MATH LET REVIEWer

NUMERATION SYSTEM

A system of reading and writing numbers is a numeration system. This consists of symbols and rules or principles on how to use these symbols. Our system of reading and writing numbers is the decimal system or the Hindu-Arabic system. Ten digits are used- 0,1,2,3,4,5,6,7,8, and 9. This system

is based on groups of ten. It uses the place value concept.

MIXED DECIMALS

In reading mixed decimals the whole number is read first, followed by the word and to locate the decimal point, then the decimal. In reading decimal, read the numbers to the right of the decimal point as if it were whole number. Then say the name of the position of the last digit on the right as the same of the decimal. We read the number in the second place value chart as sixty-eight thousand, three hundred fourteen and three hundred sixty-eight million nine hundred forty-two thousand fifteen billionths. The ths in the name of a digit to the right of the decimal point distinguishes it from the name of the corresponding digit to the left of the decimal point. There are no commas in the decimal.

ROUNDING OFF NUMBERSWhen portion of the dropped begins with 0, 1,2,3,4 or less than 5, the last digit to be retained is unchanged.Example:34, 214.01834, 214rounded off to the nearest ones34, 210rounded off to the nearest tens34, 200rounded off to the nearest hundreds

When the portion to be dropped begins with 6,7,8,9, or a digit greater than 5, the last digit to be retained in increased by 1.Example:P98. 726P 98.73rounded off to the nearest centavoP 99rounded off to the nearest pesosP 100rounded off to the nearest ten pesosWhen the portion to be dropped is exactly 5, 50, 500 .5 etc. or exactly half of the preceding place value, we round it off to the nearest even number.Example:P 47.235P 47.24rounded off to the nearest centavosP 25,000P 20,000rounded off to the nearest ten thousands (half of the preceding place value)

The digits dropped in the whole number are replaced by zero or zeros. Using the examples above.Examples:34, 214.01834, 210rounded off to the nearest ones34, 200rounded off to the nearest hundreds34, 000rounded off to the nearest thousands

SIMPLE AVERAGES

To get the simple average, we get the sum of all given values and divide the sum by the number of values.Example: Find the average of the following grades: 85%, 78%, 87%, 80%, 83%, 90%.Solution:85%78%503 / 6 = 83.5%87%80%83%90%_ 503%

WEIGHTED AVERAGE

To get the weighted average, we multiply the quantities by the measures involved. Then we divide the sum of the products by the sum of the quantities.Example: A sewer made 10 bags. On 3 bags, she spent 2 hours each; on 4 bags, 3 hours each; and on 3 bags, 1 hour each. What was the average time spent on each bag?Solution:3 bags x 2 hours= 6 hours4 bags x 3 hours= 12 hours3 bags x 1 hour= 3 hours10 bags= 21 hours21 hours / 10 bags = 2.1 hours or 2 hours and 6 minutes.

FRACTIONS

A fraction is one or more of the equal parts into which a whole is divided. The terms of the fraction are numerator and denominator. The numerator is the number above the line showing how many of the equal parts are expressed or taken. The denominator is the number below the line showing into how many equal parts the whole is divided. The line between the numerator and the denominator is the vinculum which means divided by.

KINDS OF FRACTIONS

Proper fractionImproper fractionMixed numberSimilar fractionsDissimilar fractionsDecimal fraction

OTHER TERMS

Least common denominator (LCD)Lowest-terms fractionReciprocal of a fraction

LAWS OF FRACTIONS

The value of a fraction does not change if its terms are multiplied by the same number except zero.Example: x 3/3 = 3/6

The value of a fraction does not change if its terms are divided by the same number except zero.Example:3/9 / 3/3= 1/3

CONVERSION OF FRACTIONSImproper fraction to a whole or mixed numbers. To change the improper fraction to a whole or mixed number, we divide the numerator of the fraction by its denominator. The remainder becomes the numerator and the divisor the denominator of the fraction.Mixed number to an improper fraction. To change a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction. Then, we add the product to the numerator of the fraction over the denominator.Lower terms fraction to higher terms fraction. To raise a fraction from lower to higher terms, we multiply both numerator and denominator by the number that will result to the specified denominator of bigger value.Higher terms fraction to lowest terms fraction. To reduce a fraction from higher to lowest terms, we divide both the numerator and denominator by the greatest common factor.Dissimilar fractions to similar fractions. To change dissimilar fractions we find the LCD and divide it be each denominator. Then we multiply each quotient by the numerator and place it over the LCD

ADDITION OF FRACTIONSIn adding similar fractions, we add the numerator and write the sum over the common denominator.In adding dissimilar fractions, we make the fractions similar before addition.In adding mixed numbers, we add the whole numbers first then combine the two. If the whole numbers are not very large we can change the mixed numbers to improper fractions; then we add.

SUBTRACTION OF FRACTIONS

In subtracting similar fractions, we change the fraction to similar fractions first before we subtract. In subtracting dissimilar fractions, we change the fraction to similar fractions first before we subtract. In subtracting mixed numbers, we subtract the whole numbers first and then the fractions. We can also subtract them by changing the mixed numbers to improper fractions. This will be difficult if the whole numbers are very large.

When the value of the fraction in the subtrahend is bigger than the value in the minuend, we borrow 1 unit from the whole number of the minuend, change it to similar fractions and add it to the fraction. Then we proceed to subtraction.If we change both mixed numbers to improper fractions, we can proceed to subtraction at once without borrowing.When the fractions in the mixed numbers are dissimilar, we change them first to similar fractions; then we subtract.

MULTIPLICATION OF FRACTIONSIn multiplying fractions, we multiply the numerators together to form a new numerator. Then we multiply the denominators to form new denominator.

FRACTION AND A WHOLE NUMBER

In multiplying a fraction and a whole number, we multiply the whole number by the numerator of the fraction and write the product over the denominator of the fraction.

FRACTION AND MIXED NUMBER

In multiplying a fraction and a mixed number, we change the mixed number to an improper fraction and then we proceed to multiplication.

A MIXED NUMBER AND A WHOLE NUMBERIn multiplying a mixed numbers, we change the mixed numbers to improper fractions; then, we proceed to multiplication.

MIXED NUMBER AND MIXED NUMBER

In multiplying mixed numbers, we change the mixed numbers to improper fractions; then we proceed to multiplication.

CANCELLATION

In canceling factors, we divide a pair of numerator and denominator by the same number just as when we reduce a fraction to lowest terms.

DIVISION OF FRACTIONSIn dividing fractions, we invert the divisor and multiply. All rules pertaining to multiplication of fractions will be applied.

RATIO A ratio is a relation between two like numbers or quantities expressed as a quotient or fraction.We can also reduce ratios to their lowest terms in the same manner that fractions are reduced.

PROPORTIONSA proportion is a statement that two ratios are equal. There are four terms in a proportion: the first and fourth terms are called extremes; the second and the third are known as the means. Hence, the ratio 6:24 which is equal to the ratio 1:4 can be expressed as a proportion 6: 24 = 1:4, 6 and 4 are the extremes while 24 and 1 are means. To check whether our proportion is correct, the rule is: the product of the means equals the product of the extremes.

CONVERSION TECHNIQUES

To reduce a decimal to a common fraction, we write the given decimal number disregarding the decimal point as the numerator of a common fraction with a denominator of the power of 10 of the given decimal.Examples:.7 = 7/10There is one decimal place so the denominator is 10.To reduce a common fraction to a decimal, we divide the numerator by the denominator.Example: = 0.5

To change percent to a decimal, we move the decimal point two places to the left and drop the percent sign. If the percent is in fractional units, we change first the fraction to decimal before moving the decimal point.Example:40% = .40To change a decimal to a percent, we move the decimal point two places to the right and add the percent sign.Example:.23 = 23%

To change a percent to a fraction, we drop the percent sign and replace it by 100 as denominator. If the percent is in decimal, we move the decimal point two places to the left after dropping the percent sign. Then we convert the decimal to its fractional equivalent. If the percent is in fraction, divide it by 100 and drop the percent sign. Example:27% = 27/100

To change a fraction to a percent, we divide the numerator by the denominator, then we move the decimal point of the quotient two places to the right and add the percent sign. For a mixed number, we change it first to an improper fraction before performing the indicated division.Example:3/5 = .60 = 60%

PERCENTAGE PROBLEMSA percentage is the result obtained by taking a certain percent of a number. Percentage is equal to the base times the rate. The base is the number on which the percentage is computed. The rate is the number indicating how many percent or hundredths are taken.

PERCENTAGE FORMULASP = R X BR = P/B B = P/RThe base (B) is usually preceded by the preposition of in word problems. Of indicates multiplication. The word is is symbolized by the equal sign (=). Other words may be used instead of as as many as , as large as, as great as, as much as.

PERCENTAGE FORMULAS cont..The rate ( R ) is identifiable because it is usually in the form of a percent. However, it can also be in decimal, or in a fraction.The percentage (P) refers to the equal quantity or number of items represented by the rate.

PERCENTAGE FORMULAS cont..Sample Problems:What number is 5% of 480P=R x BP=.05 x 480P=24

PERCENTAGE FORMULAS cont..Percentage is unknown. 5% is changed to a decimal (.05%) before multiplying it to 480. Another way is to change 5% to fraction ( 5 /100 or 1/20) before multiplying it to 480 and we arrive at the same product.

PERCENTAGE FORMULAS cont..P 26 is what part of P 130?P=R x BR=P / BR=26 / 130R=1 / 5

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASENote: We dont multiply or divide a number by percent. We always change the percent to a decimal or a fraction first before multiplying or dividing.

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE ContTo find the percentage of increase or decrease, we multiply the base by the rate and add the product to the base if it is an increase but subtract the product from the base if it is a decrease. B + ( B X R ) = PERCENTAGE OF INCREASE B ( B X R ) = PERCENTAGE OF DECREASE

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE ContExample: P 120 increased by 12% is how much?B R= P of increaseP 120 + ( P 120 x .12) = P 134.40

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE ContTo find the rate of increase or decrease, get the difference between the two given related values and divide it by the base or original quantity. Change the fraction to percent if it is needed.

LARGER VALUE SMALLER VALUE = RATE OF INCREASE OR DECREASEBASE OR ORIGINAL QUANTITY

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE ContExample: Find what percent, more than 25 is 30?30 - 25 = 5 = 20% 2525To determine the base when a number that is a fractional part or percent greater than or smaller than that of the unknown value, we divide the given number or percentage by the sum (if greater than) or the difference (if smaller than) between 1 and the given fraction or 100% and the given rate.

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE ContP = BASE OF INCREASE1 + FRACTIONP = BASE OF INCREASE1 - FRACTIONP = BASE OF INCREASE100% + Given %P = BASE OF INCREASE100% - Given %

Example: 2/5 greater than amount is P 70?

P 70= P 70= P 70 = P 70 x 5/7 = P 50 1 + 2/5 5/5 + 2/5 7/5

PROBLEMS INVOLVING PERCENTAGE INCREASE OR DECREASE Cont

DIVISIBILITY RULESRule 1. Divisibility by 0 or 1Any number is divisible by 1. Any number divided by 1 is the number itself. 0 divided by any number is 0. Any number divided by 0 is undefined.

DIVISIBILITY RULESRule 2. Divisibility by 2.A number is divisible by 2 if, and only if, the last digit is even, that is, if it is 0,2,4,6,8,.

Rule 3. Divisibility by 3.A number is divisible by 3 and if only if its digital root (sum of the digits) is divisible by 3.

DIVISIBILITY RULESRule 4. Divisibility by 4A number is divisible by 4 if, and if only of, the last two digits form a number which is divisible by 4.

Rule 5. Divisibility by 5.A number that ends in 0 or 5 is divisible by 5.

DIVISIBILITY RULESRule 6. Divisible by 6.A number that is divisible both by 2 and 3 is divisible by 6.

DIVISIBILITY RULESRule 7. Divisibility by 7To test if the number is divisible by 7 follow these steps.For three or more than three digit number.Multiply the last digit (unit digits) by 2.Get the difference of these product ( from step 1) and the remaining number (without the units digits)Repeat step 1 and 2 until you reached a two-digit difference. If this difference is divisible by 7, then the number is divisible by 7.

DIVISIBILITY RULESRule 7. Divisibility by 7If the number consist of more than 3 digits you can use these alternative rule.Divide the number into periods of three digits each beginning with the last digits.Add together the odd numbered periods, and then all the even numbered periods.Take the difference of the two sums. If this difference is divisible by 7, then the number is divisible by 7.

DIVISIBILITY RULESRule 8. Divisibility by 8A number is divisible by 8 if and if only if the last three digits form a number that is divisible by 8.

DIVISIBILITY RULESRule 9. Divisibility by 9.A number is divisible by 9 if, and only if the digital root ( sum of the digits) is 0 or 9. When the digital root of a number is not 0 or 9, that digital root is the remainder when the number is divided by 9.

DIVISIBILITY RULESRule 10. Divisibility by 10.A number ending in 0 is always divisible by 10.

Rule 11. Divisibility by 11.To test if the number is divisible by 11, alternatively subtract and add the digits from the last digits to the first, that is from right to left. If the end result is divisible by 11, then the number is divisible by 11, otherwise the end result is the remainder when the number is divisible by 11. OR

DIVISIBILITY RULES

DIVISIBILITY RULESContA number is divisible by 11 if the difference (large minus smaller) between the sum of the odd-numbered digits and the sum of the even numbered digits is divisible by 11.

DIVISIBILITY RULESRule 12. Divisibility by 12.A number is divisible by 12 if, and only if, it is divisible by 3 and 4.

DIVISIBILITY RULESRule 13. Divisibility by 11,7 and 13To test if the number with more than three digits is divisible by 7, 11 and 13, follow the step by step procedure:Starting from the right end of the number, divide the number into periods of three digits.

DIVISIBILITY RULESRule 13. Divisibility by 11,7 and 13 ContAdd the odd numbered periods together and the even numbered periods together.Find the difference of the two sums. If that difference is divisible by 7 or 11 or 13, then the number is divisible by whichever divisor that divides it evenly.

DIVISIBILITY RULESRule 14. Divisibility by 25; 75 and 125A number is divisible by 25 if, and only if, the last two digits form a number divisible by 25. A number divisible by 75 if it is divisible by 3 and 25. A number is divisible by 125 have the last three digits divisible by 125.

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITEAny counting number a can be expressed as the product of b and c, which are both counting numbers.That is, a = b x cThen a is a multiple of b and of c.Consider 42 and 68. Then,42 = 6 x 7= 2 x 21= 3 x 14= 1 x 42

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITEThus, 42 is a multiple of 1, 2, 3, 6, 7, 14, 21 and 42.68= 2 x 34= 4 x 17= 1 x 68

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITEWhat are the factors of 42 and 68?We mentioned previously that 1 and the number itself are factors of the given number. This time we will be considering the kinds of numbers according to their factors.Consider the following counting numbers:43 1081 1297

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITELet us list the factors of the above counting numbers by expressing them as the product of two counting numbers.43 =43 x 110=2 x 5 = 10 x 181= 9 x 9 = 27 x 3 = 81 x 112=4 x 3 = 2 x 6 = 12 x 1 97=97 x 1

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITEWe see that 43 and 97 have factors 1 and themselves. While 10, 81, and 12 have different factors other than 1 and themselves. Thus we come up with the following definitions:if a counting number has 1 and itself as its only factors, then the counting number is called a prime number.If a counting number has factors other than 1 and itself, then the counting number is called a composite number.

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITEExample: Factors the following number: 13, 9, 2, and 21.13=13 x 19=3 x 3 = 9 x 12=2 x 121= 3 x 7 = 21 x 1

FACTORS, MULTIPLIES, GCF, LCM, PRIME OR COMPOSITENote that 13 and 2 are prime numbers, while 9 and 21 are composite numbers.A prime number is a number greater than one, which has only two factors, one and itself.The prime numbers less than 50 are 2, 3,5,7,11,13,17,19,23,29,31,37,41,43 and 47.

Composite Numbers and Prime FactorsA whole number which is not prime, and is greater than 1 is called a composite number. It can be expressed as a product of two smaller factors. Examples are 4,6,8,10.The number 0 and 1 are special numbers which are neither prime nor composite.

Greatest Common Factor (gcf) or greatest common divisor (gcd)of two or more whole number is the greatest whole number that is a factor or a divisor of the given numbers. This is less than or equal to the smallest number given. If the greatest common factor of the given number is 1, then the numbers are said to be relatively prime.

Greatest Common Factor (gcf) or greatest common divisor (gcd) contConsider all the factors of 12 and 36.12 = 1, 2, 3, 4, 6, 1236 = 1, 2, 3, 4, 6, 9, 12, 18, 36The factors of 12 and which are also the factors of 36 are called common factors. So 1, 2, 3, 4, 6 and 12 are the common factors of 12 and 36The largest common factor, 12, is called the Greatest Common Factor or GCF

Greatest Common Factor (gcf) or greatest common divisor (gcd) contRemarks:The number 1 is neither prime or composite.The number 2 is the smallest prime number, and it is the only even prime number.

Greatest Common Factor (gcf) or greatest common divisor (gcd) contThe first ten prime number are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Since we can always determine more prime numbers (although this may be quite a chore), we say there are an infinite number of primes.Prime and composite are terms that relate only to counting numbers.

There are couple of ways to determine the prime factorization of a number. One of these methods is described as follows:Method 1: Factor TreeFind two numbers whose product is the given number.if each of these factors is prime, then you have found the prime factors of the given number.If the factors in step 2 are not prime numbers, find two numbers whose product is equal to the factors in step 2.Repeat step 3 until all the factors are prime numbers.Greatest Common Factor (gcf) or greatest common divisor (gcd) cont

Example: Write the prime factorization of 24.Solution:Find the numbers that when multiplied, give 24. Two possible numbers are 4 and 6.Since neither factor is prime, we continue with step 3Two numbers that give a product of 4 are 2 and 2. Go to step 2.

Greatest Common Factor (gcf) or greatest common divisor (gcd) cont

Method 2: DecompositionRepeatedly divide the given number by prime numbers like 2,3,5, and so on, until a quotient of 1 is obtained. All remainders obtained in these division must be 0.Write the prime factorization by using all the divisors from step 1 as factors.

Greatest Common Factor (gcf) or greatest common divisor (gcd) cont

GCFIf c is the greatest whole number that can divide a and b, such that the quotient is again a whole number, then c is the greatest common factor (GCF) of a and b. We denote GCF of a and b as (a, b) = c.

Greatest Common Factor (gcf) or greatest common divisor (gcd) cont

LCMThe least common multiple (LCM) of any two or more whole numbers is the smallest whole that they can devide such that the quotient is again a whole number. We denote the LCM of a and b as (a,b) = c.

Greatest Common Factor (gcf) or greatest common divisor (gcd) cont

TOPICS ON ANGLES AND LINESDefinitionBetweeness: Let A, B, and C be the three points on a line, B is between A and C if aa. B is also between A and C if /AB/ +/ BC/ = /AC/.Line segment. The line segment determined by the points P and Q is the set consisting of P and Q and all the points between them. The length of the line segment is the distance between points P and Q.

TOPICS ON ANGLES AND LINES ContRay PQ. This is the union of the line segment PQ and all the points on the line PQ, such that Q is between P and these points. Point P is the endpoint of the ray.Ray opposite ray PQ: This is the set consisting of the point P and all the points of the line PQ which do not belong to ray PQ. Opposite rays is always collinear.

Collinear: Two or more points are collinear if and only if there is a line containing all these points.Coplanar: Two or more points are coplanar if and only if there is a plane containing these points.Midpoint: R is the midpoint of line segment PQ if R is on line segment PQ and PR = RQTOPICS ON ANGLES AND LINES Cont

TOPICS ON ANGLES AND LINES ContAngle: This is a set of points in the union of two noncollinear rays which intersect at their endpoints. The rays are the sides of the angle and the common endpoint is the vertex.Right,Acute and Obtuse angles: An angle is right if it measures 90 degrees. It is acute if its measure is less than 90. It is obtuse if its measure is more than 90.

TOPICS ON ANGLES AND LINES ContVertical angles: These angles are vertical when their union forms two intersecting lines.Complementary angles: Two angles having 90 as the sum of their measures.Supplementary angles: Two angles having 180 as the sum of their measures.

Adjacent angles: Two angles which are coplanar, have common vertex and a common side, and the intersection of their interiors is empty.Congruent angles: Angles with the same measures.Congruent line segments: Segments with the same length.

TOPICS ON ANGLES AND LINES Cont

Perpendicular: A line, ray, or segment is perpendicular to another line, ray or segment if they intersect and form a right angle. The point of intersection is called the foot of the perpendicular.Parallel lines: Two lines which are coplanar and do not intersect.

TOPICS ON ANGLES AND LINES Cont

Skew lines: Two lines which do not lie on one plane.Transversals: This is a line that intersects two other at different points. Note: Different kinds of angle formed: alternate interior angles, alternate exterior angles, and corresponding angles.Concurrent lines: Three or more lines that meet at point.TOPICS ON ANGLES AND LINES Cont

Angle Included between two sides: This is the angle formed by the two sides.Right triangle: A triangle with one right angle. The sides that are perpendicular (sides forming the right angle) are called the LEGS, and the side opposite the right angle is hypotenuse. The hypotenuse is longer than any of the legs of the right triangle.Topics on Triangle

Obtuse triangle: A triangle with one obtuse angle.Acute triangle: A triangle which has three acute angles.Isosceles triangle: A triangle with two sides congruent. The congruent sides are called the LEGS and the third side, the BASE. The VERTEX angle is the angle formed by the legs; the two remaining angles are the Base angles.Topics on Triangle

Scalene triangle: Triangle having no sides congruent.Equilateral triangle: A triangle with all sides congruent.Equiangular triangle: A triangle with three congruent angles. The measure of each is 60 degrees.Topics on Triangle

Altitude of a triangle: This is a line segment drawn from the vertex of the triangle perpendicular to the line containing the opposite side.Median of the triangle: This is a line segment from the vertex to the midpoint of the opposite side.

Topics on Triangle

Triangle:A = b hP = a + b + c

Square:A = s2P = 4sFORMULAS for Areas and Perimeters of geometrical figures:

Rectangle:A = L x wP = 2L + 2w

Trapezoid:A = h (b1 + b2)FORMULAS for Areas and Perimeters of geometrical figures:

Rhombus:A = d1d2

Circle:A = r2C = 2 r

Area of equilateral triangleA = s2/4FORMULAS for Areas and Perimeters of geometrical figures:

Algebra Basic Rules:Commutative Property of additiona + b = b + aCommutative Property of Multiplicationab = baAssociative Property of Addition(a + b) + c = a + ( b + c)Associative Property of Multiplication( ab) c = a ( bc )

Distributive propertya ( b + c ) = ab + bc( a + b) c = ac + bcAdditive Identity Propertya + 0 = 0 + a = aMultiplicative Identity Propertya .1 = 1 . a = 1

Algebra Basic Rules:

Additive Inverse Propertya + (-a) = 0Multiplicative Inverse Propertya . 1/a = 1, a 0

Algebra Basic Rules:

Properties of Exponentsxm . x n = xm+n( xy)n = xnyn(xm)n = xmnxm = xm n ; m > n ym(x/y)n = xnynx0 = 1x-n = 1/ xn ; x 0

INTRODUCTION to PROBABILITY THEORYTechniques in Counting:The Fundamental Principle. If an event E1 can happen in n1 number of ways and another event E2 can happen in n2 number of ways, then the number of ways both events can happen in the specified order = n1 n2 ways.Permutation is an arrangement or objects wherein order is taken into account.

Permutation of objects taken all at a time.n P n = n! Permutation of n objects taken r at a time.n P r = Circular Permutation ( one position must be fixed)( n-1) P (n-1) = (n 1) !

INTRODUCTION to PROBABILITY THEORY

Permutation of n objects not all distinctINTRODUCTION to PROBABILITY THEORYCombination is a selection of objects with no attention given to the order of the objects.

Combination of n objects taken all at the same timen C n= 1

Combination of n objects taken r at a time.

n C r =INTRODUCTION to PROBABILITY THEORY

Combination in a seriesn C 1 + n C 2 + n C 3 + n C n = 2n 1Classical Definition of ProbabilityThe probability of the occurrence of an event (called success) isPr (E) = Where n = no. of successesN =total number of possible outcomesINTRODUCTION to PROBABILITY THEORY

Scientific Notation A number is expressed in scientific notation when it is written in the form a x 10nwhere 1 is less than/equal to a < 10 and n is an integer.

RADICAL EXPRESSIONSIf n is a positive integer greater than 1, a and b are real numbers such thatxn = a, then x is the nth root of a.Let n be a positive integer greater than 1, and let a be a real number such that is denoted in real numbers. Then, a1/n =

RadicalsIf n is a positive integer greater than 1, and a and b are real numbers such that bn =a, then b is an nth root of a.Example:7 is a square root of 49 because 72 = 49; furthermore-7 is also a square root of 49 because (-7)2 = 49

RadicalsLet n be a positive integer greater than 1, and a be a real number. The principal nth root of a is denoted by , and has the following defining properties:If a then is the positive nth root of a.If a < 0 and n is odd, then is the negative nth root of a , and = 0If a and n is even, then has no meaning in real numbers.

Radicals

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