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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).
Analytical Methods
Published in Alexander D. Poularikas, Handbook of Formulas and Tables for Signal Processing, 2018
N natural number, N = {0,1, 2, 3, } sometimes 0 is omitted. Z integers, (Z+ positive integers), Z = {0, ±1, ±2, ±3, } Q rational numbers, Q = {p / q : p, q z, q 0}, Q is countable, i.e., there exists a one-toone correspondence between Q and N R real numbers, R = {real numbers} is not countable. Real numbers which are not rational are called irrational. Every irrational number can be represented by an infinite non-periodic decimal expansion. Algebraic numbers are solutions to an equation of the formanxn+...+ao = 0, ak z. Transcendental are those numbers in R which are not algebraic. (Example: 4/7 is rational; 5 is algebraic and irrational; e and are transcendentals.) C Complex numbers, C = {x + jy : x, y R} where j 2 = -1.
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
After T4 explained all steps of the procedure for converting repeating decimals into rational numbers in the form of a/b (establishment of C-4 and C-5), one of the students in the classroom asked her to tell which numbers are not rational numbers. T4 did not ignore the student’s question (establishment of C-8) and responded by providing the number π as a non-example of a rational number (establishment of C-7). The explanations provided by T4 for the number π are given as follows: You can transform a decimal number into a rational number … The number π is not a rational number. Do you remember the number π from the topic of circles? It repeats forever irregularly as 3.14 … There is no regularity. Since it does not repeat with a certain logic, I cannot write it as a decimal number. For this reason, it cannot be a rational number. As a matter of fact, all decimal numbers excluding repeating decimals and the normal decimal numbers are irrational numbers. However, note that irrational numbers are very rare.