Why doesn't √-1×√-1 always equal 1 in complex numbers?

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  • Thread starter Thread starter Gourav kumar Lakhera
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Discussion Overview

The discussion revolves around the properties of square roots in the context of complex numbers, specifically addressing why the multiplication of square roots of negative numbers does not yield expected results, such as √-1 × √-1 equaling 1. The scope includes conceptual clarification and mathematical reasoning related to complex number operations.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asserts that √-5 × √-5 equals 5, suggesting a misunderstanding of the properties of square roots in complex numbers.
  • Another participant corrects this claim, stating that √-5 equals i√5, leading to the conclusion that √-5 × √-5 equals -5, not 5.
  • There is a question posed about why √-1 × √-1 does not equal 1, with reference to the fact that i^2 equals -1.
  • A participant notes that the rule √a × √b = √(ab) only holds when both a and b are nonnegative real numbers, implying that the initial assumptions about square roots are flawed.

Areas of Agreement / Disagreement

Participants express disagreement regarding the application of square root properties to negative numbers, with some clarifying misconceptions while others maintain their original assertions. The discussion remains unresolved regarding the implications of these properties in complex numbers.

Contextual Notes

There are limitations in the understanding of square root operations involving negative numbers, particularly regarding the assumptions made about the applicability of certain mathematical rules in the context of complex numbers.

Gourav kumar Lakhera
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As we know that √-5×√-5=5 i.e multiplication with it self
My question is that according to this √-1×√-1=1.but it does not hold good in case of i(complex number).
I.e i^2 =-1. Why?
 
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Gourav kumar Lakhera said:
As we know that √-5×√-5=5
No, this is not true. ##\sqrt{-5} = i\sqrt{5}## so ##\sqrt{-5} \cdot \sqrt{-5} = i^2 (\sqrt{5})^2 = -5##, not 5 as you show above.
Gourav kumar Lakhera said:
i.e multiplication with it self
My question is that according to this √-1×√-1=1
This isn't true, either, for the same reason as above.
Gourav kumar Lakhera said:
.but it does not hold good in case of i(complex number).
I.e i^2 =-1. Why?

You are apparently using the rule that ##\sqrt a \sqrt b = \sqrt{ab}##. That rule holds only when both a and b are nonnegative real numbers.
 
Thnkuu buddy
 

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