MATHEMATiCSA REVIEWER IN MATHEMATICS

Created for by Mr. Boyet Bandoja Aluan

Study Guide to accompany

Mathematics THE SCIENCE OF NUMBERS

To the readerMathematics is an incredible exciting subject,

but in order to appreciate new discoveries , discussions and skills it is necessary to have grasp of the underlying concepts and ideas. This work is designed to give you a comprehensive overview of important mathematical phenomena. It is also serves as your references in your future studies.

This is designed to supplement , not to replace, your text and your instructor/reviewer. Important concepts and ideas have been synthesized into short, easy-to-read segments that provide an overview of different topics. This will also serve as your reviewing tools.

Strategies for studying MathematicsEach individual has his her own unique study pattern. However, the author believe

that there are some successful strategies that are universal and here they are:

1. Preview a topic in this material– it is important to note the organization of the topics and how they are interrelated.

2. Reading a text is an active process. We recommend that you always have a pencil and pare available. Jotting note is an excellent mental trigger when it comes to review.

3. As you read, take time to review all figures , tables and formulae.4. Once you read the material, summarize it in your own words, go

back to the topic/concepts again and scan it.5. Finally when you are comfortable with the topics or chapter move

to the question part of this material. Work through all the question before checking your answers. Any question that you cannot answer or understand should be flagged as material you need to reviewed. Remember that your goal here is to understand the materials not to answer particular question.

6. Learn from your mistakes7. Get help before its too late

Mathematicsa way of describing relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. It is truly the language of science

Branches of Mathematics

A. ArithmeticB. Algebra

C. GeometryD. Trigonometry

E. CalculusF. Probability and Statistics

G. Set Theory and LogicH. Number TheoryI. Systems Analysis

J. Chaos Theory

AAGTCPSNSC

Arithmetic, one of the oldest branches of mathematics, arises from the most fundamental of mathematical operations: counting. The arithmetic operations—addition, subtraction, multiplication, division, and placeholding—form the basis of the mathematics that we use regularly. In many countries arithmetic is the primary area of mathematical study during the first six years of school.

Algebra is the branch of mathematics that uses symbols to

represent arithmetic operations. One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations. For example, we can represent the numbers 2 and 3 by the symbols x and y. From observation we know that it does not matter in which order we add the numbers (2 + 3 = 3 + 2), and we can represent this equivalence as the equation x + y = y + x. The equation is valid no matter what numbers x and y represent. Because algebra uses symbols rather than numbers, it can produce general rules that apply to all numbers. What most people commonly think of as algebra involves the manipulation of equations and the solving of equations.

Geometry

 is the branch of mathematics that deals with the properties of space. Students in high school study plane geometry—the geometry of flat surfaces—and may move on to solid geometry, the geometry of three-dimensional solids. But geometry has many more fields, including the study of spaces with four or more dimensions.

Astronomy was one of the earliest sciences to implement the ideas of geometry. Astronomers built mechanical devices consisting of gears and fixed spheres that described the orbits of celestial bodies with astonishing accuracy. German mathematician Johannes Kepler used geometry in the late 16th and early 17th centuries to argue that the universe was not Earth-centered and to prove that planets revolved around the Sun in elliptical orbits.

 TrigonometryThe study of triangles in plane geometry led to

trigonometry. Originally trigonometry was concerned with the measurement of angles and the determination of three parts or a triangle (sides or angles) when the remaining three parts were known. If we know two angles and the length of one side of a triangle, for example, we can compute the other angle and the length of the remaining sides. Trigonometry uses triangles because all shapes in plane geometry can be broken down into triangles.

Calculus is the branch of mathematics concerned with

the study of rates of change, slopes of curves at given points, areas and volumes bounded by curves, and similar problems. Scientists apply calculus to numerous problems in physics, astronomy, mathematics, and engineering. In recent years calculus has also been applied to problems in business, the biological sciences, and the social sciences. The development of calculus in the 17th century made possible the solution of many problems that had been insoluble by the methods of arithmetic, algebra, and geometry.

Probability and Statistics

Probability and statistics deal with events or experiments where outcomes are uncertain, and they assess the likelihood of possible outcomes. Probability began in an effort to assess outcomes in gambling.

Set theory Set theory is the branch of mathematics

that seeks to establish statements that are true of sets, regardless of the kind of objects that make up the set. A set is a group of objects with a well-defined criterion for membership so that we can say definitely whether an object belongs to the set or not. The terminology and many of the results of set theory are used in symbolic logic, geometry, the theory of probability, and mathematical analysis.

Number theory Number theory is the branch of mathematics that deals with the properties of numbers, primarily integers—whole numbers that may be positive, negative, or zero. One of the earliest problems studied in algebra was the division of integers: Is it possible to write an integer as the product of smaller integers? The integer 6, for example, can be written as 2 x 3. If an integer can be written in this way, it is called a composite number; if not, it is called a prime number. The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, and 23. There are an infinite number of prime numbers.

Systems Analysis

The mathematical study of systems is called systems analysis. It plays a vital role in the understanding of communications networks and computing networks.

Chaos theory

Chaos theory, a relatively new area of mathematics, concerns the analysis of unpredictable systems that are extremely sensitive to initial conditions. One important example of a chaotic system is climate. Global climate modeling is an area of mathematical research that seeks to develop models for predicting the weather, given accurate data from weather satellites orbiting Earth.

Let us start with numbersReal Numbers

-7, - Ã-1/4, 0,1,2/3, pi, 4

Irrational Numbers-Ã, pi, square root of 15

Rational Numbers-7, -1/4, 0, 1, 2/3, 4

Non-integers-1/4, 2/3

Integers…-3, -2, -1, 0, 1, 2, 3 …

Whole Numbers0, 1, 2 , 3, …

Negative Integers-…-3, -2, -1

Natural Numbers1, 2, 3,…..

Zero0

Properties of integersFor addition and subtraction

1. Commutative property-----b + e = b + e(except for subtraction)2. Closure b + e = integer3. Associative property--------(b+e)+y= b+ (e+y)4. Identity --b+ 0 =b5. Inverse --- --b+ (-b) = 0 ; -b is also called additive

inverse 6. Distributive --b(e+y) = be + by

For multiplication and division7. Closure --by = integer 8. Commutative -- by = yb9. Associative --(by)e = b(ye)10. Identity ---b x 1 = b11. Inverse ----b(1/b) = 1 ; 1/b called multiplicative

inverse12. Distributive ----b(e+y) = be + by13. Multiplication property zero -- b(0) = 0See illustrations

Test your self

Properties of equalities of integersconsidering b, r and w as integers or real numbers

1. Reflexive property b = b2. Symmetric property if b= r, then r

= b3. Transitive property if b = r, and r = w,

then b= w4. Substitution property if b = r, then b can be replaced by r in any

expression involving b

5. Addition/subtraction property if b= r, then b+w = r+w; b-c = r- c

6. Multiplication/ division property if b = r, then bw = rw ; b/w = r/w

7. Cancellation property if b+w = r +w, then b = r ; if bw = rw and w not= 0 then b=r

Properties of zero

1. b+ 0 = b and b – 0 = b2. b(0)= 03. 0/b = 0 with b not = 04. b/0 is undefined as stated on fraction (

see types of fraction)5. If br =0, then b =0 or r= 0. this is

known as Zero Factor property

Four fundamental Operations

Addition—putting together e.g. 1+1 = 2Subtraction– getting the difference between minuend and subtrahend e.g. 15-6 =9.Multiplication– inverse of division and repeated addition e.g. 2*5 = 10 as 2+2+2+2+2Division– inverse of multiplication or repeated subtraction e.g. 15/3 =5.

Signed Numbers

It is the used of either positive(+) or negative(-) signed before the numbers.Examples -3, -18, -50…. And for positive are +1, +56, +100 …..

Operation on signed numbers1. Addition– to add numbers having the same sign, add their absolute

values and prefix the common sign.

1. +2+28 = +30, -8+-2 =-10, +56+-78 =-222. Subtraction– to subtract one signed number from the others, just

changed the sign of the subtrahend then proceed to algebraic addition.

3. Multiplication and division—two cases a. like sign and b. Unlike sign

+56 +56- - + 3 - 3 +53

operation

minuend

subtrahend

(+6)*(+5) =+30(-6)*(-5) = +30(+30)/(+5) = +6(-30)/(-5) = +6

(-6)*(+5) = -30(-30)/(+5) = -6 Case a, always

positiveCase b, always negative

Fraction A number in the form a/b here a and b are integers and b not equals to zero a is a numerator and b is the denominator

e.g. 1/3, 2,6, 10/5, - 8/9…

Types of fractiona. Simple fraction– a fraction in which the numerator and denominator are both

integers, and also called as common fraction e..g 2/3, -6/7.b. Proper fraction-- s one where the numerator is smaller than denominator e.g.

½, 2/3.c. Improper fraction—is one where numerator is greater than denominator e.g.

5/2, 10/2.d. Unit fraction– fraction with unity for its numerator and positive integer for its

denominator e.g. ½, 1/8,…e. Simplified fraction– fraction whose numerator and denominator are integers

and their greatest common factor is 1 e.g 1/5, 8/11, f. Integer represented as a fraction-- Fraction whose denominator is one e.g. 5/1,

2/1…g. Reciprocal– a fraction result from interchanging the numerator and

denominator e.g 4 become ¼, 2/5 become 5/2.h. Complex fraction– fraction in which the numerator or denominator or both are

fractions e.g. 3/4/7/8,-8/1-14/5..i. Similar fractions- two or more fraction whose denominator are equal e.g. 1/8

and 5/8..j. Zero fraction– a fraction whose numerator is zero and also equals to zero e.g.

0/5, 0/9…k. Undefined fraction– a fraction whose denominator is zero. Note nothing can be

divided by zero e.g. 8/0, 10/0l. Indeterminate fraction– fraction which has no quantity or meaning 0/0m. Mixed numbers– number that is a combination of an integer and a proper

fraction. e.g. 5/1/2, 9/8/11.

1. Simplification of fraction

If numerator and denominator are relatively prime or in lowest terms. To change fraction to lowest term cancel the GCF of both.

a. 15/25 = {5(3)}/{5(5)} cancel 5-the gcf. The result will be 3/5.

1. To multiply two fraction, multiply their numerators to numerators to get new numerators and denominator to denominator of the other fraction to get new denominator. a. Example a/b *c/d = ac/bd

2. To divide fraction, get the reciprocal of the divisor to proceed multiplication process a. Example a/b ∕c/d = a/b * d/c = ad/bc

2. Multiplication/ division of fraction

In multiplication “of” means to multiply

3. Like and unlike fraction

a. Like fractions have he same denominator, examples are 33/5, 9/5…

b. Unlike fractions have unlike denominator examples are 3/8, 2/5…

4. Changing unlike to like fractionsDetermine the least common denominator and change each to an equivalent fraction whose denominator is the LCD.Example.

¾, 2/5, 1/3¾ = ¾*5/5*3*3 = 45/602/5 = 2/5*3/3*4/4 = 24/601/3 = 1/3 * 4/4* 5*5 = 20/60

6.Changing mixed to improper fraction

a/b+c/d = (ad+bc)/bd

5. Addition/ subtraction of fraction

To change mixed to improper, multiply denominator then add to numerator, the answer will be the new numerator. The just copy the denominator. Do it reverse for changing improper to mixed number. Or by dividing numerator to its denominator.

Decimal another form of fraction wit to based on 10.

Decimal fraction read asa. 3.1 as three and one tenthsb. 3.10 as three and ten hundredthsc. 3.103 as three and one hundred three

thousandthsd. 3.2156 as three and two thousand fifty-six ten-

thousandths.

As you may notice whole numbers is read according to their proper place value and so with decimal, which was according to their places. First read as normal then add the place value of the last digit of the decimal.

NOTE THAT HE PLACE VALUE OF THE DECIMAL PLACES STARTS AT TENTHS AND SO ON.

Types of decimala. Repeating example 4/9 = 0.4444444444…..b. Terminating example 25/100 = 0.25c. Non-terminating example 0.212121212121…….

Conversionsa. Fraction to decimal divide the numerator by the denominator b. Decimal to fraction multiply the given number by a fraction

equal to one whose numerator and denominator is a power of 10, the power of which equals the number of the decimal places of the given number. Example .25 = 25/100 = 1/4

c. Percent to decimal percent means per hundred thus 50% = 50/100 = .50

d. Decimal to percent reverse the process above. Move the decimal points two places to the right. Example 0.357 = 35.7%

e. Fraction to percent change first to decimal then decimal to percent. Example ½ = .5 = 50%

f. Percent to fraction drop the percent sign, divide the given number by 100 and simplify. Example 30% = 30/100 = 3/10

What is a difference between prime and composite numbers?

Answer: prime numbers are integers greater than 1 that is divisible only by 1 and itself, while Composite are positive integers that have more than two factor. Note that 1 is the only natural numbers that is neither prime nor composite.

5

1 5

10

2 5

1 5

Prime no. Composite no.

Factors of 5.

Factors of 10.

Second ,Factors of 5.

Prime numbers between 1 and 1000

All the Prime Numbers between 1 and 1,000

2 3 5 7 11 13 17 19 23 29

31 37 41 43 47 53 59 61 67 71

73 79 83 89 97 101 103 107 109 113

127 131 137 139 149 151 157 163 167 173

179 181 191 193 197 199 211 223 227 229

233 239 241 251 257 263 269 271 277 281

283 293 307 311 313 317 331 337 347 349

353 359 367 373 379 383 389 397 401 409

419 421 431 433 439 443 449 457 461 463

467 479 487 491 499 503 509 521 523 541

547 557 563 569 571 577 587 593 599 601

607 613 617 619 631 641 643 647 653 659

661 673 677 683 691 701 709 719 727 733

739 743 751 757 761 769 773 787 797 809

811 821 823 827 829 839 853 857 859 863

877 881 883 887 907 911 919 929 937 941

947 953 967 971 977 983 991 997

Prime Numbers Between 1 and 1,000

Note: The number two(2) is the only even prime numbers

Note: THE SUM OF PRIME NOS FROM 1-50= 15, 1-100= 25, 1-150= 35, 1-200= 46, 1-1000= 168

Types of prime numbersa. Twin primes– set of two consecutive odd primes, which differ by

two e.g. (3,5)(5,7)(11,13)(17,19).b. Symmetric primes– pair of prime numbers that are the same

distance from given number in a number line also called Euler primes e.g

c. Emirp– prime numbers that remains primes when digits are reversed. E.g 11, 13, 17, 31, 71…

d. Relatively prime numbers– number whose greatest common factor is 1.

e. Unique product of power of prime– is a number whose factors are prime numbers raised to a certain power. ex. 350 eqls 2 cube*32*51

Number symmetric1 none2 none3 none4 3,55 3,76 5,77 3,118 5,113,139 7,11;5,1310 7,13;3,1711 5,17;3,191211,13;7,15;5,19

Perfect numbers–

is an integer that is equal to sum of all its possible divisors, except the number itself. Example: 6(the factor are 1,2,and 3),28,496. The preceding can be computed using the formula.

Exponent

an exponent is a number that gives the power to which a base is raised see example

Radical

Refers to the symbol that indicates a root , it was first used in 1525 by Cristoff Rudolff in his Die Coss. In

the expression inside the symbol n is called the index, a expression inside the symbol called radicand while symbol is called radical. We cannot present yet the properties of radicals using this format so click this link for further explanation. Properties of radicals

n

a

Special Products

Are the expressions where the values can be obtained without execution of long multiplication. With x, y, and z are real numbers the following are the special products.

1. Sum and difference of the same terms or difference of two square

=

2. Square of a binomial = = 3.Cube of a binomial = =

4. Difference and sum of two cubes

=

=

5. Square of a trinomial

=

What is proportion?Proportion is a statement that two ratios are equal

Properties of proportion:1. If a/y = x/d, then a:x = y:d2. If a/b = c/d, then a/c = b/d3. If a/b = c/d then b/a = d/c4. If a/b = c/d, then (a-b)/b = (c-d)/d.5. If a/b = c/d, then (a+b)/b= (c+d)/ d6. If a/b = c/d, then (a+b)/(a-b) =

(c+d)/(c-d).

In number 1, quantities a and d are called extremes while x and y are called means. If x = y, then is value is known as mean proportional. In the ratio x/y, the first term x is called the antecedent while the second term is called the consequent.

a : x = y : d

a : x = a / x

extremes

means

antecedent

consequent

How to find the LCDleast common denominatorRefers to the product of several prime numbers occurring in

the denominators, each taken with its greatest multiplicity

What is the least common denominator of 8, 9, 12 and 15?

8 = 2*2*2 or 2^3 LCD = (2^3)(3^2)(5)9 = 3*3 or 3^2 LCD = 36012 = 3* 2*2 or 3*2^215 = 3*5

HOW TO FIND THE LCMleast common multiple

A common multiple is a number that two other numbers will divide into evenly. The least common multiple (lcm) is a lowest multiple of two numbers

What is the least common multiple of 15 and 18?

15 = 3*518 = 3^2*2

LCM = (3^2)*5*2LCM = 90

HOW TO FIND THE GCFgreatest common factor

A factor is number that divides into a larger number evenly. The greatest common factor is the largest number that divides into two or more numbers evenly.

What is the greatest common factor of 70, and 112?

70 = 2*5*7112 = 2^4 *7

Common factor are 2 and 7GCF = 2*7GCF = 14

DIVISIBILITY RULES

A. By 2 – last digit is 0 and even. Ex 6 B. By 3– sum of the digits are divisible by 3. ex 3174C. By 4– if the last 2 digits form a number divisible by 4 ex

20024D. By 5– ends with 0 and 5 ex. 35E. by 6– number is divisible by both 2 and3 ex. 2538F. By 7– difference get by subtracting twice the last digit from

the number formed by remaining digit is divisible by 7. ex. 217

G. By 8 -- last digit form a number divisible by 8 ex. 1024H. By 9– sum of the digits is divisible by 9 ex 3127878I. By10– ends with 0 ex. 2000010J. By 11– if the difference between the sum of the digits in the

even and the sum of the digits in the odd is divisible by 11 ex. 12345674

K. By 12– if the number is both divisible by 4 and 3 ex. 215436

Percentage, base and ratepercentage (p) is the amount taken from a given number,. The base (b) is the given number from which the percentage is taken. Rate (r) is the number bearing the percent notation %

P = br, to get percentage R = P/b

To get the rateB = P/rTo get the base

Example 1. 40% of the 50 apples bought were spoiled. How many apples were spoiled