Distance between two complex numbers

In summary, the conversation discusses the correct notation and understanding of representing complex numbers on an Argand diagram. It is confirmed that vector PQ should be written as |vector PQ| and not vector PQ, as the former represents the magnitude while the latter is a vector. The conversation also touches on the use of algebra in the notation.
  • #1
naav
18
0
Hi...i was wondering if someone could confirm if what i have below is correct...thanks...sorry i can't present a diagram...

z(1) = x + iy and z(2) = x(2) + iy(2) are represented by the vectors OP and OQ on an argand diagram...(O is the origin)...imagine the argand diagram...the upper left hand quadrant...(OQ has an argument of say 30 degrees and OP has an argument of 45 degrees - these pieces of information are not relevant anyway)...

is the following correct...

vector OP + vector PQ = vector OQ...

then vector PQ = vector OQ - vector OP

then vector PQ = |z(2) - z(1)|...

1. was wondering if my notation and understanding here is correct...i used algebra in the second line so i was wondering if that is legit...?...

2. is it correct to say in the last line the vector = the magnitude
...
 
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  • #2
naav said:
1. was wondering if my notation and understanding here is correct...i used algebra in the second line so i was wondering if that is legit...?...
That's fine.
2. is it correct to say in the last line the vector = the magnitude
That's wrong. You should have:

[tex]\vec{PQ} = z_2 - z_1[/tex]

[tex]|\vec{PQ}| = |z_2 - z_1|[/tex]

Or, in plain text:

vector PQ = z(2) - z(1)
|vector PQ| = |z(2) - z(1)|
 
  • #3
Hi...thank you very much...

i said in my earlier post...

then vector PQ = |z(2) - z(1)|...

and it was said that it should be...

|vector PQ| = |z(2) - z(1)|

1. isn't that the same thing...

that vector PQ = the magnitude of [z(2) - z(1)]...?...
 
  • #4
No, it is not the same thing: |vector PQ| is a number (the length of the vector PQ), not a vector.

Likewise "vector PQ" is a vector while "the magnitude of [z(2)-z(1)]" is a number.
 
  • #5
thank you very much...
 

1. What is the formula for finding the distance between two complex numbers?

The distance between two complex numbers, z1 and z2, can be calculated using the Pythagorean theorem: d = √((x2-x1)^2 + (y2-y1)^2), where x1 and y1 are the real and imaginary parts of z1, and x2 and y2 are the real and imaginary parts of z2.

2. Can the distance between two complex numbers be negative?

No, the distance between two complex numbers is always a positive value as it represents the length of the line segment connecting the two numbers on the complex plane.

3. What is the relationship between the distance between two complex numbers and their magnitudes?

The distance between two complex numbers is equal to the difference between their magnitudes. This can be shown using the formula for finding the distance and the definition of magnitude: d = √((x2-x1)^2 + (y2-y1)^2) = √((|z2|-|z1|)^2).

4. How does the distance between two complex numbers change if one of the numbers is multiplied by a constant?

The distance between two complex numbers is multiplied by the absolute value of the constant. This is because multiplying a complex number by a constant only changes its magnitude, not its direction, and the distance is based on the difference in magnitudes between the two numbers.

5. Can the distance between two complex numbers be zero?

Yes, the distance between two complex numbers can be zero if they are the same number. This means that their real and imaginary parts are equal, and they are located at the same point on the complex plane.

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