Rings of constants of the form k[f]
Rings of constants of the form k[f]

The Fibonacci zeta function - Department of Mathematics and Statistics
The Fibonacci zeta function - Department of Mathematics and Statistics

ADDING AND COUNTING Definition 0.1. A partition of a natural
ADDING AND COUNTING Definition 0.1. A partition of a natural

Quadratic Formula (C)
Quadratic Formula (C)

A counting based proof of the generalized Zeckendorf`s theorem
A counting based proof of the generalized Zeckendorf`s theorem

... We can assume that they have no term in common (otherwise we can remove the terms from both). In this case we can find a term Gi in one representation so that all terms, even the largest Gj in the other representation, are smaller. But by the above argument the second representation results in a sum ...
1.2 Exponents and Radicals Definition 1.1 If x is any real number
1.2 Exponents and Radicals Definition 1.1 If x is any real number

7.4 The polar form
7.4 The polar form

1 Imaginary Numbers 2 Quiz 24A 3 Complex Numbers
1 Imaginary Numbers 2 Quiz 24A 3 Complex Numbers

Chapter 2
Chapter 2

FACTORIZATION OF POLYNOMIALS 1. Polynomials in One
FACTORIZATION OF POLYNOMIALS 1. Polynomials in One

TRUE/FALSE. Write `T` if the statement is true and `F` if the
TRUE/FALSE. Write `T` if the statement is true and `F` if the

Does the Cantor set contain irrational algebraic numbers? by
Does the Cantor set contain irrational algebraic numbers? by

Complex Numbers
Complex Numbers

Polynomials for MATH136 Part A
Polynomials for MATH136 Part A

... God, it has been said, gave us the natural numbers 1,2,3,... and man created the rest. When we just had the natural numbers 1,2,3,... we had to say that the equation 2x = 1 has no solutions. So fractions were invented. Then 2x = 1 could be solved but x 2 = 2 could not. So along come the irrational n ...
Adding and Subtracting Polynomials
Adding and Subtracting Polynomials

Topic 1: Algebra • Meaning of terms variable and function • Use of
Topic 1: Algebra • Meaning of terms variable and function • Use of

Equivalent form of implication
Equivalent form of implication

1.3 & 1.4 Solving Equations and Inequalities
1.3 & 1.4 Solving Equations and Inequalities

Reals
Reals

HERE
HERE

... (O,90°), a rotation of 90º about the origin (O). Thus, if the point for i0 could be represented as (1,0), and if the point for i could be represented as (O,90°)((1,0))=(0,1), then the point for i can be thought of as the image of the point for i0 under (O,45°), a rotation of 45° (figure 2). Moreo ...
2.3 Roots (with HW Assignment 7)
2.3 Roots (with HW Assignment 7)

ON PSEUDOSPECTRA AND POWER GROWTH 1. Introduction and
ON PSEUDOSPECTRA AND POWER GROWTH 1. Introduction and

... We may take N = 2n + 3. This shows that matrices can have identical pseudospectra and yet their second and higher powers have norms that are completely unrelated to each other. 1.2. What about diagonalizable matrices? The matrices A, B in the Greenbaum–Trefethen example are nilpotent, as are those c ...
Chapter 4 Glossary
Chapter 4 Glossary

High School – Number and Quantity
High School – Number and Quantity

1 Introduction 2 History 3 Irrationality
1 Introduction 2 History 3 Irrationality

Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.