Can two different points in the complex plane represent the same complex number? Give reasons for you answer

Answer

To determine whether two different points in the complex plane can represent the same complex number, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Complex Plane**:
- The complex plane consists of a horizontal axis (real axis) and a vertical axis (imaginary axis).
- Any complex number \( z \) can be represented as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

2. **Defining Points in the Complex Plane**:
- Let’s consider two points in the complex plane: \( z_1 \) and \( z_2 \).
- Assume \( z_1 \) is represented by the coordinates \( (x_1, y_1) \) and \( z_2 \) by the coordinates \( (x_2, y_2) \).

3. **Condition for Equality of Complex Numbers**:
- For two complex numbers to be equal, their corresponding real and imaginary parts must also be equal.
- Therefore, \( z_1 = z_2 \) if and only if \( x_1 = x_2 \) and \( y_1 = y_2 \).

4. **Analyzing Different Points**:
- If \( z_1 \) and \( z_2 \) are two different points, then by definition, at least one of the coordinates must differ.
- This means either \( x_1 \neq x_2 \) or \( y_1 \neq y_2 \) (or both).

5. **Conclusion**:
- Since the coordinates of \( z_1 \) and \( z_2 \) cannot be the same if they are different points, it follows that \( z_1 \neq z_2 \).
- Therefore, two different points in the complex plane cannot represent the same complex number.

### Final Answer:
No, two different points in the complex plane cannot represent the same complex number because their coordinates (real and imaginary parts) must be equal for the complex numbers to be equal.

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