Abstract
This paper proves that all mathematical quantities including fractions, roots or roots of root, transcendental quantities can be expressed by continued nested radicals using one and only one integer 2. A radical is denoted by a square root sign and nested radicals are progressive roots of radicals. Number of terms in the nested radicals can be finite or infinite. Real mathematical quantity or its reciprocal is first written as cosine of an angle which is expanded using cosine angle doubling identity into nested radicals finite or infinite depending upon the magnitude of quantity. The finite nested radicals has a fixed sequence of positive and negative terms whereas infinite nested radicals also has a sequenceof positive and negative terms but the sequence continues infinitely. How a single integer 2 can express all real quantities, depends upon its recursive relation which is unique for a quantity. Admittedly, there are innumerable mathematical quantities and in the same way, there are innumerable recursive relations distinguished by combination of positive and negative signs under the radicals. This representation of mathematical quantities is not same as representation by binary system where integer two has powers 0, 1, 2, 3…so on but in nested radicals, powers are roots of roots.
Highlights
A real mathematical quantity or its reciprocal can be expressed as cosine of an angle
Since a combination may consist of a number of positive and negative terms depending upon the magnitude and sign of the quantity, there may be infinite combinations of positive and negative terms of recursive relation and such infinite number of combinations will express quantities infinite in number
Since cosine angle doubling identity, involves integer 2 and only 2, but with different combinations of positive and negative signs depending upon the magnitude of the quantity, recursive relation of signs decides the magnitude
Summary
A real mathematical quantity or its reciprocal can be expressed as cosine of an angle. Cos(2x) appears in right hand side and this cos(2x) using the same identity, can be expressed in cos(4x) and cos(4x) in cos(8x) so on and cos(2k–1x) in cos(2kx) where is k is any integer. Angle x on being doubled continuously, there comes a stage when cos(2k x) equals cos(x) or –cos(x). On successively putting value of cos(x) in right hand side, equation proceeds infinitely and takes the form (Landau, 1992). Equation is recursive in nature as cos(x) appears both in left and right hand side. On successively substituting the value of cos(x), equation takes the form. Angle x is known from magnitude of quantity being expressed in continuous nested radicals, signs positive or negative of cos(2x), cos(4x), cos(8x), ... Angle x is known from magnitude of quantity being expressed in continuous nested radicals, signs positive or negative of cos(2x), cos(4x), cos(8x), ... etc can be known from value of angle x and will be mentioned in the above equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
