Is it wrong or correct that the complex number represented by a vector remains unchanged after the translation of the vector? For example, I read a sentence in a book that "no matter where the vector is translated, the complex number it represents is the same". I don't understand it. For example, the complex number represented by the vector (1,1) is 1 + I. after the vector is translated 2 units along the X axis, the complex number it represents should be 3 + I. how can it remain unchanged?

Is it wrong or correct that the complex number represented by a vector remains unchanged after the translation of the vector? For example, I read a sentence in a book that "no matter where the vector is translated, the complex number it represents is the same". I don't understand it. For example, the complex number represented by the vector (1,1) is 1 + I. after the vector is translated 2 units along the X axis, the complex number it represents should be 3 + I. how can it remain unchanged?

You've confused the coordinates of a vector with the coordinates of a point
No matter where the vector (1,1) is translated (the starting point and the ending point change), its coordinates are always (1,1)
It's wrong to say that moving 2 units to the right means 3 + I. It's just that the end point is relative to the origin, not the original vector
The original vector is (1,1) relative to its starting point