Infinite Nested Radicals - A Way to Express All Quantities Rational, Irrational Transcendental By a Single Integer Two

  • Narinder Kumar Wadhawan Civil Servant, Indian Administrative Service Retired, Panchkula, Haryana, India
  • Priyanka Wadhawan TIET, Thapar University, Patiala, Punjab-147004, India
Keywords: Infinite Nested Radicals, Recursive Relation, Cosine Angle Doubling Identity, Integer Two, Mathematical Quantity, Modulus


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.


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Herschfeld, A.: On Infinite Radicals. The American Mathematical Monthly 42(7), 419–429, (1935). DOI:

Landau, S.: Simplification of Nested Radicals. SIAM Journal on Computing 21(1), 85–110 (1992).

Landau, S.: Simplification of Nested Radicals. 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, NC, USA, 1989, 314-319, DOI:

Rick, K.: π to Thousands of Digits from Vieta’s Formula. Mathematics Magazine 81(3), 201–207, (2008).

Weisstein, Eric., Nested Radical. From MathWorld--A Wolfram Web Resource.

Zimmerman, S. and Ho, C.: On Infinitely Nested Radicals. Mathematics Magazine 81(1), 3–15, (2008).

How to Cite
Wadhawan, Narinder Kumar, and Priyanka Wadhawan. 2020. “Infinite Nested Radicals - A Way to Express All Quantities Rational, Irrational Transcendental By a Single Integer Two”. Mathematical Journal of Interdisciplinary Sciences 9 (1), 1-9.