Real Number System

Real Number System

→ Consist of all the rational and irrational numbers. The real numbers are “all the numbers” on the number line.
i.e -25, -14, -3, 7/4, 0, 5/6, 1, √2,
Real numbers can be classified either rational or irrational.
Rational Number
→Any number that can be put in the form p / q where p and q are integers and q≠0. They can always be expressed by
using terminating decimals or repeating decimals.
i.e -25, -14, -3, 7/4, 0, 5/6, 2/3, 2……
Terminating Decimals
→ Decimals that contain a finite number of digits.
i.e 36.8, 0.125, 4.5
Repeating Decimals
→ Decimals that contain a infinite number of digits.
i.e . 0.333… 7.689689…
Irrational Numbers
→Are numbers which cannot be expressed as a quotient of 2 integers.
→Non repeating & non terminating decimals
i.e. π, ẹ, √2, ⁵√11
Integers
→Consist of the natural numbers, 0, and the opposites of the natural numbers.
i.e. …-2, -1, 0, 1, 2, 3…..
Non Integer Fractions
→ Fraction whose numerator is not a multiple of the denominator.
i.e ½, ¼, -11/5, 33/32
Whole Numbers
→Consist of the natural numbers, and 0.
i.e. …0, 1, 2, 3, 4…..
Natural Numbers
→It is also referred as counting numbers.
i.e 1, 2, 3,4……
Other Vocabulary Associated with the Real Number System
• …(ellipsis)—continues without end
• { } (set)—a collection of objects or numbers. Sets are notated by using braces { }.
• Finite—having bounds; limited
 • Infinite—having no boundaries or limits
• Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.
ROMAN NUMERALS
Roman numerals are expressed by letters of the alphabet:
ROMAN VALUE
NUMERAL

I 1

V 5

X 10

L 50

C 100

D 500

M 1000

There are four basic principles for reading and writing Roman numerals:
1. A letter repeats its value that many times (XXX = 30, CC = 200, etc.). A letter can only be repeated three times.
2. If one or more letters are placed after another letter of greater value, add that amount.
VI = 6 (5 + 1 = 6)
LXX = 70 (50 + 10 + 10 = 70)
MCC = 1200 (1000 + 100 + 100 = 1200)
3. If a letter is placed before another letter of greater value, subtract that amount.
IV = 4 (5 – 1 = 4)
XC = 90 (100 – 10 = 90)
CM = 900 (1000 – 100 = 900)
Several rules apply for subtracting amounts from Roman numerals:
a. Only subtract powers of ten (I, X, or C, but not V or L)
For 95, do NOT write VC (100 – 5).
DO write XCV (XC + V or 90 + 5)
b. Only subtract one number from another.
For 13, do NOT write IIXV (15 – 1 - 1).
DO write XIII (X + I + I + I or 10 + 3)
c. Do not subtract a number from one that is more than 10 times greater (that is, you can subtract 1 from 10 [IX] but not 1
from 20—there is no such number as IXX.)
For 99, do NOT write IC (C – I or 100 - 1).
DO write XCIX (XC + IX or 90 + 9)
Absolute Value
The absolute value of a number is the distance between that number and zero on a number line. Absolute value is
shown by placing two vertical bars around the number as follows:
| 5 | The absolute value of five is five.
| -3 | The absolute value of negative three is three.
- | - 16| The negative absolute value of negative sixteen is negative sixteen .

Divisibility Rules
Divisor Rule

2 If it is even.

Ex. 32; 128; 534; 998

3 If the sum of the digits of the numbers is divisible by 3.

Ex. 2,958 (sum of the digits is 24), 17, 034 (sum of the digits is 15), 995, 919

4 If the number formed by its last 2 digits is divisible by 4.

 Ex. 256 (56 is divisible by 4); 22, 540 (40 is divisible by 4); 61, 604

5 If the ones digit is either 5 or 0.

Ex. 535; 2, 770; 19,315; 9,999,990

6 If it is an even number and is divisible by 3.

Ex. 72 ( even and sum of the digits is 9); 1,878 (even and sum if digits is 24);

545, 922

7 If the difference obtained from subtracting twice the last digit and the remaining digits is a
multiple of 7.

Notes: steps are repetitive.

Ex. 358, 631

1(2)= 2 → 35, 863 – 2 = 35, 861 → 1(2) = 2 → 3,286 – 2 = 3,584 → 4(2) = 8 →

358 – 8 = 350;

43, 673

8 If the last three digits if a number is divisible by 8.

Ex. 1,072 (72 is divisible by 8); 18, 808 (808 is divisible by 8); 59, 792

9 If the sum of the digits is a multiple of 9.

Ex. 108 ( (sum of the digits is 9); 2,520 (sum of digits is 9); 580, 608

10 If it ends with 0.

Ex. 700; 2,450; 33,330

11 If after subtracting and adding the digits successively, the answer is divisible by 11.

Ex. 1,331 → 1 – 3 +3 – 1 = 0; 60, 511 → 6 – 0 + 5 – 1 + 1 = 11; 8,547

12 If it is divisible by both 3 and 4.

Ex. 1, 464 ( 1+4+6+4 = 15 is divisible by 3 and 64 is divisible by 4); 5,616

13 If the sum of four times the unit digit and the number formed by the other digits is divisible by
13.

Ex. 169 → 4(9) + 16 = 52 → 4(2) + 5 = 13

11, 843 → 4(3) + 1184 = 1196 → 4(6) + 119 = 143 → 4(3) + 14 = 26

Prime Factorization
Prime Factorization → is finding which prime numbers multiply together to make the original number. One common
method is by using tree method.
 2 2 2 2 3

Example: What are the prime factors of 30?

30 = 5 x 3 x 2
Composite Numbers → A composite number has more than two factors.
Ex. 10 → is considered as composite because it has 2 factors, 5 and 2.
39 → is considered as composite because it has 2 factors, 13 and 3.
91 → is considered as composite because it has 2 factors, 13 and 7.
Prime numbers →can be divided evenly only by 1 or itself and it must be a whole number greater than 1.The prime
numbers between 2 and 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 since each of these numbers has only two factors,
itself and 1.
GCF & LCM
GCF (Greatest Common Factor) → the greatest factor that is common to two or more numbers (they share it).
The greatest common factor of two (or more) numbers is the product of all the prime factors the numbers have in
common.
LCM (Least Common Multiple) → a number that is a multiple of two or more numbers. Common multiples of 2 and 3
are 0, 6, 12, 18,... The least common multiple (LCM) of two numbers is the smallest number (excluding zero) that is a
multiple of both of the numbers.
Find the GCF and LCM of 36 and 54 Find the GCF and LCM of 52 and 39
36 = 2 * 2 * 3 * 3 52= 13 * 2 * 2
54= 2 * 3*3 *3 39= 13 * 3
LCM=2* 2*3*3*3 = 108 LCM= 13*2*2*3 = 156
GCF= 2* 3* 3 = 18 GCF = 13
Fundamentals and Arithmetic
PEMDAS → is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. For any expression, all exponents
should be simplified first, followed by multiplication and division from left to right and finally addition and subtraction from left to right.
Ex. #1. 5 + (4 – 2)2 x 3 ÷ 6 – 1
● Start with the Parentheses: 4 – 2 = 2. (Even though subtraction is usually done in the last step, because it's in
parentheses, we do this first.) That leaves 5 + 22 x 3 ÷ 6 – 1 = ?
● Then Exponents: 22 = 4. We now have 5 + 4 x 3 ÷ 6 – 1= ?
● Then Multiplication and Division, starting from the left: 4 x 3 = 12, leaving us with 5 + 12 ÷ 6 – 1 = ?
● Then moving to the right: 12 ÷ 6 = 2, making the problem 5 + 2 – 1 = ?
● Then Addition and Subtraction, starting from the left: 5 + 2 = 7, leaving 7 – 1 = ?
● Finally, moving to the right: 7 – 1 = 6
Ex. #2. 30 ÷ 5 x 2 + 1
● Take Note: There are no Parentheses & Exponents
● We start with the Multiplication and Division, working from left to right.
NOTE: Even though Multiplication comes before Division in PEMDAS, the two are done in the same step, from
left to right. Addition and Subtraction are also done in the same step.
● 30 ÷ 5 = 6, leaving us with 6 x 2 + 1 = ?
● 6 x 2 = 12, leaving us with 12 + 1 = ?
● We then do the Addition: 12 + 1 = 13
Ex. #3. Simplify 3 x 7 − 11 + 15 ÷ 3 using the rules for order of operations
= 3 x 7 − 11 + 15 ÷3Multiply
= 21− 11 + 15 ÷3 Divide
= 21− 11 + 5 Subtract
 = 10 + 5 Add
= 15 Final answer
For any real numbers a, b, and c.

Property Addition Multiplication

Closure a + b is a unique real number a*b

Commutative a+b =b+a a*b = b*a

Associative (a+b) + c = a + (b + c) (a*b)*c = a*(b*c)

Identity a+0 = 0 +a a*1 = 1*a

Inverse 0
=

Distributive

Conversion and Measurement

Fraction to Decimal
Divide the numerator by the denominator.
Ex. ½ = 0.5 ; ¾ = 0.75; ¼ = 0.25 ; 3/10 = 0.3
Decimal to Fraction
*Terminating → multiply the number by a fraction (equal to one) whose numerator and denominator is a multiple of 10
such that the product is a whole number.
Ex. Convert 0.25 to fraction.
100 25 1 100 65 13 100 80 4
0.25 ( )= = 0.65 ( )= = 0.80 ( )= =
100 100 4 100 100 20 100 100 5
*Repeating decimal number
Ex. Convert 0.45 to fraction Convert 0.33 to fraction
Let n = 0.45 = 0.454545 Let n= 0.33 = 0.3333
100n = 45.4545 100n = 33.3333
- n = 0.454545 -n =33.3333
99n = 45 99n = 33.3333
45 5 33 1
n= = n= =
99 11 99 3
Percent to Decimals
Divide the number by 100%. Note that 100% = 1
Ex. Convert the following to decimal
1
a. 29% b. 5 % c. 68.73%
4
(29%) ÷100% = 29 (5 ¼%) ÷ 100% → 5.25% ÷ 100% = 0.0525 (68.73%)÷ 100% = 0.6873

Decimal numbers to percent

Multiply the decimal number by 100%. Note that 100% = 1
Ex. Convert the following to percent
a.0.31 b.0.9812 c. 1.03
0.31 x 100% = 31% 0.9812 x 100% = 98.12% 1.03 x 100% = 1.03%
Unit of measurement
Kilo- Hecto- Deca- unit Deci- Centi- Milli-

Length = meter (m)

Capacity = liter (l)
Weight = gram (g)

Percentage, Base and Rate

Percentage Amount taken from a given number
Base (B) Given number from which a percentage is taken.
Rate(r) Numbering bearing the percent notation.
Relationships P = rB B=P / r r = P/B
Rules If r = 100%, then P=B
If r <100%, then P<B
If r >100%, then P>B

SIMPLE INTEREST
I = Prt where I = amount of interest, P = Principal, r = rate of Interest, and t = time (in years)
A= P + l where A= final amount
SIMPLE DISCOUNT
D=Adrt where D = amount of Discount, A=Amount or face value, r=discount rate, t=time
P = A – D where P = Proceeds

For Discounts on Merchandise, D = Original price x discount rate

Discount price = Original Price – Amount of discount
Commissions C= Sales x rate
Ratio & Proportion
Ratio → is a comparison of two or more quantities.
Proportion → is a number sentence stating the equivalence of two ratios.
Ex. 1. The ratio of 12 days to 3 weeks is 12:21 or 4:7
2. The ratio of 3 meters to 180cm is 300:180 or 5:3.
3. The distance between two cities on a map is 24 cm. If the scale used is 1cm :10cm, how far apart are the two cities.
Let n = the actual distance of the 2 cities.
1 cm 10 km
Solution: = → n = 10x24 = 240 km
24 cm n
The two cities are 240 km away from each other.

Direct→ As one quantity increases, the other increases

Ex. 1. Find the value of n if 30:25 = 12:n 2. A car travels at an average rate of 260km in 6 hrs.
25(12)
30(n)=25(12) → n = =10 How far can it go in 9 hrs, if traveling at the same rate?
30
260(9)
260:6=n:9 → n= =390 km
6
Inverse→As one quantity increases, the other decreases.
Ex. If the food in a crate is sufficient to feed 12 flood victims in 15 days, how many days would it last for 10 days?
(12 victims)(15days)=n (10 victims)
12(15)
n= = 18 days.
10
Partitive→ one quantity is being partitioned into different proportions.
Ex. The sum of the measures of the three angles of any triangle is 180-. Find the measures of the angles of the triangle
if they are in the ratio 1:2:3.
Let n = be the number of angles.
n+2n+3n= 180 → 6n = 180 → then n = 30, 2n = 60 and 3n = 90.
6
Series and Sequences
Sequence→ is an ordered list of numbers. In technical terms, a sequence is a function whose domain is the set of natural
numbers and whose range is a subset of the real numbers.
Example: Find the next term and describe the pattern:
a. 2, 4, 6, 8, 10, ...
b. 1, 4, 9, 16, 25, ...
c. 3, 7, 15, 31, 63, ...
Solution:
a. We see that the next term is 12. We can get to the next term by adding two.
b. The next term is 36. The terms are all squares.
c. The next term is 127. These numbers are all one less than a power of two.
Series → is the sum of the sequence.
Example: Find the series of the following:
a. 2, 4, 6, 8, 10, ...
b. 1, 4, 9, 16, 25, ...
c. 3, 7, 15, 31, 63, ...
Solution:
a. 30
b. 55
 c. 119
Algebra
Algebraic expression – a collection of letters (called variables) and real numbers (called constants) combined by using
the four fundamental operation and exponentiation.
Examples: m + 10 & 6r – 3
A constant is a number that does not change.
A coefficient is a number multiplied or divided by a variable.
For example, 4m + 20 → 4 is the coefficient, m is the variable, and 20 is the constant.
Evaluate an algebraic expression – To find the value of an algebraic expression, just substitute numbers for variables and
simplify.
For example, p +10 if p = 2 then 2 + 10 = 12
2r – 7 if r = 4 then 2(4) – 7 = 1
A term is a part of an expression that are added or subtracted.
For example: 5x + 6 – 4x. This expression has 3 terms: 5x, 6, and 4x
Like terms are terms that have the same variable raised to the same power. In the expression above, 5x and 4x are like
terms.
Complete the table below.
n 3n – 5

5 10

10

21

32

x 2x2 + 6

5 56

10

15

20

Exercises
1. Evaluate the expression 3x2 + 5xy – 2 for x = 3 and y = -2
2. Identify the terms and coefficients in each algebraic expression.
1
a. 5x – 1/3 b. 6x + 4y + 9 c. x2y - + 3y
x
3. Evaluate each expression for the given values of the variable.
a. 2(a+b) if a = 16.5 and b = 12
b. 2a + b +c if a=9, b= 13 and c=6
1
c. h (b + c) if h=12.5, b=10.2 and c=8
2
Special Products
Monomial→ made up of one term. It is also a real number, a variable, or a product of a real number and one or more
5x
variables. The expressions like 3xy, 13, 27xy2 are monomials. The expressions like √ ❑ , 7x and are not monomials.
y
A monomial cannot have a variable under a radical sign, in an exponent or in a denominator.
* The coefficient or numerical coefficient of a monomial is the real number factor, and the degree of a monomial is the
sum of the exponents of its variables. The degree of a nonzero real number is zero. The zero monomial, 0 , has no degree.

I.e. State the degree and the coefficient of each monomial.

Monomial -13x 0.5x2 23 2a3b2
Degree 1 2 0 5
Coefficient - 13 0.5 23 2
Binomial→ made up of two terms.
i.e. 3x +4y
Trinomial→ made up of three terms.
i.e. 8x3 – 9y4 - 7
Multinomial or Polynomial →algebraic expression composed of many terms. It is also a monomial or sum of
1
monomials. Examples are: 4x2 + 3y -5x2 + 6x +2 xy , 7a – 5b
2
Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms".
*The degree of a monomial is the sum of the exponents of its variables.
* The degree of a polynomial is the degree of the term of highest degree.

Product of Monomials
To find the product of two monomials, some rules on exponents need to be considered.
Product of Power
3 5 8
(a )(a ) = a
Power of a Power
(x3)4 = x12
Power of a Product
3 3 3
(ab) = a b

Product of a Polynomial and a Monomial

The product of a polynomial and a monomial can be found by using the associative and distributive properties, observing
at the same time the rules of exponents for multiplication.
The expression 3y (y2 + 2y - 4) becomes 3y (y2) + 3y (2y) + 3y (- 4), and simplifies further to 3y3 + 6y2 – 12y.
Simplify the algebraic expressions.
a. 2 b(b3 +3 b2−4 b+ 2)
b. x 3 (6 x 2+ 4)−4 ( x 3−1)
c. 4 b(3 b 3−2 b2−b+2)+ 33
Product of sum & difference of binomials (The FOIL Method)
Although two binomials can be multiplied using the distributive property, a shorter method is used more frequently.
a. Find the product of (a+ b)(a−b)
b. Find the product of (2 x+ 3 y)(2 x− y )

Product of Sum & Difference of Two Terms.

The product of sum and difference of two binomials is equal to the square of the first term minus the square of the second term.

2 2
(x + y )(x− y )=x − y

In solving binomials, box plot method / grid method is used. It is in a form of 3 x 3 square table (one face of a rubik’s
cube)

Cube of a Binomial
Sum of cubes:
The sum of a cubed of two binomial is equal to the cube of the first term, plus three times the square of the first term by
the second term, plus three times the first term by the square of the second term, plus the cube of the second term.
(a + b)3 = a3 + 3a2b + 3ab2 + b3
= a3 + 3ab (a + b) + b3
Difference of cubes:
The difference of a cubed of two binomial is equal to the cube of the first term, minus three times the square of the first
term by the second term, plus three times the first term by the square of the second term, minus the cube of the second
term.
(a – b)3 = a3 – 3a2b + 3ab2 – b3
= a3 – 3ab (a – b) – b3

Sum & Difference of Cubes

3 3 2 2
x + y =( x+ y)( x −xy + y )
3 3 2 2
x − y =(x− y )(x + xy + y )

Factoring
1. ax – ay = a ( x+y)
2. x 2− y 2=( x+ y)( x− y)
3. x 2 ± xy + y 2=(x ± y )2
4. x 2+( a+b) x +ab=( x+ a)¿
5. +b ¿
6. x 3 ± 3 x 2 y +3 xy 2 ± y 3=( x ± y )3
7. x 3− y 3=x 2 ∓ xy+ y 2
Fractional Expression
b ab
1. a ( )=
c c
a+b a b
2. = +
c c c
a
3. b a
=
c bc
ab+ ac
4. =b +c
a
a ac
( )=
5. b b
c
a
b ad
6. ( )=
c bc
d
a c ad +bc
7. + =
b d bd
Exponents and Radicals
For any real number r and any positive integer n, r n means that r is used as a factor n times.

coefficient 2y5 Exponent (Power)

Algebraic term
Base
 Some properties on Integral Exponent
Product of Power

Quotient of Power

Power of a Power

Power of a Product

Power of a Quotient

Zero Exponent

Negative Exponent

Fractional Exponent 1 1
a = √❑
2
9 = √❑ = 3
2

Exercise
Use the properties of exponents to simplify each expression.
1. (5q)3
2. (2x2) 2
3. (3xy) 3
4. (abx) – 2
❑
2x
5. [ 2 ] 3
a y
6. [(x2) 3] 2
−5 5
x y
7. −5 −2
x y
2 3
2a 2 [ x ] 4
8. [ ] 3
x 2a
Linear Equation in one variable
→ Is an equation can be written in the form: ax + b= c , for a , b and c are real numbers
and a≠0.
Ex. A. 2x + 5= 15; B. 3x – 4= 11; C. 5y – 20 = 5
Translation from English Phrase to Mathematical Phrase
English Phrase Mathematical Phrase
2 2
1. The sum of the squares of x and y x +y
2. The square of the sum of x and y ( x + y ¿ ¿2
3. 5 less than a number x–5
4. A number less than 5 x<5
 x+ y
5. The quotient of the sum and difference of x and y
x− y
6. X is at most 10 x≤10
7. Y is at least 3 y≥3
8. The product of three consecutive integers x (x+1)(x+2)
9. Age of a man 5 yrs. Ago x–5
10. A speed 3kph faster x+3
Quadratic Equation
Quadratic Equation is an equation involving a second degree polynomial in one variable. It is written in the
general form ax 2 +bx +c=0, where a, b and c area real nos. & a≠0. If the graph of a linear
equation is a line, the graph of a Quadratic Equation is a parabola. The root of a
quadratic equation is its solution.
Quadratic Formula x=
−b ± √❑
❑

Inequality in one variable

Properties
If a < b then a+c < b+c and a – c <b – c.
If a<b and c>0, then ac <bc and a/c < b/c
If a<b and c<0, then ac >bc and a/c > b/c.

Geometry
Angles
Complementary angles → the sum of 2 or more angle is 90o.
Supplementary angles → the sum of 2 or more angle is 180 o.
Right angle →the measure of the angle is 90 o.
Obtuse angle → the measure of the angle is greater than 90o but less than 180 o.
Acute angle → the measure of the angle is greater than 0 o but less than 90 o.
Polygon No. of sides Measure of interior Perimeter Area
angle
triangle 3 180 o P = a+b+c A= ½ bh

Square 4 360 o P = 4s A= s2.

4 360 o P = 2L + 2W A = LW
Quadrilateral

4 360 o P= a+b+c+d A= ½ (b1 + b2)h

Trapezoid

- - C = 2πr = πd A = πr2.
Circle
Statististics
Mean → the average set of data.
Ex. Find the mean of the following grades in College Trigonometry: 69, 89, 74, 97, 81.
69+89+74 +97+81
=82
5

Median→ the middle number when a data set is arranged from least to greatest.
Ex. Find the median of the following heights of basketball players(in terms of cm):
171, 175, 173, 167, 181, 193, 178
Arranged first from lowest to highest ( in ascending form)

167, 171,173, 175, 178, 181,193

175cm is the median
Mode →the number(s) that occur(s) most often in a data set.
Ex. Find the mode of the following scores in College Algebra Quiz No. 1

18,13, 16, 16, 14, 9, 15, 11, 16, 10, 7, 19, 18, 15
Therefore, the mode in given set is 16.

Data Sufficiency
1. How much did Aling Maria earn from selling 5 bilaos of puto?
A. Each bilao for P25.00
B. She earn 20% commission on each bilao
A. Statement A alone is sufficient to answer the question.
B. Statement B alone is sufficient to answer the question.
C. Statement A and B are necessary to answer the question.
D. Either statement 1 or 2 alone is sufficient to answer the question.
E. Insufficient data are given to answer the question.

Correct Answer: C. Both statement are needed to establish the total sales of the 5 bilaos of puto and the amount of
commission she got to know Aling Seniang's total earning.