1. GLOSSARY OF EXPRESSIONS AND DEFINITIONS

A SET is any well-defined collection of objects.

The objects are said to be the ELEMENTS or MEMBERS ofthe set.

The curly brackets { } are a short way of writing: „ the set of..."

POSITIVE INTEGERS

The set of NATURAL NUMBERS is

COUNTING NUMBERS

The set of NEGATIVE INTEGERS is 1}

The set of WHOLE NUMBERS is

The set of INTEGERS is

The set of RATIONAL NUNBERS is 2/3, 4/7, 5/8,...}

Rational number is x /y, the RATIO of x to y where x and y

represent whole numbers, and y is not 0 .

The of IRRATIONAL NUMBERS is { 3,14..., e = 2,71

Irrational numbers cannot be expressed as the ratio of two integers.

The set of REAL NUMBERS includes the INTEGERS, the RATIONAL numbers, and

the IRRATIONAL numbers.

EVEN NUMBERS : positive integers divisible by 2.

If x is a positive integer, then 2x gives an even number.

For example: 6, 18 , 94 , 1818 are all even.

ODD NUMBERS : positive integers, which are one more, or one less than some multiple

of two.

If x is a natural number, then ( 2x + I ), or (2x - I ) are odd numbers.

For example: 9, 75, 43, 17 are all odd.

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CONSECUTIVE NUMBERS are those integers which follow one an other.

For example: 4 , 5 , 6 , 7 are consecutive integers.

If x is an integer, the NEXT integer after x is (x + l),

The NUMBER LINE.

The RECIPROCAL of a number : for each NON-ZERO real number x , there exists a

reciprocal I / x , such that x.(l / x) = I

For example, the reciprocal of 7 is 1/7 . The reciprocal of 3/5 is 5/3 ,

The FRACTION : The symbol x/ y is a fraction x over y .

NUMERATORFRACTION =

DENOMINATOR

(the upper number : numerator, the lower number : denominator).

The PROPER FRACTION : the numerator is LESS than the denominator.

For example, 7/9 , 45 /76 , 101 /234 are proper fractions.

The MPROPER FRACTION : The numerator is GREATER than or EQUAL to thedenominator.

For example, 9/5, 17/4, 102 /76 , 86 /86 .

Improper fractions are equal to, or greater than whole numbers.

The MIXED

For example,

: a whole number and a fraction.

, 76 h , 35 1/4

Mixed numbers CONVERTED into fractional form •

10 43For example, 3—

11 11

4 39or 5—

Improper fractions CONVERTED into mixed numbers :

87For example

7or

101 210 10 9 9

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EQUAL FRACTIONS : we obtain sets of equal fractions by multiplying or dividing the

numerator and the denominator by the SAME NUMBER.

1 6 18 72For example,

12 - 36 1442

The DECIMAL SYSTEM : The decimal number system is based on units often.

There are 10 DIGITS

The digits in every number have a PLACE VALUE.

For example, 8342

That means, the 8 stands for 8 thousands, the 3 for 3 hundreds, the 4 for 4 tens, and

the 2 for 2 units.

Any number by changing-over ( by reversing) the digits becomes a new number.

For example reversing the digits, 24 becomes 42 .

The DECMAL FRACTION we a DECIMAL POINT after the unit's digit.

Rational numbers can be represented as PERIODIC (repeating) decimal fractions.

For example, 5 / 6 = 0.8333

or 22 / 5 = 4.4 is a terminating decimal because the repeating element is 0 .

Irrational numbers can be represented as NONPERIODIC decimals. We can calculate only

the approximate value for an irrational number.( The calculators use rational

approximations for irrational numbers in all calculations.)

For example, n = 3,14...., or e = 2,71...

ALGEBRAIC EXPRESSIONS

VARIABLES : A letter used to represent an arbitrary element of a given set is called a

variable.

For example, a , b , c , x , y , and z may represent arbitrary elements of the set of REAL

NUMBERS.

The VALUE : When we replace a variable by a particular number, that number is calledthe value of the variable.

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The ALGEBIU\IC EXPRESSION

An expression is called algebraic expression if it can be obtained from a FINITE (limited )

NUMBER of variables together with real numbers by using a FINITE NUMBER OF

TIMES the operations of addition, subtraction, multiplication, division or taking the nth

root( n an integer) .

For example, 3x2yz5 + 17a4bc7

14 /81 xy

23or 7 34ab — 56xyc + —

are all algebraic expressions.

The TERM : The parts of an algebraic expression connected by the + or - sign are called

the terms of the algebraic expression.

The COEFFICIENTS : The numerical factor in the terms is called the coefficient. The

coefficient is normally placed first in the term followed by the variables.

For example,

algebraic expression

terms

coefficients

15ab3c — 73xyz 8 + 17 /4 abc

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15ab3 c ; -73xyz8 ; and 17 /4 abc12

15 • -73 and 17/4

LIKE or SIMILAR TERMS : terms that contain the same variables raised to the samepowers are called like terms.

For example, 3x , 5x, 67x are like terms

32xyz2 , 23xyz , 35 /71xyz are like terms,

whereas

4xy2 , 23 yz , 46 y2z are UNLIKE terms.

LIKE TERMS are ADDED and SUBTRACTED by adding or subtracting the coefficientsof like terms.

For example, 4xy - 46yz3 + 37 x +26xy + 12yz3 78 = 30xy — 34yz3 + 37x 78.

PARENTHESES , BRACKETS , BRACES : (

In an algebraic expression two or more terms are often grouped by the symbols ofparentheses or brackets to indicate that the enclosed terms are to be regarded altogether asone entity.

For example, ( 3xy —45 c)

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Operations

ADDITION

32+7=39

Terms: 32 and 7

The sum : 39

SUBTRACTION

15-9=6

The difference : 6

MULTIPLICATION

6 x 12=72

Factors: 6 and 12

The product: 72

DIVISION

42 +7=6

The dividend: 42

The divisor : 7

The quotient: 6

read

or

or

read

or

or

read

or

or

read

or

or

" 32 plus 7 is ( equal to ) 39

" the sum of 32 and 7 is 39 "

32 added to 7 gives 39 '

15 minus 9 is ( equal to) 6"

the difference between 15 and 9 is 6 "

15 and 9 differ by 6"

" 6 times 12 is 72 "

6 multiplied by 12 is ( equal to ) 72 "

" the product of 6 and 12 is 72 "

" 42 divided by 7 gives 6 "

"the quotient of 42 and 7 is 6 "

"if we divide 42 by 7 we get 6 "

45 + and the REMAINDER is 3 .

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2. ALGEBRA REVIEW

Properties of the operations

COMMUTATIVE PROPERTIES (LAWS)

and(2.1)

ASSOCIATIVE PROPERTIES (LAWS)

and x x z) = (x x Y) xz(2.2)

DISTRBUTIVE PROPERTIES (LAWS)

and (2.3)

Identity elements

0 is the identity element for addition

Ixx=xxl I is the identity element for multiplication

Inverses

For every real number x , there is a unique real number called the ADDITIVE

INVERSE of x , denoted by x) , such that

( - x) is the negative of x.

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For every real number x # 0 , there is a unique real number called the

1

MULTIPLICATIVE INVERSE of x , denoted by (or l/x ), such that

l/x=l

1

(or l/x) is called the reciprocal of x .

Order of operations convention

Multiplication and division have priority over addition and subtraction unless parenthesis

are used to indicate otherwise.

For example : 5x7 + 12 = 35 + 12=47

5 x (7 + 12) = 5

or 27+ 8/2+6 x 27+4+12=43

27 + (8/2 + 6) x 6) x 2=27 +10 x 2-27+20 — 47

Rules for zero

0 x x = x x 0=0 , for every real number x .

If x/y=0 , then x = 0 .

If x x y = 0 , then either x = 0 , or y = 0 , or both.

The quotient x /0 is not defined. Division by 0 is UNDEFINED.

Rules for negatives

and

Multiplying or dividing LIKE signs give PLUS .

Multiplying or dividing UNLIKE signs give MINUS.

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Multiplication of algebraic expressions

Multiplication and addition of algebraic expressions are connected by the distributive law.

(see ( 2.3) ). Using the distributive law, we multiply each term of one expression by each

term of the other.

For example

(2x +

=

- 7) = 2x2 - 14x + 5x-35 =2x 2 - 9x -35

Important identities (product formulas)

+2.a.b (a —b) 2 = a 2 —2.ab +b 2

(a + b)(a - b) = a2 —b2

The common factor

Whenever two or more terms have a common factor, we can use the distributive law to

express a sum or difference of terms as a product of factors.

Ifwe take out common factors, we NSERT BRACKETS.

For example: ab + 7b - 4bc = b(a + 7 - 4c) ;

40a- 28b + = 4(10a - 7b + 1 lcd) .

EXERCISES

1. Simplify the following algebraic expressions by removing the brackets and addinglike terms:

a.) 7K -3y+7- + 5y-(x-3y+7)]

b.) 9x [3x + (x- 8y) - (x - 5y) - (x - 6y) + 24x]

c.) 4a+ 16b - [8-9b + 8a-(4b-6a)]

d.) 2x —xy +3y (x - 7xy)—

2. Take out common factors:

a.) 64ac - 8a+ 16ab

c.) 3x3 -x 2y- 3y2 + 9xy

10

+ 5xy —y) + 4x ]

b.) 6a2 -3b -6b 2 +3a

d.) x -y +x+yf.) a b

4—a—b

Natural numbers

} is the set of natural numbers, or positive integers.

PRIME NUMBERS

A natural number other than 1 , is said to be prime, if it has no positive

factors other than 1 and itself.

For example •

2, 3, 5, 11, 47, 53, 79, ... are all prime.

The number 1 has only one divisor, namely I ; and I is regarded as not

being a prime number.

NUMBERS

If a natural number is a composite number, it can be expressed as a product

of two or more natural numbers other than unity.

For example

27 = 51 =3x17 ; or 42 = are composite numbers.

The prime numbers, 1, and the composite numbers make up the SET of NATURAL

NUMBERS.

Prime factorization

It can be shown that every positive integer other than I can be expressed as a unique

product of prime numbers

For example , the prime factorization of 144 , 96 , and 150 can be found as follows

144 = = 2x2x2x18 =

96 = = 2>004 = 2x2x2x12 =

150 = 2x75 = =

= .=

It is noticeable that certain factors may be repeated. If the factor x occurs n times, we

can write it in a shorter way :

xx=x ( read x to the power n ")

( where x appears as a factor n times.)

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The raised numeral n, is called an EXPONENT , or INDEX (plural INDICES), since it

points out the number of repeated factors.

For example

144 = 24 x 32 or 5 2

96=2 5 x 3 or 150=2x3 x

The highest common factor (HCF)

The largest common factor of two or more numbers is called the HIGHEST COMMON

FACTOR. It is therefore the largest number that will divide into these numbers exactly.

For example .

The highest common factor of 144 and 96 is 24 x 3 = 16 x 3 = 48,

since 144 = 48x3 and 96=480The highest common factor of 144 , 96 and 150 is

since 144=24x6 and 106 and 150 .

The lowest common multiply (LCM)

The LOWEST CONWION MULTIPLE of two or more numbers is the smallest positive

number that is still divisible by all of the given numbers.

To find the lowest common multiple, first find the prime factorization of each number.

The LOWEST COMMON MULTIPLE is the product ofthe factors, using

the HIGHEST POWER of each which occurs in any of the original numbers.

For example:

The lowest common multiple of 144 and 96 is 25 x 32 =32 x 9=288,

since 144 = 24 x 32 and 96=2 5 x 3.

The lowest common multiple of 144 , 96 , and 150 is 7200,

since 144=2 4 x 32 and 96=2 5 x 3 and 150=2 x 3 x 52

The lowest common multiple is : 25 x 32 x52 = 7200 .

Rules for fractions

symbol is called a fraction a over b , where a and b are INTEGERS and b *.0.

lower number ofa fraction is called the DENOMINATOR,

the upper number of a fraction is called the NUMERATOR, that is

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NUMERATORfraction =

DENOMINATOR

The value of a fraction is not changed if the numerator and the denominator aremultiplied, or divided by the same number, that is

a a.cb b.c

SIMPLIFY A FRACTION: the basic technique is to cancel factors common to bothnumerator and denominator.

When cancelling the numerator and the denominator are divided by thesame number.

A fraction is said to be SIMPLIFIED when there are no remaining factors common to thenumerator and denominator. When the fraction is in its simplest form, is said to be in itsLOWEST TERMS.

For example :

25 1a.) — in its simplest form is — , after cancelling by 25.

75 3

28 7b.) in its simplest form is — , after cancelling by 4.

36 9

ADDITION AND SUBTRACTION OF FRACTIONS:

To add and subtract fractions, we have to find the LEAST COMMON

DENOMINATOR.

The least common denominator is the smallest denominator that is still

divisible by all of the original denominators. The LEAST COMMON

DENOMINATOR is the LOWEST COMMON MULTmuE ofthe original

denominators.

First we bring the fractions to the same denominator by multiplying the numerators and

denominators by the same number and then we add ( or subtract) the numerators. The

common denominator is the denominator of the sum ( or of the difference) .

For example, find the values of :

15 30

7 11

36 24

12+21+8-54 13

45 5 90 90

5 13 7.10-11.15+5.60-13.9 88 11

6 40 360 360 45

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MULTIPLICATION AND DIVISION OF FRACTIONS:

To multiply fractions we multiply the numerators together for the

numerator of the product, and multiply the denominators together for the

denominator of the product.

Before multiplying, mixed numbers are changed to improper fractions.

For example, find the values of

311 33a.)

4 25 ̄ 100

1 3 36

78727 36.27 972 243 5

8 56 56 14 14

To DIVIDE by a fraction we multiply by its reciprocal.

For example, find the values of

5a.)

7

8b.)

9

10

21

20

39

5 21

7 10

8 399 20

5.21 5-3 3 1

7-10 10 2 2

8-39 2-13 26 11

9-20 3-5 15 15

THE COMPLEX FRACTION:

When the numerator or the denominator of a fraction is itself a fraction, wecall that expression a complex fraction. We can simplify complex fractions.

For example:

5 5 5 5 5 23 115 344 4 3 12 3-23+12 81 81 81

2 21+2 23 23 233 3

Revision exercises (1)

1. Perform the indicated operations and simplify if possible:

1 32 3 a a

4 544a 2 a—x x + a 3 3

c.)x + a a—x

1 5 3y 3 9y 2 8y3

f.)5

2. Simplify each complex fraction

a 1

a.)1

7x-14

c.)7 + 3x

4

5 10b.) 31

d.)

3. Define or explain each of the following:

a.) The reciprocal of a rational number

b.) A prime number

c.) The highest common factor

d.) A rational number

e.) An odd integer

4. Find the reciprocals of the following:

9 7

25 10

2

3c.) 2—)

I 1

5. When you multiply two odd integers, is the answer even or odd?

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1 1

6. If a I and b _ —1 , find the value of c + —b a

7. Give five consecutive integers, none of which is prime.

8. Tell which of the following statements are true and which are false. Give reasons for

your answers.

a.) 57 is a prime number

c.) 67/37 is a proper fraction

b.) 51/17 is an integer

e.) The highest common factor of 36 and 108 is 6.

9. If a, b , x are positive unequal numbers and x # 0 , which of the following

statements are true and which are false:

a+bxa.)

a

x

a 2 +b 2 x 2

c.) = a + bxa+bx

a 2 —b 2 x 2

b.) = a —bxa+bx

d.) = a—ba—b a—b

1

10. — = 8 , what is the value ofx

a.) 84

b.) 84 +2

c.) 84 + 2 8

d.) 84 -2 8 +2

e.) 84 +2 8 -2

Only one of these is correct. (Do not use the calculator!)

11. If a, b, and c are positive integers less than 10, then (IOa + b)(10a + c) equals

100a(a + 1) + bc if:

b.) a+b=10

c.) b+c=10

Find the correct answer. Prove your answer.

12. Is the sum of four consecutive integers

a.) always even ? b.) always odd?

c.) always a multiple of 4 ?

Prove your answers.

13. Consider the following four statements:

2.) +3)2

x(2x + 5)

3.) (X + = + 45 — 4x 2 + 6(4x + 6)

Decide for each statement whether it is true:

a.) for just two values of x

c.) for just one value of x

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b.) for no values of x

d.) for all values of x

14. Is the sum of three consecutive integers

a.) always even ? b.) always odd ?

c.) always a multiple of three?

Prove your answers.

15. Tell which of the following statements are true and which are false.

(5AÆ —7) a.) The numbers and + 7)

are reciprocals.

b.) In the decimal system we have nine digits.

c.) Since 15 has two factors, it is a prime number.

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