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Design
Published in Wanda Grimsgaard, Design and Strategy, 2023
Irrational numbers: An irrational number is a real number that cannot be written as a fraction of two integers. There are a number of numerical ratios, a number of industrial standard sizes and a number of proportions, which include four irrational numbers of significant importance for the understanding and analysis of natural structures and processes (Bringhurst,1996), for example: Pi (π) = 3.14159... = the ratio of a circle’s circumference to its diameter√2 = 1.41421... = the diagonal of a square (A format)ϕ = 1.61803... = the golden ratio
The Reality of Experience
Published in John Flach, Fred Voorhorst, A Meaning Processing Approach to Cognition, 2019
The problem of linking the properties of mind with the properties of matter is typically referred to as the correspondence problem. As Winograd and Flores observe, the ‘rationalist tradition’ that has shaped cognitive science has tended to sidestep the correspondence problem—that is, assuming correspondence and then focusing exclusively on the ‘rules’ governing the relations among objects of the mental representation: Rationalist theories of mind all adopt some form of ‘representation hypothesis,’ in which it is assumed that thought is the manipulation of representation structures in the mind. Although these representations are not specifically linguistic (that is, not the sentences of an ordinary human language), they are treated as sentences in an ‘internal language’14 A third ontological position, typically referred to as Idealism, assumes that reality is based exclusively in Mind. In terms of the Venn diagram, this suggests that there is no reality outside of the circle of Mind. Idealism has historical roots, through Plato to early explorations in mathematics associated with irrational numbers like Pi. An irrational number has no concrete specification, yet it is fundamental to concrete objects like circles. From the idealist position, the concrete circles that we experience in our everyday lives are imperfect realizations of a more basic reality that can only be accessed through mathematics. In Plato’s terms, our experiences in the physical world are mere shadows on the cave wall that correspond to a reality based on a rational ideal (e.g., mathematics).
Functions of One Complex Variable
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
The known number system is a result of gradual development. The natural numbers (positive integers 1, 2, …) were first used in counting. Negative integers and zero (0, –1, –2, …) then arose to allow solutions of equations such as x + 3 = 2. In order to solve equations such as bx = a for all integers a and b, with b different from zero, rational numbers (or fractions) were introduced. Irrational numbers are numbers which cannot be expressed as a/b, with a and b integers and b different from zero, such as √2, π, e.
A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages
Published in Dynamical Systems, 2023
Let be an integer and be irrational. Let . If is a probability space and are commuting measure preserving automorphisms, then for any with , there exists for which There exists a measure preserving automorphism and a set satisfying such that for all we have
Middle-school mathematics teachers’ provision of non-examples and explanations in rational number instruction
Published in International Journal of Mathematical Education in Science and Technology, 2022
It is worth noting that presenting the students with a limited variety of examples and non-examples is particularly troublesome for students’ understanding of rational and irrational numbers (Zazkis, 2005). Additionally, transparency is the main characteristic of pedagogically useful examples (Bills et al., 2006) and infinite non-repeating decimal numbers (e.g. 0.257843 …) are transparent representations of irrational numbers (Zazkis & Sirotic, 2010). Therefore, it is crucial that the students are introduced to transparent representations of irrational numbers in order for them to decide on the rationality and irrationality of decimal numbers. However, it has long been reported that learners have difficulty discriminating between repeating and non-repeating infinite decimal numbers (e.g. Kidron, 2018; Toluk Uçar, 2016; Zaslavsky & Zodik, 2014; Zazkis & Sirotic, 2010). For instance, in-service mathematics teachers in Zaslavsky and Zodik (2014) and pre-service middle-school mathematics teachers in Toluk Uçar (2016) erroneously believed that infinitely repeating decimal numbers are irrational numbers. Thus, it is unfortunate that the three teachers (T1, T2, and T3) did not consider it significant to use transparent non-examples in their classrooms.
Planar defect in approximant: the case of Cu-Al-Sc alloy
Published in Philosophical Magazine, 2021
Tsutomu Ishimasa, Yeong-Gi So, Marek Mihalkovič
Since the discovery of quasicrystals [1], their structural features and physical properties have been intensively studied. In the course of these studies, it became important to understand the relationship between quasicrystals and their crystal approximants. The latter is a series of periodic structures consisting of local atomic arrangements similar to the corresponding quasicrystal and is related to the rational approximation of irrational numbers such as the golden ratio [2]. In particular, defects formed in approximants are thought to provide important hints for understanding structural relationships [3]. For a cubic approximant formed in Al-based alloy, Quiquandon et al. analysed the {110} planar defect [4]. However, knowledge of such defects is currently very limited.