By Mark V. Lawson
Algebra & Geometry: An advent to school arithmetic offers a bridge among highschool and undergraduate arithmetic classes on algebra and geometry. the writer exhibits scholars how arithmetic is greater than a suite of equipment by way of offering vital rules and their historic origins in the course of the textual content. He features a hands-on method of proofs and connects algebra and geometry to varied purposes. The textual content specializes in linear equations, polynomial equations, and quadratic types. the 1st a number of chapters conceal foundational themes, together with the significance of proofs and houses generally encountered whilst learning algebra. the remainder chapters shape the mathematical center of the publication. those chapters clarify the answer of alternative sorts of algebraic equations, the character of the options, and the interaction among geometry and algebra
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Extra resources for Algebra & geometry: an introduction to university mathematics
In Chapter 5, we shall study prime numbers. These are the natural numbers n, excluding 0 and 1, which are divisible only by 1 and n. You can check that the polynomial p(n)=n2蜢n+41 has the property that its value for n=1,2,3,4,舰,40 is always prime. Even the first few cases might be enough to lead you into believing that it always took prime values. It does not, of course, since when n=41 it is clearly not prime. All this means that, in effect, we have to prove an infinite number of statements: 02 is even, and 22 is even, and 42 is even 舰.
In addition, if (a,b,c) is a Pythagorean triple so too is (ॕa,ॕb,ॕc) where ॕ is any whole number. Perhaps surprisingly, this problem is in fact equivalent to one in geometry. Suppose that a2+b2=c2. Exclude the case where c=0 since then a=0 and b=0. We can therefore divide both sides by c2 to get (ac)2+(bc)2=1. Recall that a rational number is a real number that can be written in the form uv where u and v are whole numbers and v蠀0. It follows that (x,y)=(ac,bc) is a point with rational coordinates that lies on the unit circle, what we call a rational point.
If (1) is true then we are led to the following two conclusions. From (2), the only natural number to divide both x and y is 1. From (7) and (12), 2 divides both x and y. This is a contradiction. Thus (1) cannot be true. Hence 2 cannot be written as an exact fraction. This result is phenomenal. It says that no matter how much money you spend on a computer it will never be able to calculate the exact value of 2, just a good approximation. We now make an important definition. A real number that is not rational is called irrational.