3. Mathematics 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.
4. Each 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 mistakes
7. Get help before its too late
5. a 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
6. A. Arithmetic
B. Algebra
C. Geometry
D. Trigonometry
E. Calculus
F. Probability and Statistics
G. Set Theory and Logic
H. Number Theory
I. Systems Analysis
J. Chaos Theory
7. 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.
8. 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.
9. 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.
10. The 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. The mathematical study of
systems is called systems analysis. It
plays a vital role in the
understanding of communications
networks and computing networks.
16. 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.
18. For addition and subtraction
1. Commutative property-----b + e = b + e(except for subtraction)
2. Closure b + e = integer
3. Associative property--------(b+e)+y= b+ (e+y)
4. Identity --b+ 0 =b
5. Inverse --- --b+ (-b) = 0 ; -b is also called additive inverse
6. Distributive --b(e+y) = be + by
For multiplication and division
1. Closure --by = integer
2. Commutative -- by = yb
3. Associative --(by)e = b(ye)
4. Identity ---b x 1 = b
5. Inverse ----b(1/b) = 1 ; 1/b called multiplicative inverse
6. Distributive ----b(e+y) = be + by
7. Multiplication property zero -- b(0) = 0
See illustrations
Test your self
19. 1. Reflexive property b=b
2. Symmetric property if b= r, then r = b
3. Transitive property if b = r, and r = w, then b= w
4. 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
20. 1. b+ 0 = b and b – 0 = b
2. b(0)= 0
3. 0/b = 0 with b not = 0
4. 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
21. Addition—putting together e.g. 1+1 = 2
Subtraction– 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+2
Division– inverse of multiplication or repeated subtraction
e.g. 15/3 =5.
22. It is the used of either positive(+) or
negative(-) signed before the numbers.
Examples -3, -18, -50…. And for positive are +1,
+56, +100 …..
23. 1. 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 =-22
2. Subtraction– to subtract one signed number from the others, just changed the
sign of the subtrahend then proceed to algebraic addition.
minuend
+56 +56
operation - -
+3 - 3
subtrahend
+53
3. Multiplication and division—two cases a. like sign and b. Unlike sign
(+6)*(+5) =+30 (-6)*(+5) = -30
(-6)*(-5) = +30 Case a, always positive
(-30)/(+5) = -6
(+30)/(+5) = +6 Case b, always negative
(-30)/(-5) = +6
24. 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…
25. a. 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, 10/0
l. Indeterminate fraction– fraction which has no quantity or meaning 0/0
m. Mixed numbers– number that is a combination of an integer and a proper fraction.
e.g. 5/1/2, 9/8/11.
26. 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
In multiplication “of” means to multiply
27. 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…
Determine the least common denominator and change each to an equivalent fraction
whose denominator is the LCD.
Example.
¾, 2/5, 1/3
¾ = ¾*5/5*3*3 = 45/60
2/5 = 2/5*3/3*4/4 = 24/60
1/3 = 1/3 * 4/4* 5*5 = 20/60
a/b+c/d = (ad+bc)/bd
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.
28. Decimal fraction read as
a. 3.1 as three and one tenths
b. 3.10 as three and ten hundredths
c. 3.103 as three and one hundred three thousandths
d. 3.2156 as three and two thousand fifty-six ten-
thousandths.
NOTE THAT HE PLACE VALUE OF THE DECIMAL PLACES STARTS AT
TENTHS AND SO ON.
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.
29. a. Repeating example 4/9 = 0.4444444444…..
b. Terminating example 25/100 = 0.25
c. Non-terminating example 0.212121212121…….
a. 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
30. Prime no. Composite no.
10
5
Factors of 5. 2 5 Factors of 10.
1 5
1 5 Second ,Factors of 5.
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.
31. 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
All the Prime Numbers between 1 and 1,000
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
Note:947 SUM OF PRIME NOS FROM 977 15, 1-100=991 1-150= 35, 1-200= 46, 1-1000=
THE 953 967 971 1-50= 983 25, 997
168
Note: The number two(2) is the only even prime numbers
32. a. 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
Number symmetric
1 none
2 none
3 none
4 3,5
5 3,7
6 5,7
7 3,11
8 5,113,13
9 7,11;5,13
10 7,13;3,17
11 5,17;3,19
12 11,13;7,15;5,19
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
33. 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.
34.
35. n
Refers to the symbol that indicates a root , a 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
36. 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.
37. 1. Sum and difference of the same terms or difference of two square
=
2. Square of a binomial
=
=
3.Cube of a binomial
=
=
39. Proportion is a statement that two ratios are
equal extremes antecedent
Properties of proportion: a:x= y:d a:x= a/x
1. If a/y = x/d, then a:x = y:d
2. If a/b = c/d, then a/c = b/d means consequent
3. If a/b = c/d then b/a = d/c
4. If a/b = c/d, then (a-b)/b = (c-d)/d.
5. If a/b = c/d, then (a+b)/b= (c+d)/ d
6. 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.
40. Refers 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 = 360
12 = 3* 2*2 or 3*2^2
15 = 3*5
41. 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*5
18 = 3^2*2
LCM = (3^2)*5*2
LCM = 90
42. 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*7
112 = 2^4 *7
Common factor are 2 and 7
GCF = 2*7
GCF = 14
43. A. By 2 – last digit is 0 and even. Ex 6
B. By 3– sum of the digits are divisible by 3. ex 3174
C. By 4– if the last 2 digits form a number divisible by 4 ex 20024
D. By 5– ends with 0 and 5 ex. 35
E. by 6– number is divisible by both 2 and3 ex. 2538
F. 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. 1024
H. By 9– sum of the digits is divisible by 9 ex 3127878
I. By10– ends with 0 ex. 2000010
J. 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
44. P = br, R = P/b B = P/r
to get percentage To get the rate To get the base
Example 1. 40% of the 50 apples bought were spoiled. How many apples were spoiled
