The Identity Crisis (Element) (Published)
Citation: James H. Hughes (2022) The Identity Crisis (Element), International Journal of Mathematics and Statistics Studies, Vol.10, No.1, pp.47-71
Abstract: To prove that addition and multiplication require different identity elements, this paper introduces the idea of a multiplicative union and the different symmetries and dimensions involved.
Keywords: Dimension., Symmetry, Zero, identity element, multiplicative union, set theory.
Division by Zero (Published)
There is an exception to the rule that division by zero is undefined or prohibited, which results in a well-defined number, and follows the rules of set theory and algebra. This paper disproves the “proof by contradiction” by using a counterexample, an examination of its logic as a compound statement of two forms of statements that are not accepted as logically equivalent, an explanation of it as a sophism, and by examining some of its underlying assumptions.
Keywords: Counting Function, Empty Set, Zero, division, set theory.
Number Line Proposition: The Number Line is Countable (Published)
Where mathematics often views the number line as infinitely dense and uncountable, the proposition is offered that the number line, while infinite in extent, is countable.
Keywords: Euclid, Order, Zeno’s Paradox., density, enumerate, infinite, natural numbers, number line, precision, set theory.
Beginning Arithmetic Proposition: The Arithmetic Operator Requires Three Elements (Published)
For those not fully satisfied with abstract algebra’s presentation of group and ring theory, the proposition that the arithmetic operator requires three elements is offered as a part of a new approach to algebra.
Keywords: addition, arithmetic, division, elements, identity element, multiplication, operator, set theory.