# Complex values and size

Gold Member
complex values and "size"

The only measure of size I can think of would be to imagine a complex number as a vector and calculate its length via pythagoras. In that case, I would find i+1 and i-1 to be of equal length.. however using matlab's min function and < operand, I get two completely different answers (and different from each other too):

according to min, the first entry in A (i+1) is the "smallest"
Code:
A =

1.000000000000000 + 1.000000000000000i
-1.000000000000000 + 1.000000000000000i

>> min(A)

ans =

1.000000000000000 + 1.000000000000000i
but according to the lessthan sign, the second entry (i-1) is the "smallest"

Code:
 >> if A(1) < A(2)
dips('YES')
end
>>if A(2) < A(1)
disp('YES')
end
YES
are there alternative ways of measuring/comparing the "size" of complex values?

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there is no order among the complex numbers, but usually what I have seen is that complex functions have min and max values that refer to the min and max modulus the functions attain (this is what you were talking about for example |i+1| = |i-1| but I don't think it makes since to say if we have two complex numbers z and v, ask which one is "smaller" or the minimum of the two.

Gold Member

but with modulus you use the conjugate don't you? Hrm... guess it's the same result, just different definition. I can't think of complex numbers without thinking of it as a vector. i is just an orthogonal transform operator as far as I can tell.

by modulus i was just referring to the distance from the complex number to the origin. so in that case, yeah a complex number z would have the same modulus as its conjugate. all you really have to work with in terms of telling two complex numbers apart are its modulus and Argument, but i still can't think of that showing one numbers being greater than the other

just out of curiosity because i am just beginning to learn complex analysis but, why think of a complex number as a vector? i guess what i mean to say is, why not just think of a complex number as a complex number? i guess im just curious because i am new to the subject and i had never thought of them as vectors before. It seems interesting

Gold Member

My background is more physics, I guess the way I intuitively understand things is through spatial metaphor. "Complex number" is just a title. I know it's a real part and an imaginary part, but "imaginary part" is a terribly unhelpful name.

What is a complex number then?

Here's an explanation I heard that sat with me, I guess. It starts with the solution of:

x^2 + 1 = 0

which is

x = +/- sqrt(-1)
What does that mean? We’ve just seen that the square of a positive number is positive, and the square of a negative number is also positive, since multiplying one negative number, which points backwards, by another, which turns any vector through 180 degrees, gives a positive vector. Another way of saying the same thing is to regard the minus sign itself, -, as an operator which turns the number it is applied to through 180 degrees. Now has two such rotations in it, giving the full 360 degrees back to the positive axis.

To make sense of the square root of a negative number, we need to find something which when multiplied by itself gives a negative number. Let’s concentrate for the moment on the square root of –1, from the quadratic equation above. Think of –1 as the operator – acting on the vector 1, so the – turns the vector through 180 degrees. We need to find the square root of this operator, the operator which applied twice gives the rotation through 180 degrees. Put like that, it is pretty obvious that the operator we want rotates the vector 1 through 90 degrees.

But if we take a positive number, such as 1, and rotate its vector through 90 degrees only, it isn’t a number at all, at least in our original sense, since we put all known numbers on one line, and we’ve now rotated 1 away from that line. The new number created in this way is called a pure imaginary number, and is denoted by i.
http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/ComplexNumbers.htm

Gold Member

by modulus i was just referring to the distance from the complex number to the origin.
ah, ok. I thought the modulus was defined as:

sqrt(zz*)

where z* is the conjugate of z.

where I would just imagine i as an operator, (say transforming an i-hat to a j-hat) and you'd just take the magnitude of the vector (the distance to the origin). I mean, if you're talking about "distance" you're seeing it as a vector through two spatial dimensions aren't you?