Problem 16 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Let \(c\) be the set of complex numbers and \(z_{1}, z_{2}\) are in \(C .\) (i). Argument \(z_{1}=\) argument \(z_{2} \Rightarrow z_{1}=z_{2}\) (ii). \(\left|z_{1}\right|=\left|z_{2}\right| \Rightarrow z_{1}=z_{2}\) Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) both 1 and 2 (d) neither 1 nor 2

Step-by-Step Solution

Answer

(d) neither 1 nor 2 are correct.

Step 1: Understanding Arguments of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. It is denoted by \( \arg(z) \). If \( \arg(z_1) = \arg(z_2) \), it means that both \( z_1 \) and \( z_2 \) have the same angle, but not necessarily the same magnitude. Thus, \( z_1 \) may not equal \( z_2 \). This means statement (i) is incorrect.

Step 2: Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number \( z \) is given by \( |z| = \sqrt{a^2 + b^2} \) for a complex number \( z = a + bi \). If \( |z_1| = |z_2| \), it means that \( z_1 \) and \( z_2 \) are at the same distance from the origin, but they can be in different directions. Therefore, \( z_1 \) may not equal \( z_2 \). Thus, statement (ii) is also incorrect.

Step 3: Conclusion

Since neither statement (i) nor statement (ii) necessarily implies that \( z_1 = z_2 \), both statements are incorrect.

Key Concepts

Argument of a Complex Number

The argument of a complex number is an angle measure, and it helps us understand the direction of the number in the complex plane. If you imagine the complex plane as a regular coordinate system, the argument tells you which way to "point" from the origin to reach the complex number. It is symbolized by \( \arg(z) \), where \( z \) is the complex number. The argument is often expressed in radians and represents the angle between the positive real axis and the line that connects the origin to the point \( z \) on the complex plane. The range of the argument is typically between \(-\pi\) and \(\pi\).

• If \( \arg(z_1) = \arg(z_2) \), then both \( z_1 \) and \( z_2 \) share the same direction.
• However, identical angles don’t guarantee the numbers are equal because they might differ in their magnitudes.

This concept is crucial for understanding rotations in the complex plane and how angles affect complex number equality.

Magnitude of a Complex Number

The magnitude of a complex number, also known as its modulus, represents how "far" the number is from the origin on the complex plane. For any complex number \( z = a + bi \), the magnitude is calculated as \( |z| = \sqrt{a^2 + b^2} \). This formula looks like the Pythagorean theorem because it calculates the straight-line distance (or hypotenuse) from the origin \((0, 0)\) to the point \((a, b)\) on the complex plane.

• When \( |z_1| = |z_2| \), both numbers are equally distant from the origin.
• However, they could lie in different directions, indicating different arguments.

Understanding magnitude helps with visualizing the "size" of a complex number, which is independent of its direction on the complex plane. This concept is similar to considering absolute value in basic arithmetic—it's all about the distance.

Complex Plane

The complex plane, also known as the Argand plane, is a way to visualize complex numbers. Every complex number corresponds to a unique point on this grid, where the horizontal axis represents the real part of the number, and the vertical axis represents the imaginary part. For a complex number \( z = a + bi \), the point \((a, b)\) is plotted on the plane.

• The complex plane allows mathematicians and scientists to use geometric intuition to understand complex numbers.
• It creates a bridge between algebraic operations on complex numbers and geometric transformations like rotations and reflections.

Visualizing complex numbers on the complex plane aids in understanding arguments and magnitudes, making it easier to comprehend their behavior and relationships. By viewing complex numbers in this spatial manner, learners gain insight into both the number's magnitude and direction, which is essential for solving many mathematical problems.