Related papers: A cute proof that makes $e$ natural
We illustrate the concept of mathematical proof.
Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are…
Zero factorial, defined to be one, is often counterintuitive to students but nonetheless an interesting concept to convey in a classroom environment. The challenge is to delineate the concept in a simple and effective way through the…
The logic LAE discussed in this paper is based on an approximate entailment relation. LAE generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are…
Adjoint logic is a general approach to combining multiple logics with different structural properties, including linear, affine, strict, and (ordinary) intuitionistic logics, where each proposition has an intrinsic mode of truth. It has…
Proofs (sequent calculus, natural deduction) and imperative algorithms (pseudocodes) are two well-known coexisting concepts. Then what is their relationship? Our answer is that \[ imperative\ algorithms\ =\ proofs\ with\ cuts \] This…
Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around.…
This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
One source of beauty in mathematics is totally unexpected connections between two fundamentally different objects. For instance, is it not surprising that the time period of a real simple pendulum is linked with a function arising out of…
In this article, we give another visual proof of $\pi^e < e^\pi$.
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…
E-graphs are a prominent data structure that has been increasing in popularity in recent years due to their expanding range of applications in various formal reasoning tasks. Often, they are used for equality saturation, a process of…
A recurring debate in the philosophy of statistics concerns what, exactly, should count as a measure of evidence for or against a given hypothesis. P-values, likelihood ratios, and Bayes factors all have their defenders. In this paper we…
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this…
Irrational numbers are introduced usually already introduced in lower secondary level schools. But typically, maybe with the exception of $\sqrt{2}$, no mathematical proof of irrationality is provided. In particular it is not proven that…
Teaching precise mathematical reasoning can be very hard. It is very easy for a student to make a subtle mistake in a proof which invalidates it, but it is often hard for the teacher to pinpoint and explain the problem in the (often…
Math is widely considered as a powerful tool and its strong appeal depends on the high level of abstraction it allows in modelling a huge number of heterogeneous phenomena and problems, spanning from the static of buildings to the flight of…