Imaginary Numbers and Properties: A Puzzling Case

Click For Summary

Discussion Overview

The discussion revolves around the properties of imaginary numbers and the validity of certain mathematical operations involving square roots and complex numbers. Participants explore whether established properties hold true when extended to imaginary numbers and complex numbers.

Discussion Character

  • Debate/contested, Mathematical reasoning, Conceptual clarification

Main Points Raised

  • One participant presents a series of equations involving imaginary numbers, concluding that properties do not work with them, though this conclusion is questioned.
  • Multiple participants point out errors in the initial equations, specifically in lines 2-to-3, suggesting that the reasoning is flawed.
  • Another participant questions the general validity of the property \(\sqrt{\frac{x}{y}} \neq \frac{\sqrt{x}}{\sqrt{y}}\) for complex numbers, indicating that this holds true only under certain conditions.
  • It is noted that while the property is valid for positive real numbers, it does not apply universally to complex numbers, highlighting the need for careful consideration of definitions and contexts.
  • A later reply expands on the previous comments, suggesting that retaining all properties of the square root operation when extending it to complex numbers leads to inconsistencies.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of the initial claims regarding imaginary numbers and the properties of square roots, with multiple competing views and no consensus reached on the conclusions drawn.

Contextual Notes

Limitations include the dependence on the definitions of operations involving complex numbers and the unresolved nature of the mathematical steps presented in the initial post.

[C]alculus
Messages
2
Reaction score
0
-1/1=1/-1
sqrt(-1/1)=sqrt(1/-1)
i/1=1/i
i*(i/1)=i*1/i
i^2/1=i/i
-1/1=1
-1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?
 
Mathematics news on Phys.org
Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.
 
arildno said:
Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.
Why[tex]\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y}[/tex] ??
 
If x and y are both positive real numbers that is true, but not in general for x and y being complex numbers.
 
kntsy said:
Why[tex]\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y}[/tex] ??
To expand on GibZ's comment.
In order to extend the square root operation to work on real&complex number, and not just on the positive reals, you will not be able to retain <i>all</i> properties of the square root operation if you wish it to be self-consistent, and not self-contradictory.
 

Similar threads

High School Why are imaginary numbers called "imaginary"? If they really exist
  • · Replies 8 ·
Replies
8
Views
2K
High School Imaginary Pythagoras
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad A question about Young's inequality and complex numbers
  • · Replies 13 ·
Replies
13
Views
2K
High School Popularizing a property for n-bonacci numbers without publishing it?
  • · Replies 12 ·
Replies
12
Views
2K
High School Arithmetic and our place value system
  • · Replies 6 ·
Replies
6
Views
1K
High School Commutative & Associative property of negative numbers
  • · Replies 4 ·
Replies
4
Views
3K
High School Is imaginary "i" a purely aesthetic matter?
  • · Replies 44 ·
2
Replies
44
Views
5K
High School Can Negative Numbers Be in the Domain of the Square Root Function?
  • · Replies 4 ·
Replies
4
Views
5K
High School Cool fact about number of digits in n!
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Explaining imaginary numbers to laypeople
  • · Replies 29 ·
Replies
29
Views
4K