The Final Real Number

The Final Real Number By Marcia Buckpitt In this paper, we will define the point that closes the real number line: R and prove that it is a real number. Once we establish that a real number closes R, an imaginary number at the point at infinity is no longer useful. R is the real number line, therefore R is a line. A line is a curve by definition. If R is a line then R is a curve. A curve + a point are a closed curve by definition. Herein, N is any positive real number on R except 0. We consider the current five basic components in the realm of R to be 0, N, N+1, -N, and –N-1. infinity, –infinity are vectors not numbers. Nothing we do with numbers on R changes their relationships to each other, their position, or their value. Each N on R is fixed and unchangeable. Adding, subtracting, in fact, all mathematical manipulations of N and –N on R are methods of navigating in the stable realm: R. R is an infinite closed curve of sequential real numbers where every N is reflected by –N. Each N and –N is a unique point on R; therefore, no two numbers exist simultaneously at any one point on R. 0 is between a symmetrically-aligned infinite set of + and - real numbers. In the realm of R, symmetry requires an even number of components. If every N on R has a –N on R, then R is symmetric. If N is even, -N is even. If N is odd, –N is odd. If 0 is a real number between N and –N on a symmetric closed curve then there exists a sixth component that completes the symmetry of R. 0 has a minus zero: (-0). If 0 is positive then (-0) is negative. 0 is even by definition therefore, (-0) is even. To better understand the properties of (-0) we review some properties of 0: 0=N-N. 0 is between N and –N. 0+N=N, 0-N=-N. -N<0
Let us consider the properties of (-0): (-0) =-N=N. (-0) is opposite 0 between –N = N. (-0) +N=0, (-0)-N=0. -N> (-0)>N. 0 and (-0) are reflections of each other on opposite poles of R between N and –N. The complete closed curve R is: 0+N=N and (-0) +N=0, therefore |0+ (-0)| =|R|. If 0+N=N then N>0, (-0)>N therefore (-0)>0 and 0+ (-0) = (-0)

The Final Real Number by Marcia Buckpitt

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We know (-0) <0 at –N but that (-0) =N=-N=2N, 2N >0, (-0)>0 therefore (-0) +0= (-0). Mathematics professor Dr. Darin Doud observed that (-0) +0= (-0) is the only case where (-0) + another number does not = 0. Another way of understanding how (-0) differs from 0: At 0+ N, we go from 0 to N. At (-0) +N we go from N to 0. 0+N deflects N to N. The deflection is +N. (-0) +N deflects N to 0. The deflection is -N. 0 reflects the positive of any N added to it. (-0) reflects the negative of any N added to it. 0 and (-0) behave the same but from opposite positions. N-N =0, when added they appear to collapse to 0. A careful look reveals a pivot point where N and –N are equal distance from 0 but not added together. At N=-N we are at a point of balance on R we call (-0). We can remain at (-0) unless we add or subtract. If we add or subtract N to (-0), we get 2N. At N + (-0) we can predict what will happen by viewing 0. Just as adding N to 0 sends us a distance of N from 0; N+ (-0) sends us in a positive direction N places, however, because (-0) is a pivot point we are stopped and turned at (-0), + N sends us through –N at N to 0, resulting in 2N from 0. The complete trip from 0 to N + (-0) = 0 is a distance of 2N. We see that 0 and (-0) are bounding points on a · closed curve. |0+ (-0)| =|R|. (-0) has the same value as 0 just as -N has the same value as N, yet 0 and (-0) are no more interchangeable than are N and –N. The defining equation is: 0+N=N and (-0) +N=0 0 is between N-N and (-0) is between N=-N. The distinction between 0 and (-0) is: (-0)>0 at N and (-0) <0 at –N. The real number 0 begins R. The real number (-0) closes R.