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Related Concept Videos

Rational Expressions01:28

Rational Expressions

431
Rational expressions are algebraic fractions in which both the numerator and the denominator are polynomials. These expressions follow the arithmetic rules of numerical fractions but require extra care due to the presence of variables. A fundamental part of working with rational expressions is identifying values that make the expression undefined, typically those that result in division by zero or undefined radicals.Determining the DomainThe domain of a rational expression includes all real...
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Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

342
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
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Complex Zeros01:29

Complex Zeros

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Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
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Rationalizing Substitutions01:29

Rationalizing Substitutions

73
Integrals involving non-rational functions are often difficult to evaluate using standard techniques, especially when radicals appear in the integrand. Rationalizing substitution provides a systematic method for simplifying such integrals by converting them into rational forms that are easier to handle.Consider a rod whose linear mass density depends on a constant linear density, a characteristic length, and the distance from the left end of the rod. Determining the total mass requires...
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Radical Equations01:26

Radical Equations

435
Radical equations are mathematical expressions in which the variable is found within a radical, most commonly a square root or cube root. These equations frequently arise in science, engineering, and real-world measurements involving nonlinear relationships. To solve a radical equation, the standard procedure is to isolate the radical expression and then eliminate the radical by raising each side to a power equal to the index of the radical. This process may lead to extraneous...
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Radicals01:27

Radicals

795
Roots, often written as radicals, identify the quantity that must be raised to a specific exponent to produce a given value. A radical expression consists of two main components: the radicand, which is the value placed inside the root symbol, and the index, which indicates the degree of the root being taken. The notation n√a indicates the principal nth root of a. If n equals 2, the operation is the square root, while n = 3 defines the cube root. When n is even, a negative radicand does...
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Irrational "Coefficients" in Renaissance Algebra.

Jeffrey A Oaks1

  • 1University of IndianapolisE-mail:oaks@uindy.edu.

Science in Context
|July 13, 2017
PubMed
Summary

Algebraists historically excluded irrational coefficients until the 16th century. This study examines the shift towards their acceptance, driven by notational autonomy in algebraic mathematics.

Area of Science:

  • History of Mathematics
  • Algebraic Development
  • Mathematical Notation

Background:

  • Premodern algebra defined coefficients as counts, prohibiting irrational numbers.
  • The concept of a monomial limited coefficients to whole numbers or simple fractions.

Purpose of the Study:

  • To trace the historical acceptance of irrational coefficients in algebra.
  • To analyze the conceptual shift from rhetorical to symbolic algebra.

Main Methods:

  • Examination of primary algebraic texts from the 9th to 16th centuries.
  • Analysis of arguments for and against irrational coefficients.

Main Results:

  • Irrational coefficients were gradually introduced in 16th-century European algebra.

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  • This innovation occurred within the framework of premodern 'cossic' algebra.
  • The acceptance was linked to the increasing independence of mathematical notation from rhetorical descriptions.

    • Conclusions:

    • The allowance of irrational coefficients represented a formal innovation in algebra.
    • This development was independent of the new algebra emerging with Viète, Descartes, and Fermat.