# Thread: Complex Numbers basics problem

1. ## Complex Numbers basics problem

hi... there has been a question which always crosses my mind whenever i see complex numbers...
we say..
|x| is defined as squareroot(x^2)
so,
why do we not write |i| as i... (squareroot(i^2)=i)? (note that 'i' means iota)
now u'll say that modulus function has also been defined as distance of the argument(i, in this case) from the origin..., which u say is i unit from the origin.. but why can't be the distance of a point from the origin be 'i'(iota) units ???? whats wrong with it ???

let me frame an example here..
u draw a graph with Y-axis scale as 1 unit = 3... then in reference to this graph, do we write |3|=1 (as distance from origin=1 unit) ????? the answer is a no.... then why cant we treat iota in the same way?? here also we are defining 1 unit of y-axis as 'i'....

2. ## Re: Complex Numbers basics problem

Originally Posted by pranjvas
hi... there has been a question which always crosses my mind whenever i see complex numbers...
we say..
|x| is defined as squareroot(x^2)
so,
why do we not write |i| as i... (squareroot(i^2)=i)? (note that 'i' means iota)
now u'll say that modulus function has also been defined as distance of the argument(i, in this case) from the origin..., which u say is i unit from the origin.. but why can't be the distance of a point from the origin be 'i'(iota) units ???? whats wrong with it ???
What's wrong with it is that "distance" is a positive real number (have you ever seen a meter stick or ruler with "i" on it). In order to get that we define the absolute value of a complex number as $\displaystyle |z|= \sqrt{z\overline{z}}$ where $\displaystyle \overline{z}$ is the "complex conjugate" of z. In particular, the complex conjugate of i is -i so that $\displaystyle |i|= \sqrt{i(-i)}= \sqrt{1}= 1$.

let me frame an example here..
u draw a graph with Y-axis scale as i unit = 3... then in reference to this graph, do we write |3|=1 (as distance from origin=1 unit) ????? the answer is a no.... then why cant we treat iota in the same way?? here also we are defining 1 unit of y-axis as 'i'....
I don't understand what you are trying to say here. What does "i unit= 3" mean? It sounds like you are just trying to redefine "i" to be "3i". You can't do that and have the same number system as before. If you start with "non-sense", labeling "3" as a "1", leads to non-sense: |3|= 1!

(This reminds me of a joke Abraham Lincoln used to tell: Question: if you call a lambs tail a leg, how many legs does it have? Answer: four- calling a tail a leg doesn't make it one! In mathematics we are pretty free to "call" things what we want. But once you have labled a point on the y axis "3" you cannot then also label it "i".)

3. ## Re: Complex Numbers basics problem

Originally Posted by HallsofIvy
What's wrong with it is that "distance" is a positive real number (have you ever seen a meter stick or ruler with "i" on it). In order to get that we define the absolute value of a complex number as $\displaystyle |z|= \sqrt{z\overline{z}}$ where $\displaystyle \overline{z}$ is the "complex conjugate" of z. In particular, the complex conjugate of i is -i so that $\displaystyle |i|= \sqrt{i(-i)}= \sqrt{1}= 1$.

I don't understand what you are trying to say here. What does "i unit= 3" mean? It sounds like you are just trying to redefine "i" to be "3i". You can't do that and have the same number system as before. If you start with "non-sense", labeling "3" as a "1", leads to non-sense: |3|= 1!

(This reminds me of a joke Abraham Lincoln used to tell: Question: if you call a lambs tail a leg, how many legs does it have? Answer: four- calling a tail a leg doesn't make it one! In mathematics we are pretty free to "call" things what we want. But once you have labled a point on the y axis "3" you cannot then also label it "i".)
i m sorry, it was a typing mistake.. what i meant was "1 unit = 3"... and not "i unit = 3"