# Absolute value of imaginary numbers?

This is a discussion on Absolute value of imaginary numbers? within the A Brief History of Cprogramming.com forums, part of the Community Boards category; Originally Posted by brewbuck These numbers are quite REAL, at least so much as any other number you've ever encountered ...

1. Originally Posted by brewbuck
These numbers are quite REAL, at least so much as any other number you've ever encountered is "real," and play important roles in explaining physical phenomena.
Woah! Poor choice of words my friend. I understand the point you're trying to make, but the term real number is a well defined concept in mathematics (the set of real numbers is denoted by R)and any number with an imaginary part is most certainly NOT in the set of real numbers (denoted by C).

All real numbers are in the set of complex numbers I should probably add (since a real number is just a complex number with b=0)

2. Originally Posted by tabstop
And for that matter, there are an infinite number of norms (this is the 2-norm, but there's also the 1-norm and the 3-norm and the ... and the infinity-norm). See article.
Maybe ancient hieroglyphics would be easier to understand?
My brain just gets a stack overflow when I look at that page.

3. Originally Posted by cpjust
Maybe ancient hieroglyphics would be easier to understand?
My brain just gets a stack overflow when I look at that page.
Don't worry, because that page is irrelevant to the discussion in this thread.

4. Originally Posted by Stonehambey
Woah! Poor choice of words my friend. I understand the point you're trying to make, but the term real number is a well defined concept in mathematics (the set of real numbers is denoted by R)and any number with an imaginary part is most certainly NOT in the set of real numbers (denoted by C).

All real numbers are in the set of complex numbers I should probably add (since a real number is just a complex number with b=0)
I understand the basics.

I tried for a while to find a word that meant the same as "real" in the sense I am trying to use, but could not find a good one. I decided not to waste my time finding some obscure term just because a few mathematicians decided to make everything so incredibly confusing by picking bad terminology.

5. Originally Posted by Sang-drax
The formula
|a + ib| = sqrt(a^2 + b^2)
is true by definition. There is no proof. Given this definition, the fact that the "imaginary axis is perpendicular to the real axis" follows from this.
.....
By the way, your question is a good one. It is important to keep track of what is defined and what is derived.
This is really, really well said, and I also think the OP's questions are really good...it shows he's actually thinking about the topic at hand. In my experience you need to, at some point, simply accept the mathematics as valid. I also continually find myself asking 'why,' and as Sang-drax said you get to a point where the answer is 'because it's defined that way,' even though that answer sometimes seems arbitrary. Obviously this isn't always the case, but for something as 'tried and true' as imaginary numbers it is.

I tried for a while to find a word that meant the same as "real" in the sense I am trying to use, but could not find a good one. I decided not to waste my time finding some obscure term just because a few mathematicians decided to make everything so incredibly confusing by picking bad terminology.
Instead of saying 'real' I would've said 'useful.'

A really good book that discusses in endless detail the connection between mathematics and reality is 'Science and Sanity' by Korzybski. There's a famous quote that goes along nicely with this:

Originally Posted by Albert Einstein!
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality
In one of my math classes we would get 'slapped on the wrist' for referring to |X| as 'absolute value' rather than the more appropriate term, 'magnitude,' precisely due to this type of confusion when dealing with complex numbers.

6. I hated maths of all kinds at university and barely passed it, but discussions such as this one show the value of that education in the basics. We built the number systems we use formally, starting with natural numbers, then extending to whole numbers, rationals, reals and finally complex numbers - why all of them are needed and why they work the way they do.

One important thing to remember is that whenever you extend your current model, you need to recheck all the assumptions you made about the old model - they may no longer hold for the new one. In effect, you'd be generalizing from a subset with special properties to the whole, and that's extremely dangerous.

For example, the assumption that given |x| = |y|, x&#178; = y&#178; is perfectly valid for real numbers. It can be and has been proven - it trivially follows from the definition of || as sqrt(a&#178; + b&#178 when b = 0, which is the defining trait of real numbers. When you extend to complex numbers, where b may be != 0, you need to prove the conjecture again, and this time you'll fail.

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