Lesson 2: Real Number System

REAL NUMBER SYSTEM

OBJECTIVES
2.1 define the properties of real
numbers
2.2 arrange real numbers in
increasing or decreasing LESSON 2
order and on a number line
2.3 perform operations on real
numbers
2.4 compute probabilities
corresponding to a given
random variable

Explore the Landscape of Number System and Divisibility Rules

You learned when you were in elementary that when we quantify things or objects, or
when we express a part from a whole, we write their numerical value or their fractional
1 3
value. Examples are 2, 10, , and so on. These symbols are numerals that are used to
2 4
express the idea of a number.
Our present system of writing numerals is called the decimal system which has ten
symbols of digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
The first nine digit we use for counting are 1, 2, 3, 4, 5, 6, 7, 8 and 9. Other values that
are greater than 9 is obtained by combining the digits of the ten symbols.
As we go through our lesson, we will try to discover what the other set of numbers are
and what are their properties.

1.1 SET OF REAL NUMBERS

The set of real numbers has many subsets. A diagram is provided in Fig. 2.1.
• Numbers that are used in counting objects comprise the set of natural numbers
(𝑁).
Its start from 1 onwards and they are located at the right side of the number line.
One (1) is the smallest number.
Example: {1,2,3,4,5,6,7, … }
• Natural numbers and 0 (zero) comprise the set of whole numbers (𝑊). They are
located in right side of the number line. Zero (0) is the smallest number.
Example: {0,1,2,3,4,5,6,7,8, … }

19 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Fig. 2.1

• For natural numbers, their negative and 0 (zero) comprise the set of integers (𝑍).
Positive integers are integers starting from 1 and so on. They are located at the right side of the
number line after 0. They are also natural numbers or counting numbers.
Negative integers are integers starting −1 and so on. They are located at left side of the number
line before 0. They are negative whole numbers.
Example: {−3. −2. −1, 0,1,2,3,4,5,6, … }
• A Fraction is a part of a whole. In the number line they are located between integers.
1 1 1 4 1
Example: {−3, 2 , 4 , 3 , 5 , 6 5}
• Numbers that can be expressed as a ratio of two integers, such as terminating decimals, repeating
decimals, and fractions, belong to the set of rational numbers (𝑄). It can be fraction, an integer,
a whole number, or even a natural number.
Terminating decimal is usually defined as a decimal number that contains a finite number of digits
after the decimal point. A terminating decimal 14.2 can be represented as repeating decimal
14.2000000 …, the repeating number is zero, the number is usually labeled as terminating. All
Terminating decimal are rational numbers that can be written as reduced fractions with the
denominator containing no prime numbers factor other than two or five.

20 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

1
̅̅̅̅, 5, 1},
Example: { , 0.001, −0.1, 0.1111
5 4
• Non-terminating and non-repeating decimals belong to a set of irrational numbers (𝑄′).
Non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no
group of digits repeating endlessly. Decimals of this type cannot be represented as fraction and as
a result are irrational numbers.
3
Example: {√2, 𝜋, 𝜑, √99, ℯ, √3 }

Example: Put a checkmark ( ) under each subset of real numbers that apply for each given number.
Number Real Rational Irrational Integers Whole Natural
1. 65
2. -21
2
3. 17 5

4. √18
5. 17.75

Number Real Rational Irrational Integers Whole Natural

1. 65
2. -21
2
3. 17 5

4. √18

5. 17.75

1.2 COMPARING AND ORDERING NUMBERS

Any rational number can be expressed as a terminating decimal or repeating non-terminating decimals.
You can easily arrange the number in ascending or descending order by locating and plot the number on
the number line.
A. -2
B. 𝜋
2
C. − 3
1
D. 11
3
E. √4

Fig. 2.2

21 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

For Ascending Order;

From the number line copy the number from left to right.
2 1 3
{−2, − , , √4, 𝜋}
3 11
For Descending Order
From the number line copy the number from right to left.
3 1 2
{ 𝜋, √4, , − , −2, }
11 3

How can you arrange the number without the use of a number line?
Without using the number line, a real number can also be arranged in ascending or descending orders. All
numbers can be changed first to decimal forms before arranging them.

Example 1
Consider the set below and arrange them in descending order.
4
{ , − 0.84, √25 , − √6, 3.27, 18}
5

{0.8, − 0.84, 5.0, − 2.45, 3.27, 18}

Therefore, the set of the numbers in decreasing order is

4
{18, √25, 3.27, , −√6}
5

Example 2
Consider the set below and arrange them in ascending order.
7
{ , − 3.16 √12, − √16, 0.75}
3

{2.33̅ , − 3.16, 3.46, − 4, 0.75}

Therefore, the set of the numbers in ascending order is

{, −4, −3.16, 0.75, 2.33̅, 3.46}

Example 3
Consider the set below and arrange them in ascending order.
2
{−√36, √3 , − 1.27, 2.66̅}
5

{−6, 0.4 1.73, − 1.27, 2.66̅}

Therefore, the set of the numbers in ascending order is
{−6, −1.27, 0.4,1.73, 2.66̅}

22 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Absolute Value of a Number

The absolute value of a number is its distance from zero. You write the absolute value of a number
by using 2 vertical bars.

| − 5| is read as “the absolute value of negative 5.” Use the number line to find the absolute value.

Fig. 2.3

−5 is 5 units from zero (0). Therefore, | − 5| = 5.

| + 5| is read as “the absolute value of positive 5.” Use the number line to find the absolute value.

Fig. 2.4

5 is 5 units from zero (0). Therefore, | + 5| = 5.

Example 4
Find the absolute value and compare it.
1. | + 8| and | − 19|
2. | − 12| and | + 3|
3. | − 10| and | + 10|
4. −| − 24 and | − 4|
Answer:
1. 8 < 19, hence | + 8| < | − 19|
2. 12 > 3, hence | − 12| > | + 3|
3. 10 = 10, hence | − 10| = | + 10|
4. −(24) < 4, hence −| − 24 < | − 4|

1.3 OPERATIONS ON REAL NUMBERS

Rational Numbers
The set of rational numbers consists of:
3 2 1 1 2 3
𝑄 = {… , − . … , − , … , − , … , 0, … , , … , , … , , … }
2 2 2 2 2 2
The three dots between two rational numbers on the list indicate that there exist other rational numbers
between the two rational numbers. Also, the three dots found at the beginning and at the end indicate that
there are infinitely many rational numbers before and after the given rational numbers.
The integers {… , −3, −2, −1, 0, 1, 2, 3, … } are rational numbers. Every rational number is the quotient of
two integers.

23 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Rational Numbers and Fraction Form to Decimal Form

Rational Number
𝑎
A rational number is a number that can be written in the form , where 𝑎 and 𝑏 are integers, and
𝑏
𝑏 ≠ 0.
Rational numbers can be in fraction form or decimal form. Although written in different forms,
fractions and decimals may be equivalent.

Consider why the following decimal numbers are rational.

63
a. 0.63 is rational because it can be written as .
100
57
b. −5.7 is rational because it can be written as − .
10

Rational numbers in decimal form can be expressed as a fraction as shown by the examples in the table.
Decimal Process Fraction in Simplest Form
100 35 7
0.35 0.35 × =
100 100 20
10 8 4
0.8 0.8 × =
10 10 5
Let 𝑛 = 0. 2̅
𝑛 = 0. 2̅ (Equation 1)
Multiply by 10 since there is
only one repeated digit.
(𝑛 = 0. 2̅)10
10𝑛 = 2. 2̅ (Eq 2) 2
0.2̅
Subtract (Eq1) from (Eq 2) 9
10𝑛 = 2. 2̅
− 𝑛 = 0. 2̅
9𝑛 = 2
2
𝑛=9
̅̅̅̅
0. 33 ̅̅̅̅
It can be written as 0.3333 1
̅̅̅̅(Eq 1)
Let 𝑛 = 0. 33 3
𝑛 = 0.3333 ̅̅̅̅
Multiply by 100 since there are
two repeated digits.
(𝑛 = 0.3333̅̅̅̅)100
100𝑛 = 33. 33̅ (Eq 2)
̅̅̅
Subtract (Eq1) from (Eq 2)
100𝑛 = 33. 33 ̅̅̅̅
− ̅̅̅̅
𝑛 = 0. 33
99𝑛 = 33
33
𝑛 = 99

24 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

What do you do when the rational number is not a decimal fraction? How do you convert from one form to
the other?

To express rational numbers in fraction form to decimal form, divide the numerator by the denominator.
Consider the following fractions which are to be changed to decimal numbers.
Fraction Process Decimal Type of Decimal

3 Non-repeating,
0.375
8 Terminating

0.33̅
The bar over the
1 digit 3 means the Repeating, Non-
3 digit 3 is terminating
repeating and
non-terminating

Operation on Rational Number in Fraction Form

All the rules which play to operations with integers hold for operations with rational numbers in fraction
form. Recall that when using the number line, any movement from the initial point going to the right is
positive, and any movement going to the left is negative.

Addition and Subtraction of Rational Numbers in Fraction Form

Adding or Subtracting similar fractions is the same as adding or subtracting whole numbers. That is, adding
the numerators and keeping the same denominator.
For all integers 𝑎, 𝑏, and 𝑐, (𝑐 ≠ 0)
𝑎 𝑏 𝑎+𝑏
Addition: 𝑐
+𝑐 = 𝑐
𝑎 𝑏 𝑎−𝑏
Subtraction: − =
𝑐 𝑐 𝑐

Example 5
Find the sum or difference of the given similar fraction.
2 3
1. (4) + (4)
2 3 (2) + (3)
( )+( )=
4 4 4
2 3 7 3
( ) + ( ) = or 1
4 4 4 4

25 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

7 5
2. (6) − (6)
7 5 (7) − (5)
( )−( )=
6 6 6
7 5 2 1
( ) − ( ) = or
6 6 6 3

To add or subtract dissimilar fractions, find their LCD and change them to similar fractions then follow the
rules for adding or subtracting similar fractions.
𝑎 𝑐 𝑎𝑑 𝑏𝑐
± = ±
𝑏 𝑑 𝑏𝑑 𝑏𝑑
𝑎𝑑 ± 𝑏𝑐
=
𝑏𝑑
where 𝑏 and 𝑑 are non-zero and 𝑏𝑑 is the LCD.

Example 6
Perform the indicate operation and express the answers in simplest form.
5 4
1. (6) + (3)
𝐿𝐶𝐷 = 6
5 4 (5) + (2)(4)
( )+( )=
6 3 6
5 4 5+8
(6) + (3) = 6
5 4 13 1
(6) + (3) = 6
or 2 6

2 2
2. (3) − (5)
𝐿𝐶𝐷 = 15
2 2 (2)(5) − (2)(3)
( )−( )=
3 5 15
2 2 10−6
(3) − (5) = 15
2 2 4
(3) − (5) = 15

Multiplication and Division of Rational Numbers in Fraction Form

The rules that apply to multiplication and Division of Integers also apply to multiplication and division of
rational numbers. The following are rules that you must remember.
1. To multiply rational numbers in fraction form simply multiply the numerators and multiply the
denominators.
𝑎 𝑐 𝑎𝑐
In symbol, × = were: 𝑏 and 𝑑 are NOT equal to zero, ( 𝑏 ≠ 0; 𝑑 ≠ 0 )
𝑏 𝑑 𝑏𝑑
2. To divide rational numbers in fraction form, you take the reciprocal of the second fraction (called
the divisor) and multiply it by the first fraction.
𝑎 𝑐 𝑎 𝑑 𝑎𝑑
In symbol, 𝑏 ÷ 𝑑 = 𝑏 × 𝑐 = 𝑏𝑐
, were: 𝑏, 𝑐, and 𝑑 are NOT equal to zero.

26 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Example 7
Multiply or divide the following fraction and write your answer in simplest form.
3 5
1. 8
×6
Method 1: Method 2: Cancel the common factors.
3 5 15 3 5 1 5
8
× 6
= 48 8
×6=8×2
𝟓 1 5 𝟓
= × =
𝟏𝟔 8 2 𝟏𝟔

7 1
2. 6 8 × 5 3
Change mixed fraction to improper fraction
55 16 55 2
× = ×
8 3 1 3
55 2 𝟏𝟏𝟎
× =
1 3 𝟑

15 18
3. ÷
4 8
18 8
The reciprocal of 8
is 18
15 18 15 8
÷ = ×
4 8 4 18
15 8 5 2
× = ×
4 18 1 6
5 2 𝟏𝟎 𝟓
1
× 6 = 𝟔 or 𝟑

Operation on Rational Numbers in Decimal Form

Decimal notation is simplified a more convenient way in writing fractions, more convenient because it
follows our usual base ten place value notation. Decimals are read just like fractions.

Addition and Subtraction of Rational Numbers in Decimal Form

Decimals are added and subtracted just like whole numbers: align the decimal points, add/ subtract the
numbers in columns and insert the decimal point in the answer immediately beneath the decimal points in
the numbers being added or subtracted.
Example 8
Perform the indicated operations.
1. 9.75 + 2.882 + 12.1
9.750
2.882
+ 12.100
𝟐𝟒. 𝟕𝟑𝟐

2. 45.9010 − 26.4281
45.9010
− 26.4281
𝟏𝟗. 𝟒𝟕𝟐𝟗

27 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Multiplication and Division of Rational Numbers in Decimal Form

In the multiplication of decimals, multiply the numbers “without the decimal points”. Then, add the number
of places in both factors and put the same numbers of the places in the result.

Example 9
Multiply 3.24 × 0.56
3.24
× 0.56
1944
+ 1620
1.8144
The total number of the decimal places is 4, then you are going to count 4 places from right to left. Thus,
𝟏. 𝟖𝟏𝟒𝟒.

This can also be justified by writing each factor in expanded form and applying appropriate properties.

The process of dividing decimals is different from that of multiplying decimals. Although the process of
dividing whole numbers is also applied, there is still a necessary step required before the division is applied.
There is a need to change a decimal divisor into a whole number divisor.
To divide decimals,
1. Multiply both the divisor and the dividend by the same multiple of 10 to make a whole number
divisor:
2. Divide as in whole number division;
3. Put the decimal point in the quotient directly above the decimal point in the dividend; then
4. Check by multiplying the quotient and the divisor.

Example 10

28 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Example 11
Find the quotient of 36.4 ÷ 1.6

Integers
Integers is a special set of whole numbers comprised of zero, set of whole numbers, set of natural
numbers (or the counting numbers) and their additive inverses. Integers are the subset of the real
numbers denoted by the 𝑍.

As set, integers can be represented as:

𝑍 = {… , −3, −2, −1, 0, 1, 2, 3, … }

On a number line, these are the set of integers:

Fig. 2.5

Integers are composed of two parts: the sign and the numerical part. The sign indicates a direction which
can be positive or negative. The positive numbers may or may not be preceded by a plus (+) sign, but
negative numbers must always be preceded by a negative (– ) sign.

Graphing integers on a number line will help us to compare them and arrange them in order. The numbers
on the number line are arranged in increasing order from left to right. Of the two numbers, the one that is
located on the right of the other in the number has a greater value. You can see on a number line that all
positive numbers are greater than zero and all negative numbers are less than zero.

Operations on Integers
Addition of Integers
You can add integers by applying some rules which depend on whether the integer has the same signs or
different signs.

29 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Rules for Adding Integers

• For like signs: (+__ ) + (+__ ) or (−__ ) + (−__ )

Find the sum of their absolute values and use the common sign of the integers.
• For unlike signs: (+__ ) + (−__ ) or (−__ ) + (+__ )
Find the difference of their absolute values and use the sign of the integer with the greater
absolute value.

Example 12
Find the sum of the following.
a. (−12) + (−3) d. (−8) + (−9) + (−17)
b. 5 + (−9) e. 14 + (−3) + 11 + (−6)
c. 4 + 8 + 16

Solution:
a. (−12) + (−3) Signs are the same.
= |−12| + |−3| Find the absolute value.
= 12 + 3 Add the absolute value
= 𝟏𝟓 Since the addends are both negative, the sum is
negative.

b. 5 + (−9) Signs are different.

= |5| + |−9| Find the absolute value.
=5−9 Since the signs are different, subtract the absolute
values
=4
= −𝟒 Since the negative addends has a greater absolute
value, the sum is negative.

c. 4 + 8 + 16 Signs are the same

= |4| + |8| + |16| Find the absolute value.
= 4 + 8 + 16 Add the absolute value
= 28
= 𝟐𝟖 Since the addends are all positive, the sum is positive.

d. (−8) + (−9) + (−17) Signs are the same

= |−8| + |−9| + |−17| Find the absolute value.
= 8 + 9 + 17 Add the absolute value
= 34
= −𝟑𝟒 Since the addends are all negative, the sum is
negative.

30 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

e. 14 + (−3) + 11 + (−6) Signs are different.

= 14 + 11 + (−3) + (−6) Find the sum of the integers with the same sign.
= (14) + (11) = (−3) + (−6)
= |14| + |11| = |−3| + |−6| Find the absolute value.
= 14 + 11 =3+6 Add the absolute value.
= 25 = −9

= 25 + (−9) Add the absolute value.

= |25| + |−9|
= 25 − 9 Since the signs are different, subtract the absolute
values
= 𝟏𝟔 Since the positive addends has a greater absolute
value, the sum is positive.

Subtraction of Integers
The addition and subtraction of integers are related to each other.

Rules for Subtracting Integers

• To subtract integer, add its opposite or for all numbers 𝑎 and 𝑏,

𝑎 + (−𝑏) = 𝑎 − 𝑏

Example 13
Find the difference of the following.
a. 12 − (−5) c. (−8) − (−5) − (−16)
b. (−7) − (−19) d. (−9) − (−4) − 18 − 6

Solution:
a. 12 − (−5)
= 12 − (5) Get the opposite of (−5)
= 12 + 5 Change subtraction(– ) to addition (+)
= 𝟏𝟕

b. (−7) − (−19)
= (−7) − (19) Get the opposite of (−19).
= −7 + 19 Change subtraction(– ) to addition (+)
= 7 − 19 Following the rule, we have for addition, we are
going to get the absolute value of the addends. Since,
the signs are different, we are going get their
difference.
= 𝟏𝟐 Since the positive addends has a greater absolute
value, the sum is positive.

31 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

c. (−8) − (−5) − (−16)

= (−8) − (5) − (16) Get the opposite of (−5) and (−16)
= (−8) + (5) + (16) Change subtraction(– ) to addition (+)
= (−8) + 21 Find the sum of the integers with the same sign
= (−8) − 21 Follow the rule we have in finding the sum of integers
with different sign.
= 𝟏𝟑 Since the positive addends has a greater absolute
value, the sum is positive.

d. (−9) − (−4) − 18 − 6
= (−9) − (4) − (−18) − (−6) Get the opposite of (−4), 18 and 6
= (−9) + (4) + (−18) + (−6) Change subtraction(– ) to addition (+)
= (−33) + 4 Get the sum of the integers with same signs
= 33 − 4 Follow the rule we have in finding the sum of integers
with different sign.
= −𝟐𝟗 Since the negative addends has a greater absolute
value, the sum is negative.

Multiplication of Integers
There are also rules in multiplying integers. Here are the easy-to-remember rules you need to remember.

Rules for Subtracting Integers

• When multiplying two numbers with the same sign, the product is
positive.
o (+)(+) = (+) (−)(−) = (+)

• When multiplying two numbers with different signs, the product is

negative.
o (+)(−) = (−) (−)(+) = (−)

• Any number multiplied by 0 gives a product of 0.

o (0)(𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟) = (0) (𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟)(0) = (0)

• For a product with no zero factors:

o If the number of negative factors is odd, the product is negative;
o If the number of negative factors is even, the product is negative.

Example 14
Find the product of the following.
a. (3)(5) d. (13)(−7)
b. (−7)(−8) e. (−5)(8)(3)(0)
c. (−4)(12) f. (−1)(−2)(−3)(4)

32 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Solution:
a. (3)(5)
= (3)(5) Get the product of the two integers.
= 𝟏𝟓 Since the integers have the same sign, the product is
positive.

b. (−7)(−8)
= (−7)(−8) Get the product of the two integers.
= 𝟓𝟔 Since the integers have the same sign, the product is
positive.
c. (−4)(12)
= (−4)(12) Get the product of the two integers.
= −48 Since the integers have different sign, the product is
negative.
d. (13)(−7)
= (13)(−6) Get the product of the two integers.
= −𝟓𝟏 Since the integers have different sign, the product is
negative.
e. (−5)(8)(3)(0)
= (−120)(0) Get the product of all the integers.
=𝟎 The integers have different sign and since there is
zero, the product is zero.

f. (−1)(−2)(−3)(4)
= (12) Get the product of all the integers.
= −𝟏𝟐 Since the number of negative factors is odd, the
product is negative.

Division of Integers
Multiplication and division are relation operations. Therefore, the rules in finding the quotient of integers
are same as the rules in finding the product of integers.

Rules for Subtracting Integers

• When dividing two numbers with the same sign, the quotient is always positive.
(+) (−)
(+)
= (+) (−)
= (+)
• When dividing two numbers with different sign, the quotient is always negative.
(+) (−)
(−)
= (−) (+)
= (−)
• When dividing a number by zero, and dividing zero by a number, we have
these rules:
(𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟) (0)
(0)
= undefined (𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟)
=0

33 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Example 15
Find the quotient of the following.
125 −78 144 −225 0 −121
a. 5
b. −13 c. −12 d. 25
e. 7 f. 0

Solution:
125
a. 5
125
= 5 Get the quotient of the two integers.
= 𝟐𝟓 Since the integers have the same sign, the quotient is
positive.

−78
b. −13
−78
= −13 Get the quotient of the two integers.
=𝟔 Since the integers have the same sign, the quotient is
positive.

144
c. −12
144
= Get the quotient of the two integers.
−12
= −𝟏𝟐 Since the integers have different sign, the quotient is
negative.
−225
d. 25
−225
= 25 Get the quotient of the two integers.
= −𝟗 Since the integers have different sign, the quotient is
negative.

0
e. 7
0
=7 Get the quotient of zero and the integer.
=𝟎 Since zero is divided by the integer, then the quotient
is zero.

−121
f. 0
−121
= 0 Get the quotient the integer and zero.
= 𝐮𝐧𝐝𝐞𝐟𝐢𝐧𝐞𝐝 Since the integer is divided by zero, the quotient is
undefined.

Series of Operations
In performing series of operations in mathematics, there is an agreed order of operations to follow so that
there will be a standard answer.

34 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Rules for Subtracting Integers

1. Perform all the operations within the any grouping symbol starting with the innermost pair.
2. Perform all multiplication and division in the order they appear in the expression from left to
right.
3. Perform all addition and subtraction in the order they appear in the expression form left to right.
4. If the expression involves a fraction bar, perform all operation above and below the bar
separately following the steps 1 to 3, then simplify.
5. In the absence of grouping symbols, when both multiplication or division and addition or
subtraction symbols appear, perform multiplication or division first before addition or
subtraction.

Examples 16
Perform the indicated operations.
a. 5 − 3(6 + 4)
b. 160 ÷ 2[8{5(4 − 6)}]
−36+8×2−(−5)(−2)
c.
−12−(−4)+8(−5)+33

Solution:
a. 5 − 3(6 + 4)
= 5 − 3(𝟏𝟎) Step 1
= 5 − 𝟑𝟎 Step 2
= −𝟐𝟓

b. 160 ÷ 2[8{5(4 − 6)}]

= 160 ÷ 2[8{5(−2)}] Step 1
= 160 ÷ 2[8{−10}]
= 160 ÷ 2[−80] Step 2
= 160 ÷ −160
= −𝟏

−36+8×2−(−5)(−2)
c.
−12−(−4)+8(−5)+33
−36+8×2−(−5)(−2)
= −12−(−4)+8(−5)+33 Step 1
−36+16+5×(−2)
= Step 2
−12+4+40+33
−36+16−10
= −15
Step 3
−30
=
−15
=𝟐

35 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Properties of the Operations of Integers

Let 𝒂, 𝒃, and 𝒄 be integers.

Table 2.1
Properties of Addition of Integers
Property Meaning Examples
The sum of two integers is an
Closure Property of Addition 12 + (−7) = 5
integer.
Changing the order of the addends
15 + (−6) = (−6) + 15
Commutative Property of Addition does not affect the sum.
20 = 20
𝒂+𝒃=𝒃+𝒂
Changing the grouping of the (11 + 15) + (−7) = 11 + [15 + (−7)]
Associative Property of Addition addends does not affect the sum (11 + 15) + (−7) = 11 + [15 + (−7)]
(𝒂 + 𝒃) + 𝒄 = 𝒂 + (𝒃 + 𝒄) 19 = 19
The sum of an integer and zero is the
Identity Property of Addition integer itself. 9+0 =9
𝒂+𝟎 =𝒂
The sum of nonzero integer and its
Inverse Property of Addition additive inverse is zero. 6 + (−6) = 0
𝒂 + (−𝒂) = 𝟎
Properties of Multiplication of Integers
The product of two integers is an
Closure Property of Multiplication integer. (11)(−7) = −77
𝒂𝒃 = 𝒄
Changing the order of the factors
Commutative Property of (13)(−7) = (−7)(13)
does not affect the product.
Multiplication −91 = −91
𝒂𝒃 = 𝒃𝒂
Changing the grouping of the factors
[4(−12)]3 = 4[(−12)3]
Associative Property of Multiplication does not affect the product.
−144 = −144
(𝒂𝒃)𝒄 = 𝒂(𝒃𝒄)
The product of an integer and 1 is
Identity Property of Multiplication the integer itself. −16 ⋅ 1 = −16
𝒂⋅𝟏=𝒂
The product of an integer and its
multiplicative inverse is 1. 1
Inverse Property of Multiplication 3⋅ =1
𝟏 3
𝒂 ⋅ = 𝟏, 𝒂 ≠ 𝟎
𝒂
Multiplying an integer by a sum is
−4(2 + 3) = (−4)(2) + (−4)(3)
the same as multiplying the integer
Distributive Property of Multiplication −4(5) = (−8) + (−12)
by each addend.
−20 = −20
𝒂(𝒃 + 𝒄) = 𝒂𝒃 + 𝒃𝒄
The product of any integer and zero
Multiplicative Property of Zero is zero. −19 ⋅ 0 = 0
𝒂⋅𝟎=𝟎
The product of any integer and −1 is
Multiplicative Property of −1 the opposite of the integer. (−1)14 = −14
(−𝟏)𝒂 = −𝒂

Square Roots
In exponential notation, if let 𝒎 be a number, and square it, the expression can be written as 𝑚2 .

36 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Square of Number
If 𝒎𝟐 = 𝒃, then the square of a number is 𝒃.

An exponential notation has two parts: the base and the exponent. In the expression 52
exponent
base 𝟓𝟐

A perfect square is the square of a whole number. Since 25 is a result of squaring a number, it is an example
of a perfect square number. The table shows the list of the ten perfect squares:
Table 7.2
Perfect Square Factored Form Square Root
2
1 1 = (1)(1) 1
2
4 2 = (2)(2) 2
2
9 3 = (3)(3) 3
2
16 4 = (4)(4) 4
2
25 5 = (5)(5) 5
2
36 6 = (6)(6) 6
2
49 7 = (7)(7) 7
2
64 8 = (8)(8) 8
2
81 9 = (9)(9) 9
2
100 10 = (10)(10) 10

The square root of a number is one of the two equal factors of the perfect square. The square root of 25 is
5 since (5)(5) = 25. (−5)(−5) = 25, then −5 is also a square root of 25. Every nonzero real number has
two square roots, positive and negative. The positive square root of is also called the principal square root
of a nonzero real number. On the other hand, the square root of zero is 0.

Square Root Notation

index

√𝑏 is read as “square root of 𝑏”. 𝒏

If 𝑏 = 𝑚2 then √𝑏 = 𝑚 for 𝑚 ≥ 0.
√𝒃 radical sign

radicand
Sign

The radical sign √ is use to denote that the root of a number that must be extracted. The mathematical
expression using a radical sign is called a radical expression. The index 𝒏 is a positive integer that is greater
than 𝟏 which indicates the order of the radical or which 𝑛𝑡ℎ root is required. The radicand 𝑏 is the expression
under the radical sign.

Example 17
Find the square root of the following.
a. √121 b. √225 c. √400

37 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Solution:
e. √121
Since (11)(11) = 121; (−11)(−11) = 121
The roots are 𝟏𝟏 and −𝟏𝟏.

f. √225
Since (15)(15) = 225; (−15)(−15) = 225
The roots are 𝟏𝟓 and −𝟏𝟓.

g. √400
Since (20)(20) = 400; (−20)(−20) = 400
The roots are 𝟐𝟎 and −𝟐𝟎.

The square root of a positive integer is either rational or irrational. If the radicand is a perfect square, then
the square root is rational, otherwise it is irrational. Irrational numbers when written in decimals are
nonterminating and nonrepeating decimals.

Example 18
Determine whether each square root is rational or irrational.
a. √144
b. √13
Solution:
a. √144
Since 144 is a perfect square number, it is rational.
√144 = ±𝟏𝟐
b. √13
Since 13 is not a perfect square number, it is irrational.

Irrational numbers, when written in decimal form are only approximate values.

Approximating Square Roots

Example 19
Approximate the square root of the following and plot on the number line using the principal root.
a. √13
b. √47
Solution:
a. √13
Step 1. Find two perfect square number near √13.
√9 < √13 < √16
3 < √13 < 4
The approximate value of √13 is between the integers 3 and 4.
Approximate location of √13 on the number line.

38 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Fig. 2.6

b. √47
Step 1. Find two perfect square number near √47.
√36 < √47 < √49
6 < √47 < 7
The approximate value of √47 is between the integers 6 and 7.
Approximate location of √47 on the number line.

Fig. 2.7

Example 20
Approximate the square root of the following to the nearest hundredths and then plot on the number line.
a. √17
b. √68

Solution:
a. √17
√16 < √17 < √25
4 < √17 < 5

Step 1. Estimate
√17 is approximately 5.
Step 2. Divide the radicand by the result of Step 1.
17
5
= 3.4
Step 3. Get the average of the results of Step 1 and Step 2.
5+3.4
2
= 4.2

The approximate value of √17 to the nearest hundreths is 4.2. If you want to get a more accurate result,
you can repeat the steps several times. This time, your first divisor will be the average from the Step 3.
Thus,
17 4.2+4.0476 8.2476
4.2
≈ 4.0476 2
= 2
≈ 4.1238 𝑜𝑟 𝟒. 𝟏𝟐

39 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Fig. 2.9

Solution:
b. √68
√64 < √68 < √81
8 < √68 < 9

Step 1. Estimate
√68 is approximately 9.
Step 2. Divide the radicand by the result of Step 1.
68
9
= 7.5556
Step 3. Get the average of the results of Step 1 and Step 2.
9+7.5556
2
= 8.2778

Getting a more accurate value,

68 8.2778+8.2147 16.4925
8.2778
≈ 8.2147 2
= 2
≈ 8.2463 𝑜𝑟 𝟖. 𝟐𝟓

Fig. 2.10

40 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Mental Math 1
Put a check mark under each subset of real numbers that applies for each given
number.
Number Real Rational Irrational Integers Whole Natural
1. −44.82
2. √32
̅̅̅̅
3. 34. 62

4. √121
3
5. − 4
6. 0
7. −4

Mental Math 2
Locate and plot the following set of numbers on a number line.
12
8. {2.05, −7, , √49, −1.8}
7

15 1
9. {−0.78, √14 , −5 2 , 6.75}
7

2
̅̅̅̅, √13}
10. {0.03, −√16, 3 5 , −5. 33

Mental Math 3
Perform the indicated operations.
1 6 1 1
11. (− 2 + 8) + (− 2 + 4)

12. |−43| − (61) − (−|15|)

13. 16(−0.25)(0.25)(20)(−20)

14. [(−8)(−4) − (−0.8)] ÷ (−8)

15. 80 ÷ (−4) × (−5)

41 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

TEACHER’S NOTE

I hope that you have answered the exercises without looking at your notes. Once
you’re done, refer to the answer key on the last part of this module to check your answers.
If you get a score of 12 and above, congratulations for a job well done; if not, there is
always a chance next time. You may visit the following links to further understand the topic:

• Number sets | Fractions | Pre-Algebra | Khan Academy

https://www.youtube.com/watch?v=Ksu1lo312BM
• What are Real Numbers? | Don't Memorise
https://www.youtube.com/watch?v=3YwrcJxEbZw
• Trick to Identify Terminating Rational Numbers & Non-Terminating
Recurring Decimals | Don't Memorise
https://www.youtube.com/watch?v=rVhU8Vyhz7c
• Categories of the Real Numbers | Don't Memorise
https://www.youtube.com/watch?v=IueVrMlmQ2I
• How To Find The LCM and HCF Quickly!
https://www.youtube.com/watch?v=fjdeo6anRY4
• Number Systems | Operations On Real numbers | Maths | Letstute
https://www.youtube.com/watch?v=UJZyx12Odh8
• Integers - Multiplication, Division, Adding, Subtracting, Rules, Positive &
Negative Numbers - Math
https://www.youtube.com/watch?v=nZMMTK8aJKI
• How do we Add Two Rational Numbers? Part 1 | Don't Memorise
https://www.youtube.com/watch?v=YVerO8RvBGI
• Rational Numbers on a Number Line - Part 2 | Don't Memorise
https://www.youtube.com/watch?v=WynEmwOyMjE
• Rational Numbers on a Number Line - Part 3 | Don't Memorise
https://www.youtube.com/watch?v=SfhvqcG2aJc
• What are Irrational Numbers? | Number System | Don't Memorise
https://www.youtube.com/watch?v=CtRtXoT_2Ps
• What are Square Roots? | Exponents | Best Square Root Tricks | Don't
Memorise
https://www.youtube.com/watch?v=w8vqUMIVT1c

42 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

MATH CHALLENGE

Math Challenge 1.3 (Properties of Real Numbers)

Coverage: Set of Real Numbers
Comparing and Ordering Numbers
Objective of the Activity: Answer and solve a 20-item problem set
involving the properties of real numbers.
*to be given after the discussion of subtopics 2.1 to 1.2

Math Challenge 1.4 (Problems Involving Real Numbers)

Coverage: Operations on Real Numbers
Objective of the Activity: Answer and solve a 30-item problem set
involving operations on real numbers.
*to be given after the discussion of subtopics 2.3

BYJU’s. (2021, March 22). General data protection Regulation (GDPR) guidelines BYJU’S. BYJUS.
https://byjus.com/maths/integers/

Cool math. (2019). Decimals - cool math Pre-Algebra help lessons - how to convert fractions to decimals.
CoolMath.Com. https://www.coolmath.com/prealgebra/02-decimals/04-decimals-converting-fraction-
to-decimal-01

Crisostomo, R. M., & Padua, A. L. (2018). Our world of math grade 7 (2nd ed.). Vibal Group Inc.

Explore the Landscape of Number System and Divisibility Rules. [Picture].

https://eduthrill.com/explore-the-landscape-of-number-system-and-divisibility-rules/

Giomini, M., 2020. Day 50: Multiplying Fractions / Ordering Rational Numbers / Dividing Decimals.
[online] Mrgiomini.blogspot.com. Available at: <http://mrgiomini.blogspot.com/2015/10/day-50-
multiplying-fractions-ordering.html> [Accessed 8 September 2020].

Integer number line (video lessons, examples and solutions). (2020). Www.Onlinemathlearning.Com.
https://www.onlinemathlearning.com/integer-number-line.html

43 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

Lim, Y., Nocon, R., Nocon, E., & Ruivivar, L. (2017). Math for engaged learning (2nd ed.). Sibs Publishing
House, Inc.

Lopez, E. A., & Lopez, V. L. (2013). New mathematics for grade 7. Light Bearers Publishing House.

math.com. (2005). Numbers - dividing decimals - examples. Math.Com The World of Math Online.
http://www.math.com/school/subject1/lessons/S1U1L6EX.html

Mendoza & Oronce, (2012). E-Math. Philippines: Rex Book Store, Inc.

Orines, et.al, (2017). Next Century Mathematics 7. Philippines: Phoenix Publishing House

Solving Math Problems with Number Lines on the SAT. (2014, April 3). Retrieved from
https://study.com/academy/lesson/solving-math-problems-with-number-lines.html.

Tuazon, et al., (2017). iMath 7 K to 12 Curriculum Series. Cainta, Rizal Philippines: iBook Publishing, Inc.

Virtual Nerd. (2020). How do you turn a fraction into a terminating decimal? | virtual nerd. Retrieved
August 10, 2021, from https://virtualnerd.com/common-core/grade-7/7_NS-number-
system/A/2/2d/fraction-to-terminating-decimal-conversion

Module Creator/Curator :
Ms. Marie Mar F. Fegalan
Ms. Marjorie Mae M. Hernandez, LPT
Ms. Jessa C. Luansing, LPT
Ms. Maryrose Lizette A. Reyes, LPT
Template & Layout Designer : Ms. Jessa C. Luansing, LPT
Ms. Maryrose Lizette A. Reyes, LPT
Ms. Sherline A. Villanueva, LPT

44 | M a t h e m a t i c s 7
 Lesson 2: Real Number System

ANSWER KEY

Number Real Rational Irrational Integers Whole Natural

1. −44.82

2. √32
3. 34. ̅62
̅̅̅

4. √121
3
5. − 4
6. 0

7. −4

8.

9.

10.

11. 0
12. −3
13. −40
401
14. − 100 𝑜𝑟 − 4.01
15. 100

45 | M a t h e m a t i c s 7