Is it mathematically justified to place negative and positive numbers on the same plane?
Perhaps you have additional considerations. But you didn't specify them.
In the traditional way, we place 0 at an arbitrary line point. And then we have negative real numbers on one side and positive real numbers on the other. It's precisely the same with imaginary numbers.
Let's say we accept your point of view. Then how do you place positive real numbers on a straight line? There will be a beginning somewhere. And the line is infinite at both ends. Or you want to put the numbers on thel plane as a spiral.
To the best of my knowledge, the representation of the real numbers on the so called real line, which is a straight line with a direction, indicates two things: (i) that the set of real numbers R is totally ordered by the relation
Thank you for your reply Even though Im not a great specialist in a set theory I understand that continuous line means that two numbers are infinitely close to each other Let's say we place all real numbers to be placed on the row that is fine Lets take a segment [3,4] It includes positive transcedental pi number Do we also have a negative transcedental pi on the opposite side of x-axis or - pi? The same question applies to i as we do not have minus (-i) on the opposite side of the imaginary number line Add here other transcedental numbers and the continuous line will not be so continuous after all With that said not all numbers have their negative equivalents I have to apologze if my comments are crude Im just giving my common sense opinion here to facilitate a discussion further Thanks
Continuous line means that there do not exist gaps on the line; i.e. each point on the line represents a (unique) real number; arbitrarily (or infinitesimally) close is not exactly that; this is another property that is called density and it holds for both the reals and the rationals. Rationals are dense in the reals, meaning that given any two reals x
If 2pi times i equals 0 and pi-pi=0 we solve a system of equations where pi=x
2x=0 and x-x=0 From here we have x=0 where p=x=0
Conclusion:There is no symetry here as pi=3.14.....
Euler's formula: for every real number x, exp(ix)=cosx+isinx. Hence exp(ix)=1 implies that cosx=1 and sinx=0; not necessarily x=0.
The conjunction "2pi times i equals 0 and pi-pi=0" is then false. Starting from a false statement, one can then prove anything; this is what inconsistency means in logic.
Goes like this:
e^2pii=1
e^pi=-1
Consequently:
-2pi=pi
and thats precisely what I was trying to say
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Graphs