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Mathematical Foundations
Published in Chintan Patel, Nishant Doshi, Internet of Things Security, 2018
Figure 2.1 shows the distribution of number theory. Natural numbers are represented by ℕ and contain numbers ≥ 1. Whole numbers are also natural numbers that include 0 also. Whole numbers can be represented by . Integer numbers range between −N to +N and can be represented by . Rational numbers are numbers that can be represented in the form of a/b, and they can be represented by ℚ. Prime numbers are numbers that have GCD with any other number is 1. Prime numbers are the most important numbers in the world of cryptography. Prime numbers can be represented using ℙ. Real numbers are number line that contains all rational numbers, irrational numbers, fraction numbers and also transcendental numbers like Π, 3Π, γ. Complex numbers are numbers that can be represented by c = a + ib. Where a, b ∈ ℝ. “a” in complex number is the real part of numbers, “b” is imaginary part of complex number, and i=−1. Complex numbers can be represented by ℂ. ℕ ⊂ ⊂ ⊂ ℚ ⊂ ℝ ⊂ ℂ
Number Systems
Published in Julio Sanchez, Maria P. Canton, Microcontroller Programming, 2018
Julio Sanchez, Maria P. Canton
The digits of a number system, called the positive integers or natural numbers, are an ordered set of symbols. The notion of an ordered set means that the numerical symbols are assigned a predetermined sequence. A positional system of numbers also requires the special digit zero which, by itself, represents the absence of oneness, or nothing, and thus is not included in the set of natural numbers. However, 0 assumes a cardinal function when it is combined with other digits, for instance, 10 or 30. The whole numbers are the set of natural numbers, including the number zero.
Computer Number Systems
Published in Julio Sanchez, Maria P. Canton, Software Solutions for Engineers and Scientists, 2018
Julio Sanchez, Maria P. Canton
The digits of a number system, called the positive integers or natural numbers, are an ordered set of symbols. The notion of an ordered set means that the numerical symbols are assigned a predetermined sequence. A positional system of numbers also requires a special digit, named zero. The special symbol 0, by itself, represents the absence of oneness, or nothing, and thus is not included in the set of natural numbers. However, 0 assumes a cardinal function when it is combined with other digits, for instance, 10 or 30. The whole numbers are the set of natural numbers, including the number zero.
Analysing theories of meaning in mathematics education from the onto-semiotic approach
Published in International Journal of Mathematical Education in Science and Technology, 2022
Juan D. Godino, María Burgos, María M. Gea
The construction of the set of natural numbers N and its arithmetic is at the base of the mathematics that every educator should know. The formal construction of N based on the theory of one-to-one correspondence of sets defines the arithmetic operations in different way from the definitions when N is constructed from the Peano’s axioms. These are two possible partial meanings of the natural numbers and its arithmetic, but they are not the only ones. Each partial meaning of the natural numbers is a semiotic system, characterized by a concrete configuration of operative and discursive practices. The global meaning of numbers (Figure 3) is made up of the articulation of the different subsystems determining each partial meaning. In terms of learning, however, meanings are relative, not absolute. There are degrees of meanings; degrees of what may be termed extent, exactness, depth, complexity; and growth in meanings may take place in any of these dimensions. For relatively few aspects of life, for relatively few aspects of the school’s curriculum (including arithmetic), do we seek to carry meanings to anything like their fullest development. Moreover, whatever the degree of meaning we want children to have, we cannot engender it all at once. Instead, we stop at different levels with different concepts; we aim now at this level of meaning, later at a higher level, and so on. (Brownell, 1947, p. 257)