Download e-book for kindle: Algebra & geometry: an introduction to university by Mark V. Lawson
By Mark V. Lawson
Algebra & Geometry: An creation to school arithmetic presents a bridge among highschool and undergraduate arithmetic classes on algebra and geometry. the writer indicates scholars how arithmetic is greater than a set of equipment by means of proposing very important principles and their old origins in the course of the textual content. He includes a hands-on method of proofs and connects algebra and geometry to varied functions. The textual content specializes in linear equations, polynomial equations, and quadratic varieties. the 1st numerous chapters conceal foundational themes, together with the significance of proofs and houses often encountered whilst learning algebra. the rest chapters shape the mathematical center of the publication. those chapters clarify the answer of other sorts of algebraic equations, the character of the suggestions, and the interaction among geometry and algebra
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Extra resources for Algebra & geometry: an introduction to university mathematics
This temptation should be resisted. The square root of two and the square root of minus one are both equally abstract, the only difference between them being the purely psychological one that the former is more familiar than the latter. 2 It is not only a question of extending our conception of number. There are also occasions when we want to restrict it. For example, we might want to solve an equation using only whole numbers. It turns out that the usual high-school method for solving equations does not work in this case.
47 of Euclid. We are given a right-angled triangle. We are required to prove, of course, that a2+b2=c2. Consider the picture below. It has been constructed from four copies of our triangle and two squares of areas a2 and b2, respectively. We claim that this shape is itself actually a square. First, the sides all have the same length a+b. Second, the angles at the corners are right angles by Proof 4. Now look at the following picture. This is also a square with sides a+b so it has the same area as the first square.
The next simplest kinds of equations are those in which the unknown x occurs to the power two but no more ax2+bx+c=0, where a蠀0. They are called quadratic equations in one unknown. Whereas solving linear equations is easy, ingenuity is needed to solve quadratic equations. Using a method called completing the square, it is possible to write down a formula to solve any such equation in terms of the numbers a, b and c, called the coefficients. This formula yields two, one or no solutions depending on the values of these coefficients.
Algebra & geometry: an introduction to university mathematics by Mark V. Lawson