English for Students of Mathematics Text I Rational Numbers

1. I. Read the article. Think of the proper heading for it. Explain your choice.

II. Fill in the gaps with the words given below the article.

A. From a less abstract point of view, the notion of division, or of 1 _____ , may also be considered to arise as follows: if the duration of a given process is required to be known to an accuracy of better than one hour, the number of minutes may be specified; or, if the hour is to be retained as the fundamental unit, each minute may be represented by 1/60 or by 1/60.

B. In general, the fractional unit 1/d is defined by the property d × 1/d = 1. The number n × 1/d is written n/d and is called a 2 ______ . It may be considered as the 3 _____ of n divided by d. The number d is called the 4 ____ (it determines the fractional unit or denomination), and n is called the 5 _____ (it enumerates the number of fractional units that are taken). The numerator and denominator together are called the terms of the fraction. A positive fraction n/d is said to be proper if n < d; otherwise it is improper.

C. The numerator and denominator of a fraction are not unique, since for every positive 6 ____ k, the numerator and denominator of a fraction can each simultaneously be multiplied by the integer k without altering the fractional value. Every fraction can be written as the quotient of two relatively prime integers, however. In this form it is said to be in lowest terms. The integers and fractions constitute what are called the 7 _____ numbers. The five fundamental laws with regard to the positive integers can be generalized to apply to all rational numbers.

D. From the definition of fraction it follows that the sum (or difference) of two fractions having the same denominator is another fraction with this denominator, the numerator of which is the sum (or difference) of the numerators of the given fractions. Two fractions having different denominators may be added or 8 _____ by first reducing them to fractions with the same denominator. Thus, to add a/b and c/d, the LCM of b and d, often called the least common denominator of the fractions, must be determined. It follows that there exist numbers k and l such that kb = ld, and both fractions can be written with this common denominator, so that the sum or difference of the fractions is obtained by the simple operation of adding or subtracting the new numerators and placing the value over the new denominator.

E. In order to multiply two fractions—in case one of the numbers is a whole number, it is placed over the number 1 to create a fraction—the numerators and denominators are multiplied separately to produce the new fraction's numerator and denominator: a/b × c/d = ac/bd. In order to divide by a fraction, it must be 9___ — that is, the numerator and denominator interchanged—after which it becomes a 10 ______ : a/b ÷ c/d = a/b × d/c = ad/bc.

1. denominator

2. subtracted

3. fraction

4. integer

5. common fraction

6. quotient

7. multiplication problem

8. rational

9. inverted

10. numerator

III. Explain the meanings of the following terms: common fraction, quotient, integer, inverted, multiplication problem.

IV. Which paragraph states the following information? Justify your opinion.

• the sum or difference of two fractions

• How to multiply and divide fractions

• the rational numbers

• the fraction and the integer

• how to create a fraction

• the terms of the fraction

• the notion of fraction.

V. Answer the questions:

• How can you define rational numbers? Why are they called so? In what way do they differ from the irrational ones?

• What are the 2 crucial parts of a fraction? What do they signify?

• “The five fundamental laws with regard to the positive integers can be generalized to apply to all rational numbers”. Can you explain which laws are meant in this paragraph?

• You are a lecturer who is explaining the theory about fractures. Try to make the explanation clear and concise. Illustrate it with examples, if you find it necessary.

• You are a secondary school teacher. Explain your students how to add, subtract, multiply and divide fractions. Give examples if necessary to illustrate your point.

Text II

Roots

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

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) = a0xn + a1xn − 1 + … + an − 1x + an = 0, with a0 ≠ 0, is called a root of the equation. If the coefficients lie in the complex field, an equation of the nth 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, x2 − 5 = 0 has no rational root, although its coefficients (1 and –5) are rational numbers.

B. Evidently the problem of finding the nth roots of unity is equivalent to the problem of inscribing a regular polygon of n sides in a circle. For every integer n, the nth 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 2h + 1, or 2k times such a product, or is of the form 2k. If a is a complex number not 0, the equation xn = a has exactly n roots, and all the nth roots of a are the products of any one of these roots by the nth 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 nth 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 nth 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 nth 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, …, ωn = 1 constitute all the nth 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, …, εn = 1 give all the nth 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) nth root of a—is written n√ a or a1/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 nth root of a is termed principal. For example, the principal cube root of –27 is –3.

Find the terms in the text that correspond to 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:

• A root can be expressed as a number only.

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

• If a whole number has a rational nth root, it means that it can be written as a common fraction.

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

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

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

Which paragraph provides information about:

• the general meaning of the term “root”

• rational nth roots

• the definition of the term

• the history and the origin of the term

• the roots of the equation

• 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.

Text III