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In this article we describe and analyze preservice elementary school (K–7) teachers’ understanding of prime numbers and attempt to detect factors that influ-ence their understanding. We use representation of number properties, and what can be learned by considering representations of a number, as a lens for the analysis of participants’ responses.
We introduce a versatile method for finding prime numbers that display surprisingly intricate visual patterns— hypothetically, any desired pattern is possible, with only mild distortion. We use this method to locate several examples of large prime numbers that are, in and of themselves, self-referential works of art.
Example #1: For defined by ( ) , we have the following: )The preimage of under the map )is the set }{ , because ( and ( (The preimage of }under the map {is the set , because only ) .
Since prime numbers are the building blocks of integers, it is natural to wonder how the primes are distributed among the integers. \There are two facts about the distribution of prime numbers.
• A circular prime is prime with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. • For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. Other examples are: 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933.
Definition (12.9 Image and pre-image). Let f : A → B, and let C ⊆ A and let D ⊆ B. The set f(C) = {f(x) : x ∈ C} is the “image of A in B”. This is a subset of B. The set f−1(D) = {x ∈ A : f(x) ∈ D} is the “preimage of D in A”. This is a subset of. Do not think of f−1 as the inverse function. The inverse function exists only when f is bijective.
18 Οκτ 2021 · You are expected to be able to combine the definition of “image” with the proof techniques that you already know. Example \(6.9.4\). Assume \(f : A \rightarrow B\).
