Hi, I m trying to find out, what is imaginary unit/number. i^2

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Discussion Overview

The discussion centers around the concept of the imaginary unit, denoted as "i," and its implications in mathematics. Participants explore the philosophical and historical motivations for the introduction of imaginary numbers, their utility in solving polynomials, and the geometric interpretations of complex numbers.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the philosophical basis for the introduction of the imaginary unit, suggesting that it seems arbitrary and questioning the motivations behind defining "i" as the square root of -1.
  • Another participant points out that the existence of "i" allows for the solution of many polynomials that would otherwise have fewer roots than expected, highlighting its mathematical necessity.
  • A different viewpoint emphasizes that mathematicians do not create arbitrary extensions of numbers; rather, they develop consistent frameworks that allow for operations like addition and logarithms to be applied to complex numbers.
  • One participant discusses the historical context of imaginary numbers, linking their development to Cardano's cubic formula and the need to handle negative square roots in polynomial equations.
  • A geometric interpretation is presented, where "i" is described as a 90-degree counterclockwise rotation in a two-dimensional plane, illustrating how this visualization can help understand the property that i² = -1.
  • Another participant reiterates the philosophical inquiry into whether mathematicians can define numbers as they wish, suggesting a broader exploration of what constitutes a number beyond traditional definitions.

Areas of Agreement / Disagreement

Participants express a range of views on the nature and necessity of imaginary numbers, with some emphasizing their mathematical utility while others question the philosophical implications of their definition. No consensus is reached regarding the motivations behind the creation of imaginary numbers or the nature of mathematical definitions.

Contextual Notes

The discussion reflects various assumptions about the nature of numbers and mathematical operations, with some participants highlighting the importance of consistency in definitions while others challenge the arbitrary nature of mathematical constructs.

  • #31


PlanckShift said:
1/ε is undefined. It is so a field. It's closed under linear combinations. It's commutative and associative. It has an identity element: 1+ε0 and the inverse element which I've described already. It's distributive, too.

The nonzero elements of a field are an abelian group under multiplication, so \epsilon^{-1} = \frac{1}{\epsilon} must exist if \epsilon \ne 0.
 
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  • #32


The real numbers are a field even though 0-1 doesn't exist. It's the same thing with this definition of complex numbers although elements along the "imaginary" line don't have inverses.
 
  • #33


PlanckShift said:
The real numbers are a field even though 0-1 doesn't exist. It's the same thing with this definition of complex numbers although elements along the "imaginary" line don't have inverses.

Please stop spouting nonsense and review basic mathematics.
Nonzero imaginary numbers do have inverses: i^{-1}=-i.
 
  • #34


micromass said:
Please review your definition of a field.

Definition of a field.
 
  • #35


PlanckShift said:
Definition of a field.

OK, you found the definition. Now read the definition. In particular:

Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1.
 
  • #36


But ε2 = 0 remember? Think about it. How do you find the inverse of a+bε? Rationalize the expression using ε2 = 0. Then find where the resulting expression is undefined.
 
  • #37


PlanckShift said:
But ε2 = 0 remember? Think about it. How do you find the inverse of a+bε? Rationalize the expression using ε2 = 0. Then find where the resulting expression is undefined.

But where is the inverse of ε?
Hint: It's 1/ε, which you said was undefined. It doesn't have an inverse, so your structure isn't a field.
 

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