What is the flaw in defining i as the square root of -1?

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Discussion Overview

The discussion revolves around the definition of the imaginary unit \( i \) as the square root of -1, exploring the implications and potential flaws in this definition. Participants examine mathematical operations involving square roots, particularly in the context of complex numbers, and the logical consistency of these operations.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant claims that operations with square roots are only defined for positive numbers, suggesting that the transition from \( \sqrt{-1} \) to \( (-1)^{2/4} \) is incorrect.
  • Another participant points out that stating \( 1^{1/4} = 1 \) is not well-defined since there are multiple fourth roots of 1, which raises concerns about the validity of the earlier steps.
  • A different viewpoint emphasizes that when squaring a number, the result is always non-negative, and thus the manipulation of roots involving negative numbers leads to errors.
  • One participant argues that defining \( i \) as \( \sqrt{-1} \) is logically incorrect because every number (except zero) has two square roots, and this definition does not specify which square root is intended.
  • Another participant suggests that defining complex numbers as ordered pairs of real numbers can help avoid the paradoxes associated with the definition of \( i \) and clarify the distinction between the two square roots of -1.

Areas of Agreement / Disagreement

Participants express multiple reasons for the flaws in defining \( i \) as \( \sqrt{-1} \), indicating that there is no consensus on a single explanation. The discussion remains unresolved with competing views on the implications of this definition.

Contextual Notes

Participants highlight limitations in the definitions and operations involving square roots, particularly in relation to the properties of real and complex numbers. There is an emphasis on the need for clarity in distinguishing between different roots and the implications of defining complex numbers in various ways.

Diffy
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sqrt(-1) = (-1)^(1/2) = = (-1)^(2/4) = ((-1)^2)^(1/4)) = 1^(1/4) = 1


Can someone explain the flaw to me?
 
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Well i think that the flaw here is that the operatios with square roots are defined only for positive numbers, so basically when you went from sqrt(-1) = (-1)^(1/2) to (-1)^(2/4), i think is not correct.
 
You wrote 1^(1/4) = 1, but this is not well-defined because there are four numbers whose fourth power is 1, namely {1, -1, i, -i}. This is like saying (-2)^2 = 4 and sqrt(4) = 2, hence 2=-2.
 
Diffy said:
sqrt(-1) = (-1)^(1/2) = = (-1)^(2/4) = ((-1)^2)^(1/4)) = 1^(1/4) = 1


Can someone explain the flaw to me?

No, you can't do that, when writing: [tex]\alpha ^ {\frac{b}{c}}[/tex]

Then b, and c must be relatively prime, i.e their GCD must be 1.

When you square something, it's always non-negative, and when you square (cube, or forth,...) root it, it's still non-negative. This is where the error lies.

It's the same as:

[tex]\sqrt[3]{-1} = (-1) ^ {\frac{1}{3}} = (-1) ^ {\frac{2}{6}} = \sqrt[6]{{\color{red}(-1) ^ 2}} = 1[/tex], which is clearly false.
 
looks like there are multiple reasons. Thanks everyone.
 
looks like masnevets made the most sense out of it.
 
In fact, defining i as "[itex]\sqrt{-1}[/itex]", while a convenient mnemonic, is logically incorrect- every number (except 0) has two square roots and this doesn't specify which square root of -1 i is. In the real number system, we don't have to worry about that since we can specify the square root as being the positive root. Since the complex numbers is not an ordered field we can't do that. That's even more obvious where you see i "defined" by "i2= -1". Such an equation has two roots. Which one is i?

The most logical way of handling that is to define the complex numbers as ordered pairs of real numbers: (a, b) and definining (a, b)+ (c, d)= (a+ c,We can, then, say that by (a, b) is "represented" That way, we have immediately that (0, 1)2= (0, 1)(0, 1)= (-1, 0). It is also true that (0, -1)2= (0, -1)(0, -1)= (-1, 0) but now we can distinguish between those two numbers. We can, then represent (a, b) as a(1, 0)+ b(0,1)= a+ bi by representing (1, 0) by 1 and (0, 1) by i.

With those definitions "paradoxes" like the one given here do not occur.
 

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