Verifying I2=1: Right or Wrong?

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In summary: That's why [tex]i^2= -1In summary, the conversation discusses the mistake of moving the exponent inside the radical and how it leads to paradoxes in complex exponents. The correct definition of complex numbers is also mentioned, with an explanation of why the result of i^2 is -1.
  • #1
ranjitnepal
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please tell me the solution i did is right or wrong and why?

we know,
I=√-1
I2=√-1*√-1 =√(-1)2 = √1 = 1
 
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  • #2
Please read this: https://www.physicsforums.com/showthread.php?t=637214 [Broken]
 
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  • #3
Wrong, you cannot move the exponent inside the radical. That you arrived at -1=1 is a sign of a mistake.
 
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  • #4
lurflurf said:
Wron, you cannot move the exponent inside the radical. That you arrived at -1=1 is a sign of a mistake.

A mistake? For sure. But a very interesting mistake. "Paradoxes" of these kind indicate that something very interesting is going on with complex exponents. In particular, they indicate that complex exponents (such as roots) are multivalued. Once you take this approach, all paradoxes vanish :tongue2:
 
  • #5
ranjitnepal said:
please tell me the solution i did is right or wrong and why?

we know,
I=√-1
I2=√-1*√-1 =√(-1)2 = √1 = 1

Would it be enough just to accept the definition so that we use both of these:

i*i=-1 and i=sqrt(-1)This is what happens with (-i).
(-i)*(-i)=(-1)(-1)*i*i=(1)*(-1)=-1
What went wrong there?
Apparantly i*i=-1 and (-i)(-i)=-1.

Still seem good. i*i still -1 and (-1)(-1)=1
 
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  • #6
micromass said:
"Paradoxes" of these kind indicate that something very interesting is going on with complex exponents. In particular, they indicate that complex exponents (such as roots) are multivalued.

Yeah, similarly, by naively taking the natural logarithm of both sides of equation [itex]e^{0}=e^{2\pi i}[/itex] one gets the result [itex]0=2\pi i[/itex].
 
  • #7
ranjitnepal said:
please tell me the solution i did is right or wrong and why?

we know,
I=√-1
Well, there are your first two errors! To begin with, it is "i", not "I"!
(I suspect your editor automatically changed your "i" to "I". So did mine- I had to "fool" it by typing "ia", then going back and deleting the "a"!)
More importantly, it is a mistake to write "[itex]a= \sqrt{-1}[/itex] because there is no such number before we define "i" and you can't define a new number to be something that doesn't exist to begin with! Defining "i" to be "the number whose square is -1" is better because -1, at least, does exist before we define the complex numbers. But has the difficulty that once we start working with the complex numbers we find that every number, except 0, has two square roots and this does not tell us which of the two roots of -1 "i" is.

Better is to define the complex numbers to be the set of orderd pairs of real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ b, c+ d) and multiplication by (a, b)*(c, d)= (ac- bc, ad+ bc). We can then identify the real numbers with pairs of the form (x, 0) and "i" with (0, 1).

I2=√-1*√-1 =√(-1)2 = √1 = 1
With the "ordered pairs" definition, above, [tex]i^2= (0, 1)(0, 1)= (0(0)-(1)(1), 0(1)+ 0(1))= (-1, 0)[tex] which we have already identified with the real number -1.
 

What is the significance of verifying I2=1 in scientific research?

Verifying I2=1 is important because it confirms the accuracy of mathematical calculations and ensures the validity of experimental results. It also allows for the detection of errors or discrepancies that may have occurred during the research process.

What is the process for verifying I2=1?

The process for verifying I2=1 involves using mathematical principles and calculations to determine if the equation is true. This may include solving the equation using algebraic techniques or using known mathematical identities and properties.

Why is it important to consider the units when verifying I2=1?

Units are an essential component of scientific measurements and calculations. Verifying I2=1 with incorrect units can lead to incorrect results and conclusions. Therefore, it is crucial to ensure that the units are consistent and accurately represented in the equation.

What are the common sources of error when verifying I2=1?

Common sources of error when verifying I2=1 include human error, equipment limitations, and measurement uncertainties. It is essential to identify and account for these sources of error to ensure the accuracy and reliability of the verification process.

How does verifying I2=1 impact the overall reliability of a scientific study?

Verifying I2=1 is a critical step in the scientific research process, as it ensures the accuracy and validity of the experimental results. By confirming the equation's accuracy, researchers can have more confidence in their findings and conclusions, thus increasing the overall reliability of the study.

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