hi mam fundamental concepts of algebra
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Fundamental concepts of modern algebra
Prime factorization
Some other fundamental concepts of modern algebra also had their origin in 19th-century work on number theory, particularly in connection with attempts to generalize the theorem of (unique) prime factorization beyond the natural numbers. This theorem asserted that every natural number could be written as a product of its prime factors in a unique way, except perhaps for order (e.g., 24 = 2∙2∙2∙3). This property of the natural numbers was known, at least implicitly, since the time of Euclid. In the 19th century, mathematicians sought to extend some version of this theorem to the complex numbers.
One should not be surprised, then, to find the name of Gauss in this context. In his classical investigations on arithmetic Gauss was led to the factorization properties of numbers of the type a + ib (a and b integers and i = Square root of√−1), sometimes called Gaussian integers. In doing so, Gauss not only used complex numbers to solve a problem involving ordinary integers, a fact remarkable in itself, but he also opened the way to the detailed investigation of special subdomains of the complex numbers.
In 1832 Gauss proved that the Gaussian integers satisfied a generalized version of the factorization theorem where the prime factors had to be especially defined in this domain. In the 1840s the German mathematician Ernst Eduard Kummer extended these results to other, even more general domains of complex numbers, such as numbers of the form a + θb, where θ2 = n for n a fixed integer, or numbers of the form a + ρb, where ρn = 1, ρ ≠ 1, and n > 2. Although Kummer did prove interesting results, it finally turned out that the prime factorization theorem was not valid in such general domains. The following example illustrates the problem.
Answer:
To determine the height of the tree, you can use the mirror as shown in Figure 72. The FD ray of light, reflecting at point D, falls on the viewer's eye (at point B). Determine the height of the tree if AC = 165 cm, BC = 12 cm, AD = 120 cm, DE = 4, 8uI angle1 = angle2.