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Arithmetic with H = -1: subtraction, negative numbers, division, rationals and mixed numbers

Colignatus, Thomas

H = -1 is an universal constant. H represents a half turn along a circle, like complex number i represents a quarter turn. Kids know what it is to turn around and walk back along the same path. H creates the additive inverse with x + H x = 0 and the multiplicative inverse with x x^H = 1 for x != 0. Pronounce H as "ehta" or "symbolic negative one". The choice of H is well-considered: its shape reminds of -1 and even more (-1). Pierre van Hiele (1909-2010) already proposed to use y x^-1 and drop the fraction bar y / x with its needless complexity. Students must learn exponents anyway. The negative exponent might confuse pupils to think that they must subtract something, but the use of an algebraic symbol clinches the proposal. Also 5/2 can be written as 2 + 2^H, so that it is clearer where it is on the number line. This approach also causes a re-evaluation of the didactics of the negative numbers. The US Common Core has them only in Grade 6 which is remarkably late. The negative numbers arise from the positive axis x by rotating or alternatively mirroring into H x. Algebraic thinking starts with the rules that a + H a can be replaced by 0 and that H H can be replaced by 1. Subtraction a - b >= 0 may be extended into a - b < 0 with its present didactics, e.g. 2 - 5 = 2 - (2 + 3) = 2 - 2 - 3 = 0 - 3 = -3, but there is an intermediate stage with familiar addition 2 + 5 H = 2 + (2 + 3) H = 2 + 2 H + 3 H = 0 + 3 H = 3 H, that does not require (i) the switch at the brackets from plus to minus and (ii) the transformation of binary 0 - 3 to number -3. The expression a - (-b) involves (scalar) multiplication which indicates why pupils find this hard, and a + H H b is clearer. The use of H would affect the whole curriculum. There appears to be a remarkable incoherence in mathematics education and its research w.r.t. the negative numbers, which reminds of the problems that the world itself had since the discovery of direction by Albert Girard in 1629 and the introduction of the number line by John Wallis in 1673. This notebook provides a package to support the use of H in Mathematica. The notebook and package are intended for researchers, teachers and (Common Core) educators in mathematics education. Pupils in elementary school would work with pencil and paper of course.

This page gives the notebook with packages. The PDF is here: https://zenodo.org/record/1251687. This notebook with packages is also available at (same link, major update of notebook and packages): https://www.wolframcloud.com/objects/thomas-cool/MathEd/2018-04-02-Arithmetic-with-H.nb and http://community.wolfram.com/groups/-/m/t/1313302
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