Chapter 1. Arithmetics - Wikispacesenglishformaths.wikispaces.com/file/view/Chapter+1.doc · Web viewChapter 1: Arithmetics Unit 1: The Natural numbers and The Integers _____ 1. VOCABULARY

Chapter 1. Arithmetics

Chapter 1: ArithmeticsUnit 1: THE NATURAL NUMBERS AND THE INTEGERS ________________________________________________________1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary: - Addend (n): Số hạng- Addition (n): Phép cộng - Arbitrary number (n): Số tùy ý- Arithmetics (n): Số học- Axiom (n): Tiên đề- Base (n): Cơ số- Dividend (n): Số bị chia- Division (n): Phép chia- Divisor (n): Số chia- Even number (n): Số chẵn- Factor (n): Thừa số- Factorial (n): Giai thừa- Highest common factor (n): Ước chung lớn nhất - Index (n): Chỉ số- Lowest Common Multiple (n): Bội chung nhỏ nhất- Multiple (n): Bội- Multiplication (n): Phép nhân- Multiplier (n): Số nhân- Numeral (n): Chữ số- Odd number (n): Số lẻ- Postulate (n): Định đề - Power (n): Lũy thừa- Prime (n): Số nguyên tố- Prime factor (n): Thừa số nguyên tố- Prime factorization (n): Sự phân tích ra thừa số nguyên tố.- Product (n): Tích- Quotient (n): Thương- Ratio (n): Tỷ số- Reciprocal (n): Phần tử khả nghịch- Relatively prime (n): Phần tử nguyên tố cùng nhau.- Remainder (n): Số dư - Subtraction (n): Phép trừ- Subtrahend (n): Số bị trừ- The integers (n): Số nguyên- The natural numbers (n): số tự nhiên- Unity (n): Phần tử đơn vị

English for Mathematics Page 1


1.2 Grammar review:- Tenses.- Verb forms.- Conditional sentences.- Active and passive voices.- Preposition1.3 Exercises:1.3.1. Fill in the blanks with the correct form of the word in the bracket:If a and b ____(1)___(be) natural numbers and a = bq where q is also a natural number, then we

say that q is the quotient ____(2)____(obtain) by dividing the number a by the number q and we write: q = a/b. We can also say that a is evenly divisible by b that b divides a without remainder. Any number b that divides a exactly is ____(3)___(say) to be a divisor of a. The number a itself with respect to its divisor is ___(4)____(call) a multiple. Thus, multiples of b are the numbers b, 2b, 3b, … Multiple of the number 2 (that is, number exactly divisible by 2) are said to be even. Numbers not ____(5)___(exact) divisible by 2 are said to be odd. Every natural number is either even or odd. If each of the two number x, y ___(6)___(be) a multiple of b then the sum x + y is also a multiple of b. If two or more numbers ___(7)___(not have) any common divisor different from unity, then these numbers are said ____(8)___(be) relatively prime. For example the numbers 49 and 121 ___(9)___(be) relatively prime (co-prime, or prime to each other).

1.3.2. Choose the correct word to fill in the blanks: Greater Only Is Called Procedure Divisible

Prime numbersIf a number possesses ____(1)___two factors, namely itself and unity, then it is ____(2)___prime

number. The first six numbers are 2, 3, 5, 7, 11 and 13. Unity ___(3)___ not considered to be a prime number.

Any integer ___(4)___than 1 is a prime or can be written as a product of primes. For example, 60 has the prime factors 2, 3, and 5 and its prime factorization is 60 = 2 x 2 x 3 x 5.

To write a number as a product of prime factors:1) Divide the number by 2 if possible; continue to divide by 2 until the factor you get is not

____(5)___by 2.2) Divide the result from (1) by 3 if possible; continue to divide by 3 until the factor you get is

not divisible by 3. 3) Divide the result from (2) by 5 if possible, continue to divide by 5 until the factor you get is

not divisible by 5Continue the____(6)__ with 7, 11 and so on, until all the factors are primes.1.3.3. Fill in the blanks with the correct preposition:

Raising a number to a powerRaising ___(1)___ a power is devised from repetitive multiplication. The power is also called an

index and the number to be raised to a power is called the base. For example: 10 x 10 x 10 = . Here, the number 3 is the power (index) and 10 is the base.

The arithmetic ____(2)___ powers is contained in the following set of rules:- Power unity: Any number raised to power 1 equals itself.



- Multiplication ___(3)__ numbers and the addition of powers: If two numbers are each written as a given base raised to some power then the product of the two numbers is equal to the same base raised to the sum of the powers. To multiply the powers are added. For example,

- Division of numbers and the subtraction of powers: If two numbers are each written as a given base raised to some power then the quotient of the two numbers is equal ___(4)___ the same base raised to the difference of the powers. That means, to divide the powers are subtracted. For example,

and so .- Power zero: Any number raised to the power 0 equal unity.- Multiplication of powers: If a number is written as a given base raised to some power then

that number raised to a further power is equal to the base raised to the product of the powers. ___(5)___example,

- Negative powers: A number raised to a negative power denotes the reciprocal. For example

.

Notice that in any operation with different indices the base must be the same. We cannot use these laws to combine different powers ____(6)___ different bases.

1.3.4. Find and correct the false in the following text:An ratio is an indicated division. It should be thinked of as a fraction. The language used is: “the

ratio of a to b” which means or and the symbol is a:b. In this notion a is the first term or the

antecedent, and b is the second term or the consequent. It is importantly to remember that we treat the ratio as a fraction. A proportion is a statement that two ratios is equal. Symbolically we write a:b =

c:d or .

The statement is read “a is to b as c is to d” and we call a and d the extremes, b and c the means, and d is the fourth proportional. Proportions are treat as equations involving fractions. We may perform all the operations on them that we do on equation, and many of the resulting properties may already have been meet in geometry.

1.3.5 Put the word in each sentence in the correct order to make the meaningful sentences:a) Every/ odd / natural / is /either / number / even /or.b) Any / integer /great / than / 1 / is / a / prime / or /can / be / written / as / a / product /of/ primes.c) The / natural / are / numbers / using / the / decimal / numerals/ written / 0 / to / 9.d) The / natural/collectively / integer /numbers /and /numbers /the /negative / are /called /the.e) No / same / follower two / natural / have/ numbers / the /.2. READING COMPREHENSION:2.1 THE NATURAL NUMBERS: The natural numbers (positive integer) express the quantity of certain things (alike or not alike) to

be counted; such, for instance, are the numbers one, two, ten, twenty, a hundred, and so forth. The concept of a natural number is one of the simplest and most elemental notions in mathematics. The natural numbers are written using the decimal numerals 0 to 9 where the position of a numeral in a number dictates the value it represent. When writing a number, the digits from right to left indicate in



turn the number of units, the number of tens, hundreds, thousands, and so forth. For example: 123 stands for 1 hundred and 2 tens and 3 units. The natural numbers can be graphically represented by equally spaced points on a straight line. The natural numbers are ordered – they progress from a given number are less than (< ) the given number and numbers to the right are greater than (>) the given number. If two numbers are neither less than nor greater than each other, they are equal.

If the straight line containing the natural numbers is extended to the left we can plot equally spaced points to the left of zero. These points represent negative numbers which are written as a natural number preceded by a minus sign, for example -5. The natural numbers and the negative numbers shown here are collectively called the integer. The notion of order still applies: for example -3 < 2 and -4 > -6. An integer is a positive or negative integer or the number 0. Even numbers are numbers of the form 2k, and odd numbers are numbers of the form 2k +1, where k is an integer.

Comprehension check:Answer the following questions:1. Are the natural numbers well-ordered? How about the integer?2. Is there any largest natural number? Why or why not?3. What is the smallest natural number?4. Between the natural numbers and the integer, which is the larger set? Why?5. Let a be a odd number and b be an even number, is ab an even number? Why or why not?2.2 Highest common Factor (HCF) – Lowest Common Multiple (LCM)If two numbers have factors (or divisors) in common, then the largest of these common factors is

called their highest common factor (HCF). For example: 18 has the factors 1, 2, 3, 6, 9 and 18; 30 has the factors 1, 2, 3, 5, 6, 15, 30. Consequently, the numbers 1, 2, 3 and 6 are their common factors, the largest of which is 6. The HCF of 18 and 30 is 6.

A number m is common multiple of two other numbers k and j if it is a multiple of each of them. For example: 12 is a common multiple of 4 and 6, since 3 x 4 = 12 and 2 x 6 = 12. The least common multiple (LCM) of two numbers is the smallest number that is a common multiple of both numbers.

To find the lowest common multiple of two numbers k and j:- Write k as a product of primes and j as a product of primes.- If there are any common factors delete them in one of the products.- Multiply the remaining factors; the result is the least common multiple. For example, 6 has the prime factorization 2 x3 and 10 has the prime factorization 2x5. The LCM

is, therefore, 2 x 3 x5 = 30. Notice that the common factor 2 is only used once to find the LCM. If it were used twice then a common multiple would be obtained but it would not be the lowest one.

It is clear that for two relatively prime numbers a and b the LCM is equal to the product of the number.

Comprehension check: A. Choose true or false:1. _______The highest common factor of two numbers is the highest number.2. _______The lowest common multiple of two numbers is the product of these numbers.3. _______The prime factorization of number a is written a as a product of all the prime

numbers.4. _______ To find the LCM of two relatively prime numbers, we evaluate their product.



5. _______ The common factor of two numbers a and b is always greater than their lowest common multiple.

2.3 J.E.FREUND’S SYSTEM OF NATURAL NUMBERS POSTULATESModern mathematicians are accustomed to derive properties of natural numbers from a set of

axioms or postulates (i.e. undefined and unproven statements that disclose the meaning of the abstract concepts).

The well known system of 5 axioms of the Italian mathematician, Peano provides the description of natural numbers. These axioms are:

First: 1 is a natural number.Second: Any number which is a successor (follower) of a natural number is itself a natural

number.Third: No two natural numbers have the same follower.Fourth: The natural number 1 is not the follower of any other natural number.Fifth: If a series of natural numbers includes both the number 1 and the follower of every natural

number, then the series contains all natural numbers.The fifth axiom is the principle (law) of math induction.From the axioms it follows that there must be infinitely many natural numbers since the series

cannot stop. It cannot circle back to its starting point either because 1 is not the immediate follower of any natural number is well ordered and presents a general problem of quantification. It places the natural numbers in an ordinal relation and the commonest example of ordination is the counting of things. The domain of applications of Peano’s example of ordination is the counting of things. The domain of application of Peano’s example of ordination is the counting of things. The domain of application of Peano’s theory is much wider than the series of natural numbers alone etc., the

relational fractions and so on, satisfy the axioms similarly. From Peano’s five rules we can

state and enumerate all the familiar characteristics and properties of natural numbers. Other mathematicians define these properties in terms of 8 or even 12 axioms (J.E.Freund) and these systems characterize properties of natural numbers much more comprehensively and they specify the notion of operations both arithmetical and logical.

Note that sums and products of natural numbers are written as a + b and a.b or ab, respectively.Postulate No.1: For every pair of natural numbers, a and b, in that order, there is a unique (one

and only one) natural number called the sum of a and b. Postulate No.2: If a and b are natural numbers, then a + b = b + a.Postulate No.3: If a, b and c are natural numbers, then (a + b) + c = a + (b + c)Postulate No.4: For every pair of natural numbers, a and b, in that order, there is a unique (one

and only one) natural number called the product.Postulate No.5: If a and b are natural numbers, then ab = ba.Postulate No.6: If a, b and c are natural numbers, then (ab)c = a(bc) Postulate No.7: If a, b and c are natural numbers, then a (b + c) = ab + acPostulate No.8: There is a natural number called “one” and written 1 so that if a is an arbitrary

natural number, then a.1 = a.Postulate No.9: If a, b and c are natural numbers and if ac = bc then a = b.



Postulate No.10: If a, b and c are natural numbers and if ac = bc then a = bPostulate No.11: Any set of natural numbers which (1) includes the number 1 and which (2)

includes a + 1 whenever it includes the natural number a, includes every natural number. Postulate No.12: For any pair of natural number, a and b, one and only one of the following

alternatives must hold: either a – b, or there is a natural number x such that a + x = b, or there is a natural number y such that b + y = a.

Freund’s system of 12 postulates provides the possibility to characterize natural numbers when we explain how they behave and what math rules they must obey. To conclude the definition of “natural numbers” we can say that they must be interpreted either as standing for the whole number or else for math objects which share all their math properties. Using these postulates mathematicians are able to prove all other rules about natural numbers with which people have long been familiar.

Comprehension check:Answer the following questions:1) How many axioms did the Italian mathematician Peano give? What were they?2) Which axiom is the most important? Why?3) What does Peano’s theory state in essence?4) What can we state from Peano’s five rules?5) Who developed these axioms? What did he do?6) How useful is Freund’s system of 12 postulates?3. Discussion – Speaking - Listening3.1 Discussion:3.1.1 Prove or disprove each of these following statements:a) The set of even number is closed under addition. (A number m is even if and only if m = 2k

for some integer k).b) The sum of any two odd numbers is even. c) Let n be an integer. Prove that if is odd, then n is odd d) Let b be any number and n be a positive integer, write the formula of these following

expression: “b raised to the nth power, b squared, and b cubed.” How do we call the value b and n?e) Write in words for these formulae: d equals nth root of b; 2 is the square root of 4; 3 is the cube

(or the third root) of 27.3.1.2 Work in pairs to answer this questionIf x is an even integer and y is an odd integer, which of the following must be odd?

3.2 Speaking:

Learn how to speak these following



Notice:would be called “8 to the second power” or “8 to the power 2” or simply “8 squared”

53 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed".24 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

could be called “a to the n power”. would be called "the logarithm of 8 with base 2 is 3"or "log base 2 of 8 is 3" or "the

base-2 log of 8 is 3" would be called “the natural logarithm of 10”

log100 = 2 would be called a “common logarithm” of 100 is 2 (with base 10)3.3 Listening:Listen to the tape and fill in the blanks:RoundingAll the___(1)___ of arithmetic that we have used with the___(2)___ apply to decimal numbers.

However, when performing calculations involving decimal numbers it is not____(3)__ for the end result to be a number with a large quantity of ___(4)___after the decimal point. To make such numbers more manageable,___(5)___ can be rounded either to a specified number of significant figures or to a specified number_____(6)___decimal places.

Significant figures____(7)___counted from the first____(8)___ numeral on the left of the number. When the required numbers of significant figures have counted off, the remaining ___(9)___are deleted with the following proviso: If the first number deleted is a____(10)___or more the last significant numeral is increased by 1, otherwise it is left___(11)___.

4. Translation: 4.1. Translate into Vietnamese:4.1.1 Translate each of the following sentences into Vietnamese. a) Let , with b > 0. Then there exist integers q, r such that a = bq + r, b) For all , we have:

- d |a, d|b implies d| (a+ b)- d|a implies d|ab- d|a implies db|ab- d|a and a|b implies d|b- If and d|a, then

c) Let n be an integer greater than 1. Then p be the smallest divisor of n unequal to 1. Then p is a prime. If n is not itself a prime, then .

d) There are infinitely many prime numbers.e) Every natural number > 1 can (up to a reordering of the factors) be written uniquely a a product

of prime numbers.f) Let a, b, p and let p be a prime. If p is a divisor of ab, then p divides at least one of the

numbers of a or b. 4.1.2 Arithmetic operations



1. Addition: The concept of adding stems forms such fundamental facts that it does not require a definition and cannot be defined in formal fashion. We can use synonymous expressions, if we so much desire, like saying it is the process of combining.

Notation: 8 + 3 = 11; 8 and 3 are the addends, 11 is the sum.2. Subtraction: When one number is subtracted from another the result is called the difference or

remainder. The number subtracted is termed the subtrahend, and the number from which the subtrahend is subtracted is called minuend.

3. Multiplication: is the process of taking one number (called the multiplicand) a given number of times (this is the multiplier, which tells us how many times the multiplicand is to be taken). The result is called the product. The numbers multiplied together are called the factors of the products.

Notation: 12 x 5 = 60 or 12.5 = 60; 12 is the multiplicand, 5 is the multiplier and 60 is the product (here, 12 and 5 are the factors of product).

4. Division: is the process of finding one of two factors from the product and the other factor. It is the process of determining how many times one number is contained in another. The number divided by another is called the dividend. The number divided into the dividend is called the divisor, and the answer obtained by division is called the quotient.

Notation: 48:6 = 8; 48 is the dividend, 6 is the divisor and 8 is the quotient. Division may be checked by multiplication.

4.2. Translate into English: - Một số có chữ số tận cùng là một số chẵn ví dụ như 2, 4, 6, 8, …thì chia hết cho 2.- Một số có số tạo bởi hai chữ số tận cùng chia hết cho 4 thì số đó chia hết cho 4.- Một số tận cùng bằng 0 và 5 thì chia hết cho 5. - Một số chia hết cho 3 (hoặc chia hết cho 9) nếu tổng các chữ số của số đó chia hết cho 3 (hoặc

9). - Cho a, b, p và p là số nguyên tố. Nếu p là một ước của ab thì p là ước của ít nhất một trong

hai số a, hoặc b .4.3 Translate into English and explain what happened: Cho a = b. Nhân hai vế của biểu thức trên với a được Trừ hai vế với ta được: Phân tích ra thừa số: Chia hai vế cho a – b được: .Thay a bởi b được: Chia hai vế cho b được: 2 = 1Chuyện gì đã xảy ra?5. Exercises: 5.1 - Find the factors of these following numbers: 84 and 512.- Find the multiples of these following numbers: 149 and 254- Find the HCF and LCM of these following pair numbers:a) 512 and 84 b) 314 and 52 c) 27 and 96



5.2 It takes Eric 20 minutes to inspect a car. John needs only 15 minutes to inspect a car. If they both start inspecting cars at 9.00a.m., what is the first time the two mechanics will finish inspecting a car at the same time?

5.3 Find the first 10 prime numbers? Is there any largest prime number? Which number is the smallest prime number?

5.4 Find the prime factors and the prime factorization of 124, 243 and 679?5.5 A class of 45 students will be seated in rows. Every row will have the same number of

students. There must be at least two students in each row, and there must be at least two rows. A row is parallel to the front of the row. How many different arrangements are possible?

5.6 Let n be an integer, write the formula of the factorials of n.5.7 Write in factorials form: (n+3) x (n+2) x (n+1) x n5.8 Factorize: n! + (n + 2)! – (n+1)!5.9 If the value of an investment triples each year, what percent of its value today will the

investment be worth in 4 years?

5.10 Evaluate:

5.11 Simplify

5.12 Evaluate without a calculator

5.13 Find all possible pairs of numbers a and b which satisfy . Explain your reasoning.

5.14 If a store made a profit of $23.50 on Monday, lost $2.05 on Tuesday, lost $5.03 on Wednesday and make a profit of $30.10 on Thursday, and made a profit of $41.25 on Friday. What was its total profit (or loss) for the week? Use + for profit and – for loss.

5.15 Find (a) the prime factors and prime factorization(b) all the factors(c) the HCF and LCM

of 164 and 1025. 5.16 Find (a) the factors

(b) the prime factors(c) the HCF and LCM

of 255 and 10206. Assignment: Work in groups to prove these following theorems:a) Let n be an integer greater than 1. Let p be the smallest divisor of n unequal to 1. Then p is a

prime. If n is not itself a prime then b) There are infinitely many prime numbers. c) Every natural number > 1 (up to a reordering of the factors) can be written uniquely as a

product of prime numbers.d) Use the sieve of Eratosthenes to describe all the primes less than 100.



Puzzles:

BIRTHDAYWhen asked about his birthday, a man said:"The day before yesterday I was only 25 and next year I will turn 28."This is true only one day in a year - when was he born?

Just for fun: THE USE OF FOREIGN LANGUAGE

Little mouse: - Mommy! He’s saying something that I don’t understand at all?

Mother mouse: - Silence! It’s our enemy. Don’t go out of the house. That dirty cat is threatening

us.

Little mouse: - How did you understand what he said?

Mother mouse: - Consider it a very good reason to learn a foreign language.



Unit 2: RATIONAL AND IRRATIONAL NUMBERS REAL AND COMPLEX NUMBERS______________________________________________1. Vocabulary and Grammar review:1.1 Vocabulary:- Rational numbers (n): Số hữu tỉ- Irrational numbers (n): Số vô tỉ- Real numbers (n): Số thực- Complex numbers (n): Số phức- Fraction (n): Phân số- Numerator (n): Tử số của một phân số- Denominator (n): Mẫu số của một phân số- Proper fraction (n): Phân số rút gọn- Imaginary number (n): Số ảo- Improper fraction (n): Phân số chưa rút gọn- Mixed fraction (n): Hỗn số- Decimal number (n): Số thập phân- Circumference (n): Chu vi- Real line (n): Trục số thực- Inductive method (n): Phương pháp quy nạp- Whole numbers = integer- Order field (n): Trường được sắp.- Terminate (v): giới hạn- Exponential number (n): số mũ- Plot (v): Biểu diễn1.2 Grammar review: - Tenses- Passive – active voice- Adverbial clause1.3 Exercises:1.3.1 Fill in the blanks below with a single appropriate letter to indentify each set of

numbers with the properties or description of elements which characterize that set:The set of:_____ Even numbers ________Integers



_____ Rational numbers ________ Odd numbers_____ Irrational numbers ________ Natural number______ Real numbers ________ Whole numbersA. any number equal to a terminating decimal expression.B. {…,-3, -2, -1, 0, 1, 2, 3,….}C. any number which is rational and irrational.

D. any number of the form where p and q are integers and q is not zero.

E. any integer of the form 2k, where k is an integer.F. any integer of the form 2k+1, where k is an integer.G. any number equal to an infinite decimal expression with no repeating block of digits.H. {0, 1, 2, 3, …}I. any number which can be expressed as a ratio.{1, 2, 3, ….}1.3.2 Fill in the blanks with the correct forms of the word in the bracket:The arithmetic of the real numbers poses two fundamental problems. The first problem

____(1)___(concern) division by zero. This problem cannot be satisfactorily resolved by defining what is meant by such a ______(2)__ (divide) and so we resolve the problem by saying that division by zero is just not defined. In effect, we cannot do it. The second problem concerns the square root of a negative number. This problem can be ____(3)___(resolve), but in doing so we open up a whole new panoply of numbers. To reduce the problem to its essentials we consider the square root of minus unity . To resolve the problem we state that such a number exists and we give it the numeral j where so that . Now we find that we can define the square root of any negative real number. For example .

Notice that and . Having solved the problem of defining the square root of a negative real number we now have a

further problem. What sort of number ____(4)___(be) j? It is not a real number - the real line is complete – there ____(5)___(be) no points available on the real line where we could put j. The conclusion is that it is new, different type of number. We call it an ___(6)___(imagine) number. Indeed any number of the form aj where a is a real number is called an imaginary number.

The numbers of the form z = a + jb where a and b are real numbers and , being a ____(7)___(mix) of real and imaginary numbers, ____(8)___(call) complex numbers. The real number a is called the real part of z and b is called the imaginary part of z.1.3.3 Choose the correct words in the table to fill in the blank

1. A set A of real numbers is said to be ____(1)___ if and only if, and implies .


Inductive integer rational number order field irrational


2. The real number system must have any property which possessed by a field, an order field, or a complete _____(2)___.

A real number is called a ___(3)___ if and only if, it is the quotient of two integers. A real number which is not rational is said to be ____(4)___.1.2.4 Put the words in the correct order to make a meaningful sentence:1. Every/ and / denominator/ fraction / has / a / numerator. 2. Fraction / can’t /by 0 / zero/ as / a / denominator / have / since /division /is /not /defined. 3. Two / or /number /fractions / are / same / equivalent / or /equal / if / they /represent / the /ratio.4. All /the / irrational / number / combined /with /all /the /rational /numbers /form /the /real / numbers.5. Any / its / prime / natural / can / be / number / written / as / a / product / of / factors. 6. Calculations / using / a / calculator / involving / be / the / natural / numbers / can / performed. 2. Reading:2.1 FRACTIONSEvery fraction has a numerator and denominator. The denominator tells you the number of parts

of equal size into which some quantity is divided. The numerator tells you how many of these parts are to be taken. A fraction cannot have 0 as a denominator since division by 0 is not defined. A fraction with 1 as the numerator is the same as the whole number that is its numerator.

Fraction representing values less than 1, like for example, are called proper fractions.

Fractions which name a number equal to or greater than 1, like or , are called improper factions.

These are numerals like , which name a whole number and a fractional number. Such numerals are

called mixed fractions. Complex fraction is one in which the numerator, denominator, or both are

fractions themselves; for example: . Since a fraction represents division, the complex fraction can

be written as: .

We have already seen that if we multiply a whole number by 1 we leave the number unchanged. The same is true of fractions when we multiply both integers in a fraction by the same number. For

example, . We can also use the idea that 1 can be expressed as a fraction in various ways

and so on.

Now see what happens when you multiply by . You will have .

In the above operation you have changed the fractions to its higher terms.



Now look at this . In both of the above operations the number you have

chosen for 1 is .

In the second example you have used division to change to lower terms that is to . The

numerator and denominator in this fraction are primes and accordingly we call such a fraction the simplest fraction for the given rational number.

Two fractions are equivalent or equal if they represent the same ratio or number. If you multiply or divide the numerator and denominator of a fraction by the same nonzero number, the result is equivalent to the original fraction. A fraction has been reduced to lowest term when the numerator and denominator have no common factor. If two fractions have the same numerator, the one with the lesser denominator is greater. If two fractions have the same denominator, the one with the greater numerator is greater.

Comprehension checkAnswer the following questions:

1) What will happen if you multiply a fraction by (-1)?

2) Have you changed the fraction when you multiply by ?

3) What division have you used to change to lower terms?

4) In your opinion, is there any largest fraction? Smallest fraction?2.2 Reading: REAL NUMBERSAll rational numbers have a decimal format that either terminates or contains an infinitely

repeated, finite sequence of numerals. By inference, any decimal number that neither terminates nor contains such a repeated sequence of numeral is not a rational number. It is called an irrational number – it is a number that cannot be written as one integer divided by another integer. Irrational numbers are not rare, indeed there are many more irrational than rational numbers. They do, however, present a problem. Because of there is an infinite number of numerals after decimal point. Instead, we devise other notations such as and . The irrational number (pi) is a number we shall meet when we measure a circumference of a circle. The irrational number e is called exponential number. It also is on your calculator and is given as .

All the irrational number combined with all the rational numbers form the real numbers. When all the real numbers are plotted on a line every point on the line corresponds to a number – there are no gaps and no overlaps, the line is complete and the numbers are said to be continuous. This continuous line is referred to as the real line.

Comprehension check:A) Answer the following questions:1. What is the characteristic of the rational numbers? Give 3 examples of rational numbers?2. What are the differences between rational and irrational numbers? Give 3 examples of the

irrational numbers?3. Draw the real line and plot some rational and irrational numbers on it. 4. Is there any largest real number? Why or Why not?



5. Between rational numbers and irrational number which is larger? 3. Discussion – Speaking – Listening3.1 Speaking:Read the following fractions:

Notice: In the fraction, the numerator is read by cardinal numbers while the denominator is read by ordinal number. If the numerator is greater than 2, then the denominator must be added s. For example:

: a half or one second

: a quarter or one fourth

: Three seconds

3.2 Discussion:

Prove or disprove the statement that if x and y are real number, then .

The properties and the arithmetic operations of complex numbers.3.3 Listening:Listen to the tape and fill in the blanks then answer the following questions:A___(1)___is a part of a whole and is____(2)__by one integer divided by____(3)__ integer.

Fractions are___(4)___proper, improper or mixed. Fraction which represent the same fractional part of a whole by but which have____(5)__ numerators and denominators are said to be ___(6)___. Equivalent fractions can be____(7)___from any fraction by____(8)___ or dividing both the numerator and denominator by the____(9)___ number. When the numerator and the denominator of a fraction have no____(10)___ in common the fraction is said to be in its___(10)___terms.

Answer the following questions:1) According to the text, what is a fraction?2) How can we create an equivalent fraction from a given one?3) What is the lowest term of a fraction?4. Translation:4.1 Translate into Vietnamese4.1.1 Translate each of the following sentences into Vietnamese:a) There exists infinitely many primes of the form 4m – 1b) All the rational numbers combined with all the rational numbers form the real numbers.c) When all the real numbers are plotted on a line, every point on the line corresponds to a

number. There are no gaps and no overlaps, the line is complete and the numbers are said to be continuous.

d) A percentage is a fractional part of a whole where the denominator of the fraction is equal to 100.

4.1.2 Translate the following text into Vietnamese:



- Addition and Subtraction of Fractions: To add or subtract two fractions each must be

converted to equivalent with the same denominator. For example: . Notice that the

smallest common denominator is the LCM of the original denominators.- Multiplication and Division of Fractions: Two fractions are multiplied by multiplying their

numerator and denominator independently. For example: . To divide one fraction by

another invert the divisor and multiply. For example: .

- Using “of”: The use of the word “of” is very common when dealing with fractions. For

example, half of 6 means 3. This can be written as: of 6 = 3. From this it can be seen that the word

“of” can be substituted by the multiplication sign so of 6 = .

- Roots and Fractional Powers: Just as plants generate from roots so a number can be generated by the repetitive multiplication of a root with itself. For example, and we call 4 the sixth root of 4096. We call 16 is the third root – or cube root of 4096. Fractional powers denote roots. For example, the notation for the sixth root of 4096 is: . A square root is a second root. For example . Not all roots are unique. For example 2 and -2 are both square roots of 4. Indeed, all even roots lack uniqueness.

4.2 Translate into English:1) Những số lớn hơn zero là các số dương, những số nhỏ hơn zero là các số âm.2) Nếu a, b, x là ba số nguyên dương bất kỳ, mà a + x = b + x thì suy ra a = b.3) Nếu a, b, x là ba số nguyên dương bất kỳ, mà ax = bx thì suy ra a = b. 4) Nếu A và B là các tập hợp và kéo theo thì ta nói A là tập con của B và viết .5) Nếu A là tập hợp con của B và có ít nhất một phần tử của B không phải là phần tử của A thì ta

nói A là tập hợp con thực sự của B. Hai tập hợp A và B là bằng nhau khi và chỉ khi và .6) Nâng một số lên luỹ thừa được định nghĩa là phép nhân liên tiếp. Lũy thừa còn được gọi là số

mũ and số được nâng lên lũy thừa được gọi là cơ số. Ví dụ: . Ở đây, số 3 được gọi là

số mũ và 10 được gọi là cơ số.

5. Exercises:5.1 Reduce each of the following fractions to their lowest term:

c)

5.2 Evaluate:

5.3 Calculate and simplify the expression

5.4 If x = 4, what is

5.5 Write as x raised to a power.



5.6 The sum of three integers is 66. The second is 2 more than the first, and the third is 4 more than twice the first. What are the integers?

5.7 The Smith family is traveling to a vacation destination in two cars. Mrs. Smith leaves home at noon with the children, traveling 40 miles per hour. Mr. Smith leaves 1 hour later and travels at 55 miles per hour. At what time does Mr. Smith overtake Mrs. Smith?

5.8 A box of light bulbs contains 24 bulbs. A worker replaces 17 bulbs in the shipping department and 13 bulbs in the accounting department. How many boxes of bulbs did he use?

5.9 Simplify: d)

5.10 A business is owned by 3 men and 1 woman, each of whom has an equal share. If one of the

men sells of his share to the woman, and another of the men keeps of his share and sells the rest

to the woman, what fraction of the business will the woman own?5.11 The price of ground coffee beans is d dollars for 8 ounces and each ounce makes c cups of

brewed coffee. In terms of c and d, what is the dollar cost of the ground coffee beans requires to make 1 cup of brewed coffee?

5.12 Write the expression as a complex number in standard form

a) (1 – 4i)(3+5i) b) c)

5.13 Plot the complex numbers in the complex plane.a) 1 + i b) -2 – 2i c) 3- 3i

Puzzles: Can you arrange the numerals 1 to 9 (1, 2, 3, 4, 5, 6, 7, 8 and 9) in a single fraction that equals exactly 1/3 (one third)?

Example that doesn't work: 7192/38456 = 0.187Just for fun

A LESSON IN SUMMATIONThe teacher: - If your father can do a piece of work in one hour and your mother can do it in one

hour, how long would it take both of them to do it? A pupil: - Three hours, teacher!The teacher: - Why?The pupil: I had to count the time they would waste in arguing.



Unit 3: FERMAT AND SOME OF HIS FAMOUS THEOREMS____________________________________________________

1. VOCABULARY AND GRAMMAR REVIEW:1.1 Vocabulary:Look up in the dictionary to find out the meaning of the words:- Algebra (n): Đại số- Algebraic (a): Thuộc về đại số- Algorithm (n): Thuật toán- Angle (n): Góc- Associative property (n): tính chất kết hợp- Bequeath (v): để lại- Commutative property (n): tính chất giao hoán- Concept (n): Khái niệm- Conjecture (n)(v): sự phỏng đoán - Differential equation (n): Phương trình vi phân - Distributive property (n): tính chất phân phối- Equality (n): phương trình- Equation (n): phương trình - Geometry (n): Hình học- Ideal (n): Iđêan - Line (n): đường thẳng- Notion (n): Khái niệm- Number theorist (n): Nhà toán học nghiên cứu lý thuyết số- Point (n): điểm- Proclamation (n): sự công bố- Right angle (n): Góc vuông- Theorem (n): định lý- Triangle (n): Tam giác- Triple (n): bộ ba- Unsolvability (n): Tính không giải được. 1.2 Grammar review:- Tense- Relative clause- Passive – Active voice- Comparison

1.3 Exercises:1.3.1 Put the sentences into the right order to make a complete paragraph



What is mathematics:_____ The largest branch is that which builds on ordinary whole numbers, fractions, and irrational

numbers, or what is called collectively the real number system. _____ Hence, from the standpoint of structure, the concepts, axioms and theorems are the

essential components of any compartment of maths. __(1)__ Maths as science, viewed as whole, is a collection of branches._______ These concepts must verify explicitly stated axioms. Some of the axioms of the maths of

numbers are the associative, commutative, and distributive properties and the axioms about equalities. _______ Arithmetic, algebra, the study of functions, the calculus differential equations and other

various subjects which follow the calculus in logical order are all developments of the real number system. This part of maths is termed the maths of numbers.

______ Some of the axioms of geometry are that two points determine of line, all right angles are equal, etc. From these concepts and axioms, theorems are deduced.

______ A second branch is geometry consisting of several geometries. Maths contains many more divisions. Each branch has the same logical structure: it begins with certain concepts, such as the whole numbers or integers in the maths of number or such as points, lines, triangles in geometry.

1.3.2 Fill in the blanks with the suitable words - Basic tool (n) - Fundamental notion (n) - Represent (v)- Concern (v) - Number (n) - Representation (n)- Depend (v) - Organization (n) - Show (v)- Unsolvability (n)AlgorithmsOriginally algorithms __________ (1) solely with numerical calculations; Euclid’s algorithms, for

finding the greatest common divisor of __________(2) is the best illustration. There are many properties of Euclid’s powerful algorithm which has become a _________(3) in modern algebra and number theory. Nowadays the concept of an algorithm is one of the most ________(4) in maths. Experience with computers_________(5) that the data manipulated by programs can represent virtually anything. In all branches of maths, the task to prove the solvability or ___________(6) of any problem requires a precise algorithm. In computer science the emphasis has now shifted to the study of various structures by which information ________(7) and the branching or decision making aspects of algorithms, which allow them to fall on one or another sequence of the operation _________(8) on the state of affairs at the time. It is precisely these features of algorithms that sometimes make algorithms models more suitable than traditional maths models for the _________(9) and ________(10) of knowledge.

2. Reading: 2.1 FERMAT’S LAST THEOREMPierre de Fermat was born in Toulouse in 1601 and died in 1665. Today we think of Fermat as a

number theorist, in fact as perhaps the most famous number theorist who ever lived.The history of Pythagorean triples goes back to 1600 B.C, but it was not until the seventeenth

century A.D that mathematicians seriously attacked, in general terms, the problem of finding positive integer solutions to the equations . Many mathematicians conjectured that there are no positive integer solutions to this equation if n is greater than 2. Fermat’s now famous conjecture was inscribed in the margin of his copy of the Latin translation of Diophantus’s Arithmetica. The note



read: “To divide a cube into two cubes, a fourth power or in general any power whatever into tow powers of the same denomination above the second is impossible and I have assuredly found an admirable proof of this, by the margin is too narrow to contain it”.

Despite Fermat’s confident proclamation the conjecture, referred to as “Fermat’s last theorem” remains unproven, Fermat gave elsewhere a proof for the case n = 4, it was not until the next century that L.Euler supplied a proof for the case n = 3, and still another century passed before A.Legendre and L.Dirichlet arrived at independent proofs of the case n = 5. Not long after, in 1838, G.Lame established the theorem for n = 7. In 1843, the German mathematician E.Kummer submitted a proof of Fermat’s theorem to Dirichlet. Dirichlet found an error in the argument and Kummer returned to the problem. After developing the algebraic “theory of ideals”, Kummer produced a proof for “most small n”. Subsequent progress in the problem utilized Kummer’s ideals and may more special cases were proved. It is now know that Fermat’s conjecture is true for all n < 4.003 and many special values of n, but no general proof has been found.

Fermat’s conjecture generated such interest among mathematicians that in 1908 the German mathematician P.Wolfsehl bequeathed DM 100.000 to the Academy of Science at Gottingen as a prize for the first complete proof to the problem for which the greatest number of incorrect proof was published. However, these faulty arguments did not tarnish the reputation of the genius who first proposed the proposition – P.Fermat.

Comprehension check:A. Answer the following sentences:a) How old was Pierre Fermat when he died?b) Which problem did mathematicians face in the 17 century A.D?c) What did many mathematicians conjecture at that time?d) Who first gave a proof to Fermat’s theorem?e) What proof did he give?f) Did any mathematicians prove Fermat’s theorem after him? Who were they?B. Are the statements True (T) or False (F)? Correct the false sentences.a) The German mathematician E. Kummer was the first to find an error in the argument.b) With the algebraic “Theory of ideals” in hand, Kummer produced a proof for “most small n”

and many special cases.c) A general proof has been found for all value of n.d) The German mathematician P.Wolfskehl won DM 100.000 in 1908 for the first complete proof

of the theorem.2.2 CALCULATORS:It is assumed that most calculations will be performed using a calculator. The typical electronic

hard calculator contains three types of key. The number keys, the arithmetic operations keys and the functions keys. In addition the calculator has a memory that is capable of storing a single number.

The number keys range from 0 to 9 and by pressing them in sequence natural numbers can be displayed on the calculator screen. There is also a decimal point key or decimal numbers. Some calculators will even display and manipulate fractions.

The operations keys permit addition; subtraction; multiplication; division and raising to a power. Using these arithmetic operations can be performed on the numbers entered into the calculator.



The function keys permit a variety of complicated operations to be performed on the numbers entered into the calculator. For example, the function key will produce the positive square root of the entered number.

Comprehension checkAnswer the following questions:1) What is the main idea of the second paragraph?2) How many types of keys does a typical electronic hard calculator have?3) What does the functions keys permit?4) In your opinion, is the calculator very important to studying mathematics? Why or Why not? 3. Speaking - Writing – Listening - Discussion :3.1 Discussion1) In your opinions, Why do we have to study maths? What should we do to be good at maths? 2) Which branch of maths do you like most?3) Could you tell the class one of the mathematicians that you admire most?3.2 Writing: Write a short paragraph about 150 words about this following subject. It is said that maths is the most important natural science. Do you agree or disagree with that

idea? Why or Why not? 3.3 Listening: Listen to the tape and fill in the blanks: An axis ____(1)___ a straight line with two possible directions, one of which is the direction

and_____(2)___ the other (opposite), the negative direction. The positive direction is ordinarily designated by an arrow. The number (or coordinate) axis (or number line) is and axis with the origin (starting point) designated by O, and with a scale unit. ____(3)___ numbers can be depicted by indicating points on the number line. Whole numbers (integer) are indicated by ___(4)___that are obtained by laying off the scale unit a certain number of times to the left of the origin O in the case of negative number and to the right in the case of positive number. ___(5)___is indicated by the starting point (origin). Fractional (rational) numbers are also readily depicted by points on the number line. There are infinity of whole numbers but the points depicting them on the ____(6)__ line are rather spread out – integral points on the line are spaced at unit distance from one another. Rational points on the line are very ___(7)____ together, and yet there are points on the number line that are not ___(8)__ of rational numbers.

4. Translation 4.1 Translate into Vietnamese:4.1.1 Translate each of the following sentences into Vietnamese:a) The numbers are called Fermat number.

b) Let and p be a prime. Then p divides . Furthermore, if p doesn’t divide a then p divides . (Fermat’s little theorem).

c) Any number raised to the power 0 equals unity. d) If a number is written as a give base raised to some power then that number raised to a further

power is equal to the base raised to the product of the powers.



4.1.2 Translate the following text into Vietnamese:Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de-

Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's Plane loci to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat.

From Bordeaux Fermat went to Orléans where he studied law at the University. He received a degree in civil law and he purchased the offices of councillor at the parliament in Toulouse. So by 1631 Fermat was a lawyer and government official in Toulouse and because of the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat.

Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem. This theorem states that

xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2. 4.2. Translate into EnglishĐịnh lý Fermat:Định lý này phát biểu rằng nếu a là một số nguyên và p là một số nguyên tố không chia hết a, thì p

chia hết ; hoặc theo ngôn ngữ đồng dư thì . Ví dụ, được chia hết bởi 5. Một hệ quả đơn giản được phát biểu như sau, nếu p là số nguyên tố có thể chia hết a hoặc không chia hết a thì p phải chia hết : hoặc tương đương .

5. Exercises: 5.1 Fermat posed the problem“Give a triangle find the point where the sum of the distances to the vertices is a minimum”Could you solve this problem?

5.2 The number of boys attending Union High School is twice the number of girls. If of the

girls and of the boys play soccer, What fraction of the students at Union play soccer?

5.3 If and are both integers, which of the following could be the value of N?

a) 4b) 64c) 72

5.4. Snow is falling at a rate of inch per 24 minutes. How much snow will fall in 2 hours?

5.5. As part of a special promotion, customers receive one free woozle with every five they buy. Kate buys only woozles and leaves the store with a total of 30 of them. If she spent $75, how many dollars does each woozle sell for?


http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Beaugrand.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html


5.6. Carmela drives 20 miles from her home to the store at a speed of 30 miles per hour. If she makes the return trip home at a speed of 40 miles per hour, what is the total amount of time she spent driving?

5.7. Sixty percent of a class goes on a field trip. If twelve students don’t go on the field trip, how many students are in the class?

5.8. In how many different ways can 3 people arrange themselves in a row of 4 seats?5.9. The ratio of flour to sugar in a certain recipe is 7:1. If 12 cups of flour are used, how many

cups of sugar are needed?5.10. If m and n are both negative numbers, m is less than -1 and n is greater than -1, which of the

following gives all possible values of the product mn? A) All negative numbersB) All negative numbers less than -1.C) All negative number greater than -1D) All positive numbersE) All positive numbers less than 1.

Puzzles: Farmer Crosses RiverA farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat

that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage.

How can the farmer bring the wolf, the goat, and the cabbage across the river?Just for fun

THE BIG ZERO AND THE LITTLE ZEROOnce when a little zero fell on to a big zero, the big zero exclaimed: “You stupid idiot! Aren’t you

ashamed of yourself? Don’t you know how much bigger I am than you?”The little zero replied: “What’s the point of your being bigger? We are worth the same – nil”.

______________________________________________________________________________ASSIGNMENT:1. Prove the following theorem:

“Let , with b > 0. Then there exist integer q, r such that a = bq + r, .2. Prove that , where 3. Read the following text. Translate it into Vietnamese, then use the sieve of Eratosthenes to

find out all the prime number less than 200.

“The classical method to find the first few (not too large) prime numbers is the sieve of Eratosthenes. Although this method nowadays only has historical value, we will briefly describe it. It is a nice way to find all prime numbers less than 100. First write down all numbers from 2 to 99. The principle of the sieve is a procedure in which we erase multiples of certain numbers. In each iteration, the first step consists of finding the first number which has not yet been used in this erasing process.



Call this number p. Then p is a prime number. The second step is to erase all multiples of p, except for p itself, and repeat the process.

4. Prove the following theorem: “There are infinitely many prime numbers.”

5. Prove that: “Every natural number greater than 1 can (up to a reordering of the factors) be written uniquely as a product of prime numbers.

6. Write a short paragraph about the Euclidean algorithm to find out the greatest common divisor (or the highest common factor) of the two integer numbers a and b, then give an example to illustrate this algorithm.

7. Write a short paragraph about Mersenne numbers and Fermat numbers. Give examples. 8. Prove that there exist many prime numbers of the form 4m -1. 9. Work in group to solve the problemsA. Translate each of the following sentences into Vietnamese:a) When multiplying two (or more) numbers with the same base, keep the base and add the

indices: b) When dividing two numbers with the same base, keep the base and subtract the indices.

c) Any number raised to the power of zero equals one. d) When raising a number in index form to a power, keep the base and multiply the powers.

e) If a product of factors is to be raised to a power, each factor in the bracket is raised to that

power. f) If a fraction is to be raised to a power, each number in the bracket is to be raised to that power.

g) A number raised to a negative power is the same as the reciprocal of the number raised to the positive of that power.

B. Simplify:

a) b)

10. Choose one of these following topics to write a short paragraph (more than 300 words):1. The famous mathematician that you admire most. 2. Diophantine equations and its solving method. 3. The characteristic of prime numbers. Some methods of finding a prime number. 4. Fermat numbers.5. The characteristic of irrational numbers and the surd. (A surd is an irrational number which

can only be expressed exactly using the radical or root sign ( ). For example, are surds but are not.)

6. Continuous fractions.


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Chapter 1. Arithmetics - Wikispacesenglishformaths.wikispaces.com/file/view/Chapter+1.doc · Web viewChapter 1: Arithmetics Unit 1: The Natural numbers and The Integers _____ 1. VOCABULARY