Order the paragraphs to make a logical text. Explain your choice.

Order the paragraphs to make a logical text. Explain your choice.

Roots

A. The term root has been carried over from the equation xⁿ = a to all polynomial equations. Thus, a solution of the equation f (x) = a ₀ xⁿ + a ₁ x^{n − 1} + … + a_{n − 1} x + a_n = 0, with a ₀ ≠ 0, is called a root of the equation. If the coefficients lie in the complex field, an equation of the n th degree has exactly n (not necessarily distinct) complex roots. If the coefficients are real and n is odd, there is a real root. But an equation does not always have a root in its coefficient field. Thus, x ² − 5 = 0 has no rational root, although its coefficients (1 and –5) are rational numbers.

B. Evidently the problem of finding the n th roots of unity is equivalent to the problem of inscribing a regular polygon of n sides in a circle. For every integer n, the n th roots of unity can be determined in terms of the rational numbers by means of rational operations and radicals; but they can be constructed by ruler and compasses (i.e., determined in terms of the ordinary operations of arithmetic and square roots) only if n is a product of distinct prime numbers of the form 2 ^h + 1, or 2 ^k times such a product, or is of the form 2 ^k. If a is a complex number not 0, the equation xⁿ = a has exactly n roots, and all the n th roots of a are the products of any one of these roots by the n th roots of unity.

C. A root is a solution to an equation, usually expressed as a number or an algebraic formula.

D. More generally, the term root may be applied to any number that satisfies any given equation, whether a polynomial equation or not. Thus π is a root of the equation x sin (x) = 0.

E. If a whole number (positive integer) has a rational n th root—i.e., one that can be written as a common fraction—then this root must be an integer. Thus, 5 has no rational square root because 2² is less than 5 and 3² is greater than 5. Exactly n complex numbers satisfy the equation xⁿ = 1, and they are called the complex n th roots of unity. If a regular polygon of n sides is inscribed in a unit circle centred at the origin so that one vertex lies on the positive half of the x -axis, the radii to the vertices are the vectors representing the n complex n th roots of unity. If the root whose vector makes the smallest positive angle with the positive direction of the x -axis is denoted by the Greek letter omega, ω, then ω, ω², ω³, …, ω ⁿ = 1 constitute all the n th roots of unity. For example, ω = −¹/₂ + ^√(−3) /₂, ω² = −¹/₂ − ^√(−3) /₂, and ω³ = 1 are all the cube roots of unity. Any root, symbolized by the Greek letter epsilon, ε, that has the property that ε, ε², …, ε ⁿ = 1 give all the n th roots of unity is called primitive.

F. In the 9th century, Arab writers usually called one of the equal factors of a number jadhr (“root”), and their medieval European translators used the Latin word radix (from which derives the adjective radical). If a is a positive real number and n a positive integer, there exists a unique positive real number x such that xⁿ = a. This number—the (principal) n th root of a —is written ⁿ√ a or a ^{1/ n} . The integer n is called the index of the root. For n = 2, the root is called the square root and is written √ a. The root ³√ a is called the cube root of a. If a is negative and n is odd, the unique negative n th root of a is termed principal. For example, the principal cube root of –27 is –3.

Find the terms in the text that correspond the definition.

1. Any rational number that can be expressed as the sum or difference of a finite number of units, being a member of the set...-3, -2, -1, 0, 1, 2, 3…

2. The unique set of values that yield a true statement when substituted for the variables in an equation.

3. The number that is not integrally divisible by two.

4. The number which produces a given number when cubed.

5. A statement of the equality of two expressions formulated by applying to a set of variables the algebraic operations, namely, addition, subtraction, multiplication, division, raising to a power, and extraction of a root.

6. A quantity that has both magnitude and direction but not position.

7. A branch of mathematics in which numbers, relations among numbers, and observations on numbers are studied and used to solve problems.

8. In mathematics, a factor of a number that, when multiplied by itself, gives the original number.

9. A numerical or constant factor in an algebraic term.

10. Numbers that can be expressed as integers or as the quotient of integers.

Decide whether the statements are true or false:

1. A root can be expressed as a number only.

2. 2.The integer n is called the coefficient of the root.

3. If a whole number has a rational n th root, it means that it can be written as a common fraction.

4. For every integer n, the n th roots of unity can be determined in terms of the irrational numbers through rational operations and radicals.

5. If the coefficients lie in the complex field, an equation of the n th degree has n (not necessarily distinct) complex roots.

6. If the coefficients are real and n is even, there is a real root.

Which paragraph provides information about:

ü the general meaning of the term “root”

ü rational n th roots

ü the definition of the term

ü the history and the origin of the term

ü the roots of the equation

ü the problem of finding the roots

Summarize the article trying to be short and informative.

Mention the different uses of the term root, its history and the problem of finding the roots of unity.

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