III method просмотров: 751
III method / way. The field of complex number <complex field> is built as a simple algebraic extension over the real number field <real field>. From the scientific point of view, the principle of this method is about the [following] (The way of explaining complex numbers to students lies in the following). The polynomial algebra [shows us] that (In polynomial algebra) when the b0 + b1x + b2x2 + … + bn – 1 xn – 1 polynomial of degree n ≥ 2 is irreducible over the number field P (i. e. it doesn’t have roots in this field), and a is a number, [which is] the root of this polynomial, each element b of the field P(a), formed [by] joining element a to the field P, can be presented explicitly [as]: b = b0 + b1a + b2a2 + … + bn – 1 an – 1. (2)
Suppose P = R. Let us consider the x2 + 1 polynomial, which doesn’t have real roots. Let us use i to mark the number (mark the number with i), which is the solution of the equation x2 + 1 = 0, and consider it an imaginary number. We obtain that the simple algebraic extension R(i) is the complex number field C. Though, from (2) it follows that each element of R(i) will take the form a + bi, where a, b ∈ R.
We don’t explain all these calculations to students. [To make] quadratic equations universally resolvable we extend the set of real numbers by joining the number i to it, for which i2 = –1, to it. So, complex numbers are the first-degree polynomials relative to i, i. e. the expressions of the form z = x+ yi, where x and y are real numbers. This approach is called genetic one, because it is based on historical extending of the number concept and the needs of science, which lead to this extending.
This method is used to present the information in textbooks (In this way complex numbers are presented in textbooks) [7; 10; 11]. [The intelligibility for pupils] counts in its favor, as well as [the contraction of theoretical calculations] (without information loss) and [the ability to switch] from theoretical calculations to practical ones. The connection [between] imaginary numbers and reality is an issue that can’t be covered fully in the course of school curricula. That’s why it should be taken into account while studying the basics of the complex numbers theory, in order not to let students remember this section only as a formal-and-logical game, which doesn’t relate to real world in any way.