Understanding Complex Numbers and Magnitude

A complex number $z$ is typically written in the form $z = x + iy$, where $x$ is the real part (Re($z$)) and $y$ is the imaginary part (Im($z$)), and $i$ is the imaginary unit with $i^2 = -1$. The magnitude (or modulus) of a complex number $z = x + iy$, denoted by $|z|$, is the distance of the complex number from the origin (0,0) in the complex plane. It is calculated as $|z| = \sqrt{x^2 + y^2}$.

The problem states that we have two complex numbers, $z_1$ and $z_2$, such that their magnitudes are equal: $|z_1| = |z_2|$. This condition means that $z_1$ and $z_2$ lie on a circle centered at the origin in the complex plane with the same radius.

Analyzing Statement 1: $z_1 = z_2$

The first statement claims that if $|z_1| = |z_2|$, then $z_1$ must be equal to $z_2$. For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must also be equal.

Let's test this with an example. Consider $z_1 = 3$ and $z_2 = -3$.

• $|z_1| = |3| = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$.
• $|z_2| = |-3| = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3$.

Here, $|z_1| = |z_2| = 3$. However, $z_1 = 3$ and $z_2 = -3$, so $z_1 \neq z_2$. The real parts are $3$ and $-3$, and the imaginary parts are both $0$. Since the real parts are different, the complex numbers are not equal.

Another example: Consider $z_1 = 1 + i$ and $z_2 = 1 - i$.

• $|z_1| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}$.
• $|z_2| = |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$.

Here, $|z_1| = |z_2| = \sqrt{2}$. However, $z_1 = 1 + i$ and $z_2 = 1 - i$, so $z_1 \neq z_2$ because their imaginary parts ($1$ and $-1$) are different.

These examples show that $|z_1| = |z_2|$ does not necessarily mean $z_1 = z_2$. Therefore, Statement 1 is not correct.

Analyzing Statement 2: Real part of $z_1$ = Real part of $z_2$

The second statement claims that if $|z_1| = |z_2|$, then the real part of $z_1$ must be equal to the real part of $z_2$. Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. The statement says $x_1 = x_2$. We are given $\sqrt{x_1^2 + y_1^2} = \sqrt{x_2^2 + y_2^2}$.

Let's use an example. Consider $z_1 = 1$ and $z_2 = -1$.

• $|z_1| = |1| = 1$.
• $|z_2| = |-1| = 1$.

Here, $|z_1| = |z_2| = 1$. The real part of $z_1$ is $\text{Re}(z_1) = 1$. The real part of $z_2$ is $\text{Re}(z_2) = -1$. Since $1 \neq -1$, $\text{Re}(z_1) \neq \text{Re}(z_2)$.

This example demonstrates that $|z_1| = |z_2|$ does not imply that their real parts are equal. Therefore, Statement 2 is not correct.

Analyzing Statement 3: Imaginary part of $z_1$ = Imaginary part of $z_2$

The third statement claims that if $|z_1| = |z_2|$, then the imaginary part of $z_1$ must be equal to the imaginary part of $z_2$. Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. The statement says $y_1 = y_2$. We are given $\sqrt{x_1^2 + y_1^2} = \sqrt{x_2^2 + y_2^2}$.

Let's use an example. Consider $z_1 = i$ and $z_2 = -i$.

• $|z_1| = |i| = \sqrt{0^2 + 1^2} = 1$.
• $|z_2| = |-i| = \sqrt{0^2 + (-1)^2} = 1$.

Here, $|z_1| = |z_2| = 1$. The imaginary part of $z_1$ is $\text{Im}(z_1) = 1$. The imaginary part of $z_2$ is $\text{Im}(z_2) = -1$. Since $1 \neq -1$, $\text{Im}(z_1) \neq \text{Im}(z_2)$.

Another example: Consider $z_1 = 1 + i$ and $z_2 = 1 - i$ (used before). $\text{Im}(z_1) = 1$ and $\text{Im}(z_2) = -1$. They are not equal, even though $|z_1| = |z_2|$.

These examples show that $|z_1| = |z_2|$ does not imply that their imaginary parts are equal. Therefore, Statement 3 is not correct.

Conclusion on Complex Number Equality and Magnitude

We have examined all three statements. In each case, we found counterexamples where $|z_1| = |z_2|$ but the statement is false. The condition $|z_1| = |z_2|$ only tells us that the complex numbers are the same distance from the origin. They can be located at different positions on the circle, having different real and imaginary parts (or swapped real/imaginary parts, or different signs for real/imaginary parts).

Therefore, none of the statements 1, 2, or 3 are necessarily correct based on the condition $|z_1| = |z_2|$.

Revision Table: Complex Number Properties

Additional Information: Geometric Interpretation of Complex Numbers

Complex numbers can be visualized as points in a plane called the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. A complex number $z = x+iy$ corresponds to the point $(x,y)$ in this plane.

The magnitude $|z|$ represents the distance from the origin $(0,0)$ to the point $(x,y)$. The condition $|z_1| = |z_2|$ means that the points corresponding to $z_1$ and $z_2$ lie on the same circle centered at the origin. Points on the same circle can have different coordinates (different real and imaginary parts) unless they are the exact same point.

For example, points $(3,0)$ and $(-3,0)$ are both at distance 3 from the origin, corresponding to $z_1=3$ and $z_2=-3$. Points $(1,1)$ and $(1,-1)$ are both at distance $\sqrt{2}$ from the origin, corresponding to $z_1=1+i$ and $z_2=1-i$. Points $(0,1)$ and $(0,-1)$ are both at distance 1 from the origin, corresponding to $z_1=i$ and $z_2=-i$. These examples clearly show that equal magnitude does not force the real or imaginary parts, or the numbers themselves, to be equal.