Algebraic Threefolds, Varenna, Italy 1981, Second Session: by Conte A. (ed.)

By Conte A. (ed.)

Lecture notes in arithmetic No.947

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Extra resources for Algebraic Threefolds, Varenna, Italy 1981, Second Session: Proceedings

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This shows that, in some way, i is the only “number” that we can square and get a negative value. Using this definition all the square roots above become, −9 = 3 i −100 = 10 i −5 = 5 i −290 = 290 i These are all examples of complex numbers. The natural question at this point is probably just why do we care about this? aspx College Algebra numbers and we’re going to need a way to deal with them. So, to deal with them we will need to discuss complex numbers. So, let’s start out with some of the basic definitions and terminology for complex numbers.

The degree of each term in a polynomial in two variables is the sum of the exponents in each term and the degree of the polynomial is the largest such sum. Here are some examples of polynomials in two variables and their degrees. x 2 y − 6 x 3 y12 + 10 x 2 − 7 y + 1 degree : 15 6 x 4 + 8 y 4 − xy 2 degree : 4 x 4 y 2 − x3 y 3 − xy + x 4 degree : 6 6 x − 10 y + 3 x − 11 y degree : 14 14 3 In these kinds of polynomials not every term needs to have both x’s and y’s in them, in fact as we see in the last example they don’t need to have any terms that contain both x’s and y’s.

Note as well that multiple terms may have the same degree. We can also talk about polynomials in three variables, or four variables or as many variables as we need. The vast majority of the polynomials that we’ll see in this course are polynomials in one variable and so most of the examples in the remainder of this section will be polynomials in one variable. Next we need to get some terminology out of the way. A monomial is a polynomial that consists of exactly one term. A binomial is a polynomial that consists of exactly two terms.

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