What is the absolute value of imaginary numbers, why not supernatural numbers?

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Discussion Overview

The discussion revolves around the concept of absolute value in relation to imaginary numbers, specifically questioning the nature of absolute values and the potential for defining new types of numbers, such as "queer" numbers. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the absolute value of imaginary numbers, specifically questioning |5i| and |-5i|.
  • One participant suggests that the absolute value of a "queer" number could be defined as a negative number of the same integer value.
  • Another participant questions the purpose of defining such numbers, suggesting it may lead to new mathematical discoveries.
  • Some participants discuss the definition of absolute value as a measure of distance from zero, emphasizing that distance is inherently positive.
  • A later reply mentions that defining a function that gives negative absolute values would not align with the conventional definition of absolute value.
  • One participant expresses doubt about the feasibility of creating a new number system with the properties described.
  • Another participant provides the formula for the absolute value of a complex number, confirming that |5i| and |-5i| both evaluate to 5.
  • Several participants note the age of the thread and the practice of revisiting older discussions.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of defining new types of numbers. While some agree on the established definition of absolute value, others propose alternative concepts without reaching a consensus.

Contextual Notes

The discussion includes various assumptions about the nature of absolute value and the properties of numbers, which remain unresolved. There is also a lack of clarity on the implications of introducing new number types.

bananan
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what is the absolute value of imaginary numbers, why not "queer" numbers?

the square root of -1 is "i".

the absolute value of an interger is itself, and of a negative number, it is a positive interger.

|-5| = 5
|5| = 5

what is
|5i| = ?
|-5i| = ?

why not invent a queer number?

the absolute value of a queer number is a negative number of the same interger value?


hence
|-5| = 5
|5| = 5
|5q| = -5
|-5q|= -5
 
Last edited:
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The better question is why would one do this?
 
why does i need to invent absolute value? it is because it forms a norm. norm of an element gives a measurement such that we can compare mathematcal objects.
 
DeadWolfe said:
The better question is why would one do this?

same reason mathematicians asked "why Euclid's fifth postulate?"

maybe whole new mathematics may be invented/discovered. maybe it might unify physics.
 
Somehow I duobt it.
 
The point of an absolute value is that it determines the distance from 0 (absolutely... whatever that means). So if you're on the real line, a positive number is its size from 0, and a negative number is the negative of itself from 0. On the complex plane, you just use pythagoras on the line you draw with coordinates (x,y). If you can describe a system with negative distance, then fine, it has a negative absolute value (good luck with that).

By the way, you say why doesn't someone invent a number with the properties described above; you just did. Look how much good it did us ;)

Why don't you try playing around with the different properties that emerge from such a number and see if you find anything interesting (I already have actually)
 
bananan said:
same reason mathematicians asked "why Euclid's fifth postulate?"

maybe whole new mathematics may be invented/discovered. maybe it might unify physics.
I take it, then, that you don't know why mathematicians asked that! They had very good, cogent, reasons for wondering about the fifth postulate. There would be no point in "defining numbers whose absolute value is negative" because the whole point of absolute value is that it measures the distance from the number to 0 and distance is always positive.
We DEFINE absolute value to be positive because we want it that way! That was not the case for x2.

You could, of course, define a function f(x) by "f(x)= x if x is negative, -x if x is positive" which would give exactly what you say for number we already have. Of course, it wouldn't be "absolute value", if fact, it would be -|x|.
 


Office_Shredder said:
Why don't you try playing around with the different properties that emerge from such a number and see if you find anything interesting (I already have actually)

I'm interested in what you found, actually
 


Did you notice that this thread was over 4 years old?

Do people go "prospecting" in the archives?
 
  • #10


bananan said:
why not invent a queer number?

the absolute value of a queer number is a negative number of the same interger value?


hence
|-5| = 5
|5| = 5
|5q| = -5
|-5q|= -5
Hey, I had an idea exactly like that!

HallsofIvy said:
Did you notice that this thread was over 4 years old?
Somebody came up with this idea 4 years before I did? :cry:

HallsofIvy said:
Do people go "prospecting" in the archives?
Occasionally. :biggrin:





I'll say that I am very interested in what will result from the queer number.



bananan said:
what is
|5i| = ?
|-5i| = ?
I believe that the answer to both is 5, am I correct?
 
  • #11


For some complex number C in the form x+y\mathrm{i}, |C| = \sqrt{\Re(C)^2+\Im(C)^2}=\sqrt{x^2+y^2}.

So, for plus or minus 5i, this evaluates to 5.
 
  • #12


HallsofIvy said:
Did you notice that this thread was over 4 years old?

Do people go "prospecting" in the archives?

I guess so! There was another "seemingly active thread" about closed form expressions that caught my eye today, as did the fact that some first-poster had resurrected it after years of inactivity...
 
  • #13


The Chaz said:
I guess so! There was another "seemingly active thread" about closed form expressions that caught my eye today, as did the fact that some first-poster had resurrected it after years of inactivity...

The beauty of math is that it neither lives nor dies, but suffuses all of reality itself. Euler's identity was certainly true long before it was ever discovered.
 
  • #14


That is the beauty of math?? I wish someone had told me earlier...

:cool:
 

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