demonstrates how adding two magnitudes cannot produce a magnitude greater than the sum of individual magnitudes.

In practice, this helps to measure distances and can be used in mathematical analysis to compare vector lengths.

Exploring the Reverse Triangle Inequality

While the triangle inequality deals with the addition of magnitudes, the reverse triangle inequality concerns their differences. It is represented as: \[ |z_1 - z_2| \geq ||z_1| - |z_2|| \] This inequality tells us that the absolute difference in magnitudes cannot be greater than the magnitude of the difference between two complex numbers. In simpler terms, if you subtract two complex numbers, the resulting magnitude \(|z_1 - z_2|\) will always be equal or greater than the absolute difference of their magnitudes. This makes sure that:

• The absolute distances between points (like vectors) respect their individual magnitude differences.
• It provides a bound on what the minimum distance can be when dealing with differences.

This inequality is especially useful in proofs and calculations involving norms and helps ensure that values remain within logical physical constraints.

Delving into Complex Number Properties

Complex numbers extend the real number system to include solutions to equations like \( x^2 + 1 = 0 \). They are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined by \(i^2 = -1\). Here are key properties of complex numbers:

• Magnitude/Absolute Value: This is computed as \( |z| = \sqrt{a^2 + b^2} \), representing the distance from the origin in the complex plane.
• Conjugate: The conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \), which reflects the number across the real axis.
• Addition and Subtraction: Cand be performed by adding or subtracting their respective real and imaginary parts.

These basic properties are critical in manipulating complex numbers through arithmetic operations and for solving complex equations. Recognizing and applying these properties can simplify problem solving and extend mathematical models that include complex numbers.