Two questions on complex numbers. If both 2 and (- I + i (3^.5)), respectively, are cube roots of 8, does that mean the latter expression = 2? Second

Two questions on complex numbers. If both 2 and (- I + i (3^.5)), respectively, are cube roots of 8, does that mean the latter expression = 2?

 

ok I will ask a different question:

 

\(x^2=4\\ x=+2 \;\; or \;\; x=-2\\ \text{so take away the positive bit which is just a convention}\\ \sqrt4=+2 \quad or \;\;-2\\ \text{Does that mean that +2=-2 ?} \)

 

or looking ati it slightly differently

\((-3)^2=9\\ (+3)^2=9\\ \therefore\;\;-3=+3\)

 

 

 

Second question, in the book the Mathematical Universe by W. Dunham, on page 292 he says 'It is easy to see that i ^ 2 = (-1^.5)^2 = -1'. Why is this 'easy' to see? Is it because the i squared is defined as -1?

 

He means it is easy to see because i^2=-1 but he is wrong because he has not used brackets properly.

By not using brackets properly he has not followed proper convention.

He has overlooked this because in the context of what he was talking, his meaning was clear, he thought so anyway :)

 

 

\((-1^{0.5})^2=(-1*1^{0.5})^2=(-1*\sqrt1)^2=(-1*1)^2=(-1)^2=1\)

 

but

 

\(((-1)^{0.5})^2=(\sqrt{-1})^2=(i)^2=-1 \)

 

 

I have checked and this calculator does return 1.   It is correct.

 

I like questions like this because it means you are really trying to meanings sorted properly :)

Thank you for your reply.

 

Just to be sure I understand, to the first question, your example says that the system of negative numbers provides -2 as a square root of 4.  While the other square root of 4 is 2 (which is not from the system of negative numbers), there is no reason to think the two roots are equal to each other. Similarly, as one cube root of 8 is from the system of imaginary numbers, there is no reason to think it is equal to 2, which is also a cube root of 8 but is not from the system of imaginary numbers.  In sum, roots of the same number from different systems of numbers should not be considered equal to each other. 

 

As to the other question, I still find it awkward that the author says it is 'easy' to see, I guess because typically a teacher or lecturer who uses this terminology I think  uses it to explain the outcome of a a process of some kind (add, subtract, multiply, etc) but here negative 1 is not the outcome of a process but is according to a basic definition that i squared = negative 1..   

 

Also, I'm glad you gave examples of how the calculator goes about its work because in the past I have tried to use the calculator to help understand basic issues like this, but now I see my own $20 calculator, depending on how the format of the keys are pressed, in this example of i squared, returns +1, -1, or Domain error.  So, to avoid misinterpreting the result, I really must understand the subject thoroughly when using it.

Thanks for your additonal comment.

 

My initial question could have been answered with a yes or no but you answered it with a question.  This required me to try to understand based on your question.  I was quite willing to do that in order to enable my understanding the subject better.

 

In reading about this subject elsewhere, I am pretty sure mention was made that systems of numbers (maybe 'systems' is the wrong word) evolved to accomodate issues in mathematics which could not be resolved using previously existing knowledge.  Negative numbers were one such development, imaginary numbers another.  To understand your answer/question, then, I concluded you were speaking from this perspective, ie there are different systems.

 

If someone were to ask me now if both 2 and (-1 + i (3^.5) are equal to each other, I would say each is a cube root of 8, they are not equal to each other, and I will continue to attempt to learn why they are not equal to each other..