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T.L.E 7
INFORMATION AND
COMMUNICATION
TECHNOLOGY
FIRST QUARTER
REVIEW
In the world of computers,
numbers are represented using
different systems. The most
common system is the binary
system, which uses only the digits
0 and 1 to represent all numbers
and data. Each digit in a binary
number is called a bit.
Another important system is the
hexadecimal system, which uses
the digits 0-9 and letters A-F to
represent numbers. Computers
also use the octal system, which
uses digits 0-7.
What is number system?
A number system is a method of
representing numbers using
specific rules. It provides a
consistent way to express
numerical values. In
mathematics, we use various
number systems to represent and
manipulate numbers. Here are
the four most common types:
A number system is a method of
representing numbers using
specific rules. It provides a
consistent way to express
numerical values. In
mathematics, we use various
number systems to represent and
manipulate numbers. Here are
the four most common types:
Here are the four most common
types:
Decimal Number System
(Base-10):
• The decimal system uses ten
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
• Each position to the left of the
decimal point represents powers
of 10 (units, tens, hundreds,
thousands, etc.).
Binary Number System (Base-2):
• The binary system uses only
two digits: 0 and 1.
• It is widely used in computer
science and digital
electronics.
Octal Number System (Base-8):
• The octal system uses eight
digits: 0, 1, 2, 3, 4, 5, 6, and 7.
• It is less common but still used
in some contexts.
Hexadecimal Number System
(Base-16):
• The hexadecimal system uses
sixteen digits: 0–9 and A–F
(where A represents 10, B
represents 11, and so on).
• It is commonly used in
computer programming and
memory addressing.
Computer Number
Systems and Conversion
of Computer Number
Systems
LEARNING COMPETENCIES
• Differentiate the various
computer number systems
• Apply conversion of
computer number systems
OBJECTIVES
• Explain the basics of number
systems and their
significance in computing.
• Identify the steps in the
conversion of Decimal to
Binary.
• Convert Decimal to Binary
OBJECTIVES
• Identify steps in the
conversion of binary to
decimal.
• Convert Binary to Decimal.
• Identify steps in the
conversion of Decimal to
Octal.
OBJECTIVES
• Convert Decimal to Octal.
• Identify steps in the
conversion of Octal to
Decimal.
• Convert Octal to Decimal
The number system plays a
crucial role in mathematics and
our daily lives.
1. Representation of Quantities:
• A number system provides a
way to represent quantities.
Whether it’s counting objects,
measuring distances, or
calculating time, numbers
allow us to express these
concepts precisely.
1. Representation of Quantities:
• For example, when you count
the number of apples in a
basket or measure the length
of a room, you’re using the
number system.
2.Foundation for Mathematical
Concepts:
• All mathematical concepts
and formulas are based on
the number system. Whether
you’re solving equations,
working with geometry, or
analyzing data, numbers are
fundamental.
2.Foundation for Mathematical
Concepts:
• From basic arithmetic
operations (addition,
subtraction, multiplication,
division) to advanced calculus
and algebra, numbers
underpin mathematical
reasoning.
3. Types of Numbers:
✧
The number system
encompasses various types of
numbers:
1.Counting Numbers: These start
with 1 and continue indefinitely
(1, 2, 3, …).
2.Whole Numbers: Include all
counting numbers along with
zero (0, 1, 2, …).
3.Integers: Positive and negative
whole numbers, including zero (-
3, -2, -1, 0, 1, 2, 3, …).
4. Rational Numbers: Expressible
as fractions (e.g., 3/4, -2/5).
5. Irrational Numbers: Cannot be
expressed as fractions (e.g., √2,
π).
6. Real Numbers: Encompass
both rational and irrational
numbers.
7. Even Numbers: Divisible by 2
(e.g., 2, 4, 6, …).
8. Odd Numbers: Not divisible by
2 (e.g., 3, 5, 7, …).
9. Prime Numbers: Divisible only
by 1 and themselves (e.g., 5, 7,
13).
10. Composite Numbers: Have
multiple factors (e.g., 10, 15, 28).
4. Digital Systems and Data
Representation:
✧
• Understanding number
systems is essential for digital
systems (like computers)
because they process data
using binary representation
(base-2).
4. Digital Systems and Data
Representation:
• Computers use bits (0s and 1s)
to represent information, and
this binary system relies on the
principles of the number
system.
4. Digital Systems and Data
Representation:
• The number system is not only
a mathematical tool but also
a fundamental aspect of our
everyday interactions with the
world. It allows us to quantify,
calculate, and communicate
effectively.
Unlocking of Difficulties
Number system - is a mathematical
way of representing a set of values
using digits or symbols.
Decimal - is a number that consists
of a whole part and a fractional
part separated by a decimal point.
Unlocking of Difficulties
Binary - is a number expressed in
the base-2 numeral system, in this
system, we use only two symbols:
typically, "0" (zero) and "1" (one).
Octal - is a type of numeral system
that uses a base of eight, in this
system, the digits range from 0 to 7.
Unlocking of Difficulties
Hexadecimal - is a base-16
numeral system. Unlike our
everyday decimal system (base
10), which uses ten symbols (0-9),
hexadecimal employs sixteen
symbols. These symbols represent
values from 0 to 15.
Binary Numbers
The binary number system is a
fundamental concept in
computer science and digital
electronics. It uses a base-2
numeral system, which means it
only employs two distinct
symbols: 0 (zero) and 1 (one).
Here are the key points about
binary numbers:
Representation:
10. In binary, each digit is called
a bit.
11. The binary system is used
internally by almost all modern
computers and electronic
devices because it directly
maps to electronic circuits using
logic gates.
Representation:
12. Unlike our everyday decimal
system (base 10), which uses ten
symbols (0-9), binary uses only
two symbols (0 and 1).
Conversion:
13. To convert a decimal
number to binary, follow these
steps:
14. Divide the decimal number
by 2.
15. Use the integer quotient
obtained as the dividend for the
next step.
Conversion:
16. Continue dividing until the
quotient becomes 0.
17. Write down the remainders
in reverse order to get the
binary representation.
Example: Convert 4 to Binary:
18. Let's convert the decimal
number 4 to binary:
Step 1: Divide 4 by 2.
Quotient: 2, Remainder: 0
Step 2: Divide 2 by 2.
Quotient: 1, Remainder: 0
Example: Convert 4 to Binary:
Step 3: Divide 1 by 2.
Quotient: 0, Remainder: 1
Reverse the remainders: 100
19. Therefore, 4 in binary is 100.
Bit in Binary Number:
20. A single binary digit is called
a bit.
Examples:
21. 10101 is a five-bit binary
number.
Bit in Binary Number:
22. 101 is a three-bit binary
number.
23. 100001 is a six-bit binary
number.
QUESTIONS:
1. How many different digits are
used in the binary number
system?
2. Can you explain the concept
of place value in binary
numbers?
QUESTIONS:
3. Why is the binary number
system important in computing
and digital technology?
Worked Example
DIRECTION
identify what type of the
number system.
1. 78 - ____________
2. 100112 - ____________
3. F - _______
4. 328 - __________
5. 1110102 - ___________
ACTIVITY 1
Computer Number System
OBJECTIVES
At the end of the activity, you are
expected to:
• Understand the concept of
number systems used in
computers.
• Differentiate between the
decimal, binary, octal, and
hexadecimal number systems.
OBJECTIVES
• Convert numbers from one
system to another.
DIRECTIONS:
A. Word Completion –
Converting Decimal to Binary
Directions: Supply the missing
word to complete the Steps in
Conversion of Binary to Decimal.
1. Divide the number by ______.
2. Write the quotient and the
_____on its corresponding _____.
3. Get the quotient and divide it
again by __________. Write the
quotient and remainder in its
column.
4. Continue dividing until the
quotient results to _________.
Always write its quotient and
remainder in their column.
5. Copy the remainder from the
bottom to ___________. That
would be the binary equivalent
of the decimal number.
DIRECTIONS:
B. Decimal and Binary
Directions: Convert each
decimal number to its binary
equivalent, binary to decimal
number.
conversion of number system ng meaurement
Octal Numbers
The octal number system is a
base-8 system that uses digits
from 0 to 7 to represent any
number. It is similar to other
number systems like binary and
hexadecimal. Here are some
key points about octal numbers:
Base and Digits:
29. The base of the octal system
is 8.
30. It uses the digits 0, 1, 2, 3, 4,
5, 6, and 7.
Representation:
31. Octal numbers are usually
represented by binary numbers
when grouped in pairs of three.
32. For example, the octal number
128 is expressed as 0010102 in
binary, where 1 is equivalent to 001
and 2 is equivalent to 010.
Conversion:
33. To convert decimal numbers
to octal, we use the octal
dabble method.
34. For example, to convert
56010 to octal:
Conversion:
35. 560 ÷ 8 = 70 with a
remainder of 0
36. 70 ÷ 8 = 8 with a remainder
of 6
Conversion:
37. 8 ÷ 8 = 1 with a remainder of 0
38. 1 ÷ 8 = 0 with a remainder of 1
39. So, the octal representation is
10608.
Table of Octal Digits
40. We use only 3 bits to
represent octal numbers.
41. Each group has a distinct
value between 000 and 111.
42. Octal digits range from 0 to 7.
Table of Octal Digits
Octal numbers are commonly
used in computer programming
because they provide a
compact way to represent
binary numbers (each octal
digit corresponds to three
binary digits).
QUESTIONS:
1. How many symbols are used in
octal numbering system?
2. Can you explain the concept
of place value in octal numbers?
3.Why are octal numbers often
used in computer programming?
ACTIVITY 2
OBJECTIVES
At the end of the activity, you are
expected to:
• Understand the importance of
binary numbers in computing.
• Learn the step-by-step process
of converting decimal numbers
to binary and binary to
decimal.
OBJECTIVES
At the end of the activity, you
are expected to:
• Practice conversions through
guided activities.
DIRECTIONS:
A. Complete the sentence.
Directions: Fill in the blank with
the correct words.
1. To convert a decimal number
to octal, we first need to _______
the decimal number by 8.
2. The _______ of the division is
the first digit of the octal
number.
3. The __________ of the division is
used for the next step.
4. This process of dividing by 8
and using the remainder
continues until the _________
becomes 0.
5. The octal number is formed by
writing the remainder in
________order.
Decimal Numbers
A decimal number is a number
that consists of a whole number
and a fractional part separated
by a point (often called the
decimal point).
Whole Number Part: The whole
number part represents the
integral value of the quantity. It
can be any positive or negative
integer.
For example:
•In the decimal number 42.85,
the whole number part is 42.
•In the decimal number -123.456,
the whole number part is -123.
Fractional Part: The fractional part
represents a value smaller than
one. It appears after the decimal
point.
For example:
•In the decimal number
42.85, the fractional part is 0.85.
•In the decimal number -123.456,
the fractional part is 0.456.
Decimal Point: The decimal
point separates the whole
number part from the fractional
part. It serves as a visual
indicator that we are dealing
with a decimal.
For example:
47.The decimal point in 42.85
separates 42 (whole) from 85
(fractional).
48.The decimal point in -123.456
separates -123 (whole) from 456
(fractional).
Decimal numbers allow us to
express quantities that are
whole plus some part of a
whole. They are commonly
encountered in everyday
situations, such as grocery
shopping, measurements, and
financial calculations.
QUESTIONS:
1. What is the definition of a
decimal number?
2. Can you explain how decimal
numbers are different from
whole numbers?
3. How are decimal numbers
represented on a number
line?
Worked Example
Decimal numbers are an
important part of mathematics
and are used in many everyday
situations. They represent values
that are between whole numbers,
allowing for more precise
measurements and calculations.
In this worksheet, you will practice
working with decimal numbers
through an activity.
QUESTIONS:
1. Explain the difference
between a whole number
and a decimal number.
2. Describe the steps to round a
decimal number to the
nearest hundredth.
QUESTION:
3. How would you add two
decimal numbers with
different numbers of digits to
the right of the decimal
point?
ACTIVITY 3
OBJECTIVES
At the end of the activity, you
are expected to:
• Understand the decimal and
octal number systems.
• Learn the process of
converting decimal numbers
to octal and vice versa.
DIRECTIONS:
B. Decimal and Octal
Directions: Converting Decimal
numbers to Octal and Octal to
Decimal Numbers. Choose the
correct letter of the answer.
1.What is the decimal equivalent
of the octal number 64?
a. 4 b. 14 c. 52 d. 100
2. If an octal number is 127, what
is its value in decimal?
a. 16 b. 55 c. 87 d. 102
3. Convert the octal number 345
to decimal.
a. 125 b. 197 c. 209 d.
229
4. What is the decimal value of
the octal number 777?
a. 9 b. 383 c. 504 d. 511
5. If an octal number is 476, what
is its decimal equivalent?
a. 312
b. 302
c. 256
d. 318
Hexadecimal
Number
The hexadecimal number
system often simply called
“hex,” is a base-16 numeral
system. Unlike our familiar base-
10 (decimal) system, which uses
ten symbols (0 to 9) to represent
numbers, hexadecimal employs
sixteen symbols.
These symbols include the
numerals 0 to 9 and the letters A
to F. Each digit in a
hexadecimal number
corresponds to a specific value,
allowing us to express quantities
in a compact and efficient
manner.
Base-16 Representation:
Hexadecimal uses a base value
of 16. This means that each
column in a hexadecimal
number represents a power of
16.
Base-16 Representation:
•The sixteen symbols used in
hexadecimal are: 0, 1, 2, 3, 4, 5,
6, 7, 8, 9, A, B, C, D, E, and F.
For example, the hexadecimal
number “1A” represents 1 × 16 +
10 = 26 in decimal.
Conversion to Other Systems:
• Hexadecimal numbers can be
converted to other number
systems:
• To decimal (base-10): Each
digit is multiplied by the
corresponding power of 16 and
summed.
Conversion to Other Systems:
• To binary (base-2): Each
hexadecimal digit corresponds
to a 4-bit binary representation.
• To octal (base-8): First convert
to decimal, then to octal.
• These conversions are essential
in computer science and
digital systems.
Hexadecimal provides a
convenient way to work with
binary data and efficiently
represent values in various
contexts.
Worked Example
Example:
Convert the decimal number
456 to hexadecimal
1. Divide the decimal number
by 16 and record the quotient
and remainder.
2. Repeat the division using the
quotient from the previous step
until the quotient is zero.
Example:
Convert the decimal number
456 to hexadecimal
3. The hexadecimal number is
formed by the remainders read
from bottom to top.
Steps:
1. Initial Number: 456
2. First Division:
• Divide 456 by 16.
•Quotient: 456÷16=28456 div
16 = 28456÷16=28 (integer part)
•Remainder: 456 mod
16=8456 mod 16 =
8456mod16=8
Steps:
3. So, 456 divided by 16 gives a
quotient of 28 and a remainder of
8.
4.Second Division:
• Now divide the quotient from
the first division (28) by 16.
• Quotient: 28÷16=128 div 16 =
128÷16=1 (integer part)
Steps:
•Remainder: 28 mod 16=1228 
mod 16 = 1228mod16=12
5.So, 28 divided by 16 gives a
quotient of 1 and a remainder of
12. Note that in hexadecimal, 12
is represented as 'C'.
Steps:
6. Third Division:
•Now divide the quotient from
the second division (1) by 16.
•Quotient: 1÷16=01 div 16 =
01÷16=0 (integer part)
•Remainder: 1mod 16=11 
mod 16 = 11mod16=1
Steps:
7.So, 1 divided by 16 gives a
quotient of 0 and a remainder of
1.
8.Construct the Hexadecimal
Number:
• Write the remainders from the
last to the first.
Steps:
• The remainders are 1, 12 (C in
hexadecimal), and 8.
9.Therefore, the hexadecimal
representation of the decimal
number 456 is 1C8.
SUMMARY
• Divide the decimal number by
16.
• Write down the remainder.
• Use the quotient for the next
division.
• Repeat until the quotient is
zero.
• The hexadecimal number is the
remainders read from bottom
to top.
ACTIVITY 4
OBJECTIVES
At the end of the activity, you
are expected to:
• Understand the decimal and
hexadecimal number
systems.
OBJECTIVES
At the end of the activity, you
are expected to:
• Learn the process of
converting decimal numbers
to hexadecimal and
hexadecimal to decimal
numbers.
DIRECTIONS:
Decimal Number System
Directions:
Choose the correct letter of
your answer.
1. What is the place value of the
digit 5 in the number 6.573?
a. Thousandths c. Units
b. Tenths d. Hundreds
2. Which of the following is
equivalent to 3.62?
a. 362 b. 36 c. 3.6 d. 0.362
3. What is 0.45 written as a
fraction in simplest form?
a. 45/100 c. 9/5
b. 9/20 d. 4/5
4. Which of the following
numbers is the largest?
c. 7.321 c. 732.1
d. 73.12 d. 73.21
5. If you add 2.5 and 1.75, what is
the sum?
a. 4.25 c. 4.75
b. 4.5 d. 5.25
Directions:
Converting Decimal Numbers to
Hexadecimal and Hexadecimal
to Decimal.
1. What is the hexadecimal
equivalent of the decimal
number 27?
a. 1B b. 1F c. 23 d. 2E
2. Which of the following is a
correct hexadecimal
representation of the decimal
number 50?
a. 30 b. 32 c. 35 d. 38
3. If the hexadecimal number 3D
is converted to decimal, what is
the result?
a. 53 b. 61 c. 57 d. 59
4. What is the decimal value of
the hexadecimal number A4?
a. 161 b. 164 c. 166 d. 168
5. Convert the decimal number
95 to hexadecimal.
a. 5D b. 60 c. 61 d. 65
ANSWER
1.a. 1B
2.b. 32
3.d. 59
4.c. 166
5.a. 5D
WRAP-UP QUESTIONS:
1. What are the importance of
the different computer
number systems?
2. How does the concept of the
different computer number
systems and the conversion of
decimal number to binary,
octal, and hexadecimal?
REFLECTION
1. How do you find today’s
lesson? Was it hard? Which part
of the lesson do you find difficult?
POST-TEST
POST-TEST
1.Which of the following best
describes the binary number
system?
a) A number system that uses 0-9
b) A number system that uses 0-7
c) A number system that uses 0-1
d) A number system that uses o-F
2.What is the first step in
converting a decimal number to
octal?
a) Divide the decimal number
by 8
b) Write the decimal number in
reverse order
POST-TEST
2.What is the first step in
converting a decimal number to
octal?
c) Multiply the decimal number
by 8
d) Add 8 to the decimal number
POST-TEST
2. What is the primary focus of
the Visual Arts sector in ICT?
c. Maintaining and repairing
computer systems
d. Transmitting information over
distances
POST-TEST
3. What do you do with the
remainder of each division step?
a) Discard it
b) Add it to the next division
c)Use it to form the octal
number
d) Multiply it by 8
POST-TEST
4. How do you determine the
digits of the octal number?
a) The quotients of the divisions
b) The remainders of the
divisions
c) The sum of the divisions
d) The difference between the
divisions
POST-TEST
5. What happens when the
divisor becomes 0?
a) The process is complete
b) You start over with a new
decimal number
POST-TEST
5. What happens when the
divisor becomes 0?
c) You add 8 to the octal
number
d) You multiply the octal
number by 8
POST-TEST
6. In what order should the digits
of the octal number be written?
a) In the order they were
generated
b) In reverse order
c) From largest to smallest
d) From smallest to large
POST-TEST
7. How many unique digits are
used in the octal number
system?
a) 2 b) 8 c) 10 d) 16
8.What is the decimal equivalent
of the binary number 1010?
a) 5 b) 8 c) 10 d) 15
POST-TEST
9. Which of the following is the
hexadecimal representation of
the decimal number 27?
a) 1B b) 1F c) 1A d) 1C
10.What is the octal equivalent
of the binary number 101011?
a) 43 b) 53 c) 63 d) 73
POST-TEST
ANSWER
POST-TEST
1.Which of the following best
describes the binary number
system?
a) A number system that uses 0-9
b) A number system that uses 0-7
c) A number system that uses 0-1
d) A number system that uses o-F
2.What is the first step in
converting a decimal number to
octal?
a) Divide the decimal number
by 8
b) Write the decimal number in
reverse order
POST-TEST
2.What is the first step in
converting a decimal number to
octal?
c) Multiply the decimal number
by 8
d) Add 8 to the decimal number
POST-TEST
3. What do you do with the
remainder of each division step?
a) Discard it
b) Add it to the next division
c)Use it to form the octal
number
d) Multiply it by 8
POST-TEST
4. How do you determine the
digits of the octal number?
a) The quotients of the divisions
b) The remainders of the
divisions
c) The sum of the divisions
d) The difference between the
divisions
POST-TEST
5. What happens when the
divisor becomes 0?
a) The process is complete
b) You start over with a new
decimal number
POST-TEST
5. What happens when the
divisor becomes 0?
c) You add 8 to the octal
number
d) You multiply the octal
number by 8
POST-TEST
6. In what order should the digits
of the octal number be written?
a) In the order they were
generated
b) In reverse order
c) From largest to smallest
d) From smallest to large
POST-TEST
7. How many unique digits are
used in the octal number
system?
a) 2 b) 8 c) 10 d) 16
8.What is the decimal equivalent
of the binary number 1010?
a) 5 b) 8 c) 10 d) 15
POST-TEST
9. Which of the following is the
hexadecimal representation of
the decimal number 27?
a) 1B b) 1F c) 1A d) 1C
10.What is the octal equivalent
of the binary number 101011?
a) 43 b) 53 c) 63 d) 73
POST-TEST

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conversion of number system ng meaurement

  • 3. In the world of computers, numbers are represented using different systems. The most common system is the binary system, which uses only the digits 0 and 1 to represent all numbers and data. Each digit in a binary number is called a bit.
  • 4. Another important system is the hexadecimal system, which uses the digits 0-9 and letters A-F to represent numbers. Computers also use the octal system, which uses digits 0-7.
  • 5. What is number system?
  • 6. A number system is a method of representing numbers using specific rules. It provides a consistent way to express numerical values. In mathematics, we use various number systems to represent and manipulate numbers. Here are the four most common types:
  • 7. A number system is a method of representing numbers using specific rules. It provides a consistent way to express numerical values. In mathematics, we use various number systems to represent and manipulate numbers. Here are the four most common types:
  • 8. Here are the four most common types: Decimal Number System (Base-10): • The decimal system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. • Each position to the left of the decimal point represents powers of 10 (units, tens, hundreds, thousands, etc.).
  • 9. Binary Number System (Base-2): • The binary system uses only two digits: 0 and 1. • It is widely used in computer science and digital electronics.
  • 10. Octal Number System (Base-8): • The octal system uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. • It is less common but still used in some contexts.
  • 11. Hexadecimal Number System (Base-16): • The hexadecimal system uses sixteen digits: 0–9 and A–F (where A represents 10, B represents 11, and so on). • It is commonly used in computer programming and memory addressing.
  • 12. Computer Number Systems and Conversion of Computer Number Systems
  • 13. LEARNING COMPETENCIES • Differentiate the various computer number systems • Apply conversion of computer number systems
  • 14. OBJECTIVES • Explain the basics of number systems and their significance in computing. • Identify the steps in the conversion of Decimal to Binary. • Convert Decimal to Binary
  • 15. OBJECTIVES • Identify steps in the conversion of binary to decimal. • Convert Binary to Decimal. • Identify steps in the conversion of Decimal to Octal.
  • 16. OBJECTIVES • Convert Decimal to Octal. • Identify steps in the conversion of Octal to Decimal. • Convert Octal to Decimal
  • 17. The number system plays a crucial role in mathematics and our daily lives.
  • 18. 1. Representation of Quantities: • A number system provides a way to represent quantities. Whether it’s counting objects, measuring distances, or calculating time, numbers allow us to express these concepts precisely.
  • 19. 1. Representation of Quantities: • For example, when you count the number of apples in a basket or measure the length of a room, you’re using the number system.
  • 20. 2.Foundation for Mathematical Concepts: • All mathematical concepts and formulas are based on the number system. Whether you’re solving equations, working with geometry, or analyzing data, numbers are fundamental.
  • 21. 2.Foundation for Mathematical Concepts: • From basic arithmetic operations (addition, subtraction, multiplication, division) to advanced calculus and algebra, numbers underpin mathematical reasoning.
  • 22. 3. Types of Numbers: ✧ The number system encompasses various types of numbers: 1.Counting Numbers: These start with 1 and continue indefinitely (1, 2, 3, …).
  • 23. 2.Whole Numbers: Include all counting numbers along with zero (0, 1, 2, …). 3.Integers: Positive and negative whole numbers, including zero (- 3, -2, -1, 0, 1, 2, 3, …). 4. Rational Numbers: Expressible as fractions (e.g., 3/4, -2/5).
  • 24. 5. Irrational Numbers: Cannot be expressed as fractions (e.g., √2, π). 6. Real Numbers: Encompass both rational and irrational numbers. 7. Even Numbers: Divisible by 2 (e.g., 2, 4, 6, …).
  • 25. 8. Odd Numbers: Not divisible by 2 (e.g., 3, 5, 7, …). 9. Prime Numbers: Divisible only by 1 and themselves (e.g., 5, 7, 13). 10. Composite Numbers: Have multiple factors (e.g., 10, 15, 28).
  • 26. 4. Digital Systems and Data Representation: ✧ • Understanding number systems is essential for digital systems (like computers) because they process data using binary representation (base-2).
  • 27. 4. Digital Systems and Data Representation: • Computers use bits (0s and 1s) to represent information, and this binary system relies on the principles of the number system.
  • 28. 4. Digital Systems and Data Representation: • The number system is not only a mathematical tool but also a fundamental aspect of our everyday interactions with the world. It allows us to quantify, calculate, and communicate effectively.
  • 29. Unlocking of Difficulties Number system - is a mathematical way of representing a set of values using digits or symbols. Decimal - is a number that consists of a whole part and a fractional part separated by a decimal point.
  • 30. Unlocking of Difficulties Binary - is a number expressed in the base-2 numeral system, in this system, we use only two symbols: typically, "0" (zero) and "1" (one). Octal - is a type of numeral system that uses a base of eight, in this system, the digits range from 0 to 7.
  • 31. Unlocking of Difficulties Hexadecimal - is a base-16 numeral system. Unlike our everyday decimal system (base 10), which uses ten symbols (0-9), hexadecimal employs sixteen symbols. These symbols represent values from 0 to 15.
  • 33. The binary number system is a fundamental concept in computer science and digital electronics. It uses a base-2 numeral system, which means it only employs two distinct symbols: 0 (zero) and 1 (one). Here are the key points about binary numbers:
  • 34. Representation: 10. In binary, each digit is called a bit. 11. The binary system is used internally by almost all modern computers and electronic devices because it directly maps to electronic circuits using logic gates.
  • 35. Representation: 12. Unlike our everyday decimal system (base 10), which uses ten symbols (0-9), binary uses only two symbols (0 and 1).
  • 36. Conversion: 13. To convert a decimal number to binary, follow these steps: 14. Divide the decimal number by 2. 15. Use the integer quotient obtained as the dividend for the next step.
  • 37. Conversion: 16. Continue dividing until the quotient becomes 0. 17. Write down the remainders in reverse order to get the binary representation.
  • 38. Example: Convert 4 to Binary: 18. Let's convert the decimal number 4 to binary: Step 1: Divide 4 by 2. Quotient: 2, Remainder: 0 Step 2: Divide 2 by 2. Quotient: 1, Remainder: 0
  • 39. Example: Convert 4 to Binary: Step 3: Divide 1 by 2. Quotient: 0, Remainder: 1 Reverse the remainders: 100 19. Therefore, 4 in binary is 100.
  • 40. Bit in Binary Number: 20. A single binary digit is called a bit. Examples: 21. 10101 is a five-bit binary number.
  • 41. Bit in Binary Number: 22. 101 is a three-bit binary number. 23. 100001 is a six-bit binary number.
  • 42. QUESTIONS: 1. How many different digits are used in the binary number system? 2. Can you explain the concept of place value in binary numbers?
  • 43. QUESTIONS: 3. Why is the binary number system important in computing and digital technology?
  • 45. DIRECTION identify what type of the number system.
  • 46. 1. 78 - ____________ 2. 100112 - ____________ 3. F - _______ 4. 328 - __________ 5. 1110102 - ___________
  • 48. OBJECTIVES At the end of the activity, you are expected to: • Understand the concept of number systems used in computers. • Differentiate between the decimal, binary, octal, and hexadecimal number systems.
  • 49. OBJECTIVES • Convert numbers from one system to another.
  • 50. DIRECTIONS: A. Word Completion – Converting Decimal to Binary Directions: Supply the missing word to complete the Steps in Conversion of Binary to Decimal.
  • 51. 1. Divide the number by ______. 2. Write the quotient and the _____on its corresponding _____. 3. Get the quotient and divide it again by __________. Write the quotient and remainder in its column.
  • 52. 4. Continue dividing until the quotient results to _________. Always write its quotient and remainder in their column. 5. Copy the remainder from the bottom to ___________. That would be the binary equivalent of the decimal number.
  • 53. DIRECTIONS: B. Decimal and Binary Directions: Convert each decimal number to its binary equivalent, binary to decimal number.
  • 56. The octal number system is a base-8 system that uses digits from 0 to 7 to represent any number. It is similar to other number systems like binary and hexadecimal. Here are some key points about octal numbers:
  • 57. Base and Digits: 29. The base of the octal system is 8. 30. It uses the digits 0, 1, 2, 3, 4, 5, 6, and 7.
  • 58. Representation: 31. Octal numbers are usually represented by binary numbers when grouped in pairs of three. 32. For example, the octal number 128 is expressed as 0010102 in binary, where 1 is equivalent to 001 and 2 is equivalent to 010.
  • 59. Conversion: 33. To convert decimal numbers to octal, we use the octal dabble method. 34. For example, to convert 56010 to octal:
  • 60. Conversion: 35. 560 ÷ 8 = 70 with a remainder of 0 36. 70 ÷ 8 = 8 with a remainder of 6
  • 61. Conversion: 37. 8 ÷ 8 = 1 with a remainder of 0 38. 1 ÷ 8 = 0 with a remainder of 1 39. So, the octal representation is 10608.
  • 62. Table of Octal Digits 40. We use only 3 bits to represent octal numbers. 41. Each group has a distinct value between 000 and 111. 42. Octal digits range from 0 to 7.
  • 63. Table of Octal Digits
  • 64. Octal numbers are commonly used in computer programming because they provide a compact way to represent binary numbers (each octal digit corresponds to three binary digits).
  • 65. QUESTIONS: 1. How many symbols are used in octal numbering system? 2. Can you explain the concept of place value in octal numbers? 3.Why are octal numbers often used in computer programming?
  • 67. OBJECTIVES At the end of the activity, you are expected to: • Understand the importance of binary numbers in computing. • Learn the step-by-step process of converting decimal numbers to binary and binary to decimal.
  • 68. OBJECTIVES At the end of the activity, you are expected to: • Practice conversions through guided activities.
  • 69. DIRECTIONS: A. Complete the sentence. Directions: Fill in the blank with the correct words.
  • 70. 1. To convert a decimal number to octal, we first need to _______ the decimal number by 8.
  • 71. 2. The _______ of the division is the first digit of the octal number. 3. The __________ of the division is used for the next step.
  • 72. 4. This process of dividing by 8 and using the remainder continues until the _________ becomes 0.
  • 73. 5. The octal number is formed by writing the remainder in ________order.
  • 75. A decimal number is a number that consists of a whole number and a fractional part separated by a point (often called the decimal point).
  • 76. Whole Number Part: The whole number part represents the integral value of the quantity. It can be any positive or negative integer. For example: •In the decimal number 42.85, the whole number part is 42. •In the decimal number -123.456, the whole number part is -123.
  • 77. Fractional Part: The fractional part represents a value smaller than one. It appears after the decimal point. For example: •In the decimal number 42.85, the fractional part is 0.85. •In the decimal number -123.456, the fractional part is 0.456.
  • 78. Decimal Point: The decimal point separates the whole number part from the fractional part. It serves as a visual indicator that we are dealing with a decimal.
  • 79. For example: 47.The decimal point in 42.85 separates 42 (whole) from 85 (fractional). 48.The decimal point in -123.456 separates -123 (whole) from 456 (fractional).
  • 80. Decimal numbers allow us to express quantities that are whole plus some part of a whole. They are commonly encountered in everyday situations, such as grocery shopping, measurements, and financial calculations.
  • 81. QUESTIONS: 1. What is the definition of a decimal number? 2. Can you explain how decimal numbers are different from whole numbers? 3. How are decimal numbers represented on a number line?
  • 83. Decimal numbers are an important part of mathematics and are used in many everyday situations. They represent values that are between whole numbers, allowing for more precise measurements and calculations. In this worksheet, you will practice working with decimal numbers through an activity.
  • 84. QUESTIONS: 1. Explain the difference between a whole number and a decimal number. 2. Describe the steps to round a decimal number to the nearest hundredth.
  • 85. QUESTION: 3. How would you add two decimal numbers with different numbers of digits to the right of the decimal point?
  • 87. OBJECTIVES At the end of the activity, you are expected to: • Understand the decimal and octal number systems. • Learn the process of converting decimal numbers to octal and vice versa.
  • 88. DIRECTIONS: B. Decimal and Octal Directions: Converting Decimal numbers to Octal and Octal to Decimal Numbers. Choose the correct letter of the answer.
  • 89. 1.What is the decimal equivalent of the octal number 64? a. 4 b. 14 c. 52 d. 100 2. If an octal number is 127, what is its value in decimal? a. 16 b. 55 c. 87 d. 102
  • 90. 3. Convert the octal number 345 to decimal. a. 125 b. 197 c. 209 d. 229 4. What is the decimal value of the octal number 777? a. 9 b. 383 c. 504 d. 511
  • 91. 5. If an octal number is 476, what is its decimal equivalent? a. 312 b. 302 c. 256 d. 318
  • 93. The hexadecimal number system often simply called “hex,” is a base-16 numeral system. Unlike our familiar base- 10 (decimal) system, which uses ten symbols (0 to 9) to represent numbers, hexadecimal employs sixteen symbols.
  • 94. These symbols include the numerals 0 to 9 and the letters A to F. Each digit in a hexadecimal number corresponds to a specific value, allowing us to express quantities in a compact and efficient manner.
  • 95. Base-16 Representation: Hexadecimal uses a base value of 16. This means that each column in a hexadecimal number represents a power of 16.
  • 96. Base-16 Representation: •The sixteen symbols used in hexadecimal are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. For example, the hexadecimal number “1A” represents 1 × 16 + 10 = 26 in decimal.
  • 97. Conversion to Other Systems: • Hexadecimal numbers can be converted to other number systems: • To decimal (base-10): Each digit is multiplied by the corresponding power of 16 and summed.
  • 98. Conversion to Other Systems: • To binary (base-2): Each hexadecimal digit corresponds to a 4-bit binary representation. • To octal (base-8): First convert to decimal, then to octal. • These conversions are essential in computer science and digital systems.
  • 99. Hexadecimal provides a convenient way to work with binary data and efficiently represent values in various contexts.
  • 101. Example: Convert the decimal number 456 to hexadecimal 1. Divide the decimal number by 16 and record the quotient and remainder. 2. Repeat the division using the quotient from the previous step until the quotient is zero.
  • 102. Example: Convert the decimal number 456 to hexadecimal 3. The hexadecimal number is formed by the remainders read from bottom to top.
  • 103. Steps: 1. Initial Number: 456 2. First Division: • Divide 456 by 16. •Quotient: 456÷16=28456 div 16 = 28456÷16=28 (integer part) •Remainder: 456 mod 16=8456 mod 16 = 8456mod16=8
  • 104. Steps: 3. So, 456 divided by 16 gives a quotient of 28 and a remainder of 8. 4.Second Division: • Now divide the quotient from the first division (28) by 16. • Quotient: 28÷16=128 div 16 = 128÷16=1 (integer part)
  • 105. Steps: •Remainder: 28 mod 16=1228 mod 16 = 1228mod16=12 5.So, 28 divided by 16 gives a quotient of 1 and a remainder of 12. Note that in hexadecimal, 12 is represented as 'C'.
  • 106. Steps: 6. Third Division: •Now divide the quotient from the second division (1) by 16. •Quotient: 1÷16=01 div 16 = 01÷16=0 (integer part) •Remainder: 1mod 16=11 mod 16 = 11mod16=1
  • 107. Steps: 7.So, 1 divided by 16 gives a quotient of 0 and a remainder of 1. 8.Construct the Hexadecimal Number: • Write the remainders from the last to the first.
  • 108. Steps: • The remainders are 1, 12 (C in hexadecimal), and 8. 9.Therefore, the hexadecimal representation of the decimal number 456 is 1C8.
  • 110. • Divide the decimal number by 16. • Write down the remainder. • Use the quotient for the next division. • Repeat until the quotient is zero. • The hexadecimal number is the remainders read from bottom to top.
  • 112. OBJECTIVES At the end of the activity, you are expected to: • Understand the decimal and hexadecimal number systems.
  • 113. OBJECTIVES At the end of the activity, you are expected to: • Learn the process of converting decimal numbers to hexadecimal and hexadecimal to decimal numbers.
  • 114. DIRECTIONS: Decimal Number System Directions: Choose the correct letter of your answer.
  • 115. 1. What is the place value of the digit 5 in the number 6.573? a. Thousandths c. Units b. Tenths d. Hundreds 2. Which of the following is equivalent to 3.62? a. 362 b. 36 c. 3.6 d. 0.362
  • 116. 3. What is 0.45 written as a fraction in simplest form? a. 45/100 c. 9/5 b. 9/20 d. 4/5 4. Which of the following numbers is the largest? c. 7.321 c. 732.1 d. 73.12 d. 73.21
  • 117. 5. If you add 2.5 and 1.75, what is the sum? a. 4.25 c. 4.75 b. 4.5 d. 5.25
  • 118. Directions: Converting Decimal Numbers to Hexadecimal and Hexadecimal to Decimal.
  • 119. 1. What is the hexadecimal equivalent of the decimal number 27? a. 1B b. 1F c. 23 d. 2E 2. Which of the following is a correct hexadecimal representation of the decimal number 50? a. 30 b. 32 c. 35 d. 38
  • 120. 3. If the hexadecimal number 3D is converted to decimal, what is the result? a. 53 b. 61 c. 57 d. 59 4. What is the decimal value of the hexadecimal number A4? a. 161 b. 164 c. 166 d. 168
  • 121. 5. Convert the decimal number 95 to hexadecimal. a. 5D b. 60 c. 61 d. 65
  • 122. ANSWER
  • 123. 1.a. 1B 2.b. 32 3.d. 59 4.c. 166 5.a. 5D
  • 124. WRAP-UP QUESTIONS: 1. What are the importance of the different computer number systems? 2. How does the concept of the different computer number systems and the conversion of decimal number to binary, octal, and hexadecimal?
  • 125. REFLECTION 1. How do you find today’s lesson? Was it hard? Which part of the lesson do you find difficult?
  • 127. POST-TEST 1.Which of the following best describes the binary number system? a) A number system that uses 0-9 b) A number system that uses 0-7 c) A number system that uses 0-1 d) A number system that uses o-F
  • 128. 2.What is the first step in converting a decimal number to octal? a) Divide the decimal number by 8 b) Write the decimal number in reverse order POST-TEST
  • 129. 2.What is the first step in converting a decimal number to octal? c) Multiply the decimal number by 8 d) Add 8 to the decimal number POST-TEST
  • 130. 2. What is the primary focus of the Visual Arts sector in ICT? c. Maintaining and repairing computer systems d. Transmitting information over distances POST-TEST
  • 131. 3. What do you do with the remainder of each division step? a) Discard it b) Add it to the next division c)Use it to form the octal number d) Multiply it by 8 POST-TEST
  • 132. 4. How do you determine the digits of the octal number? a) The quotients of the divisions b) The remainders of the divisions c) The sum of the divisions d) The difference between the divisions POST-TEST
  • 133. 5. What happens when the divisor becomes 0? a) The process is complete b) You start over with a new decimal number POST-TEST
  • 134. 5. What happens when the divisor becomes 0? c) You add 8 to the octal number d) You multiply the octal number by 8 POST-TEST
  • 135. 6. In what order should the digits of the octal number be written? a) In the order they were generated b) In reverse order c) From largest to smallest d) From smallest to large POST-TEST
  • 136. 7. How many unique digits are used in the octal number system? a) 2 b) 8 c) 10 d) 16 8.What is the decimal equivalent of the binary number 1010? a) 5 b) 8 c) 10 d) 15 POST-TEST
  • 137. 9. Which of the following is the hexadecimal representation of the decimal number 27? a) 1B b) 1F c) 1A d) 1C 10.What is the octal equivalent of the binary number 101011? a) 43 b) 53 c) 63 d) 73 POST-TEST
  • 138. ANSWER
  • 139. POST-TEST 1.Which of the following best describes the binary number system? a) A number system that uses 0-9 b) A number system that uses 0-7 c) A number system that uses 0-1 d) A number system that uses o-F
  • 140. 2.What is the first step in converting a decimal number to octal? a) Divide the decimal number by 8 b) Write the decimal number in reverse order POST-TEST
  • 141. 2.What is the first step in converting a decimal number to octal? c) Multiply the decimal number by 8 d) Add 8 to the decimal number POST-TEST
  • 142. 3. What do you do with the remainder of each division step? a) Discard it b) Add it to the next division c)Use it to form the octal number d) Multiply it by 8 POST-TEST
  • 143. 4. How do you determine the digits of the octal number? a) The quotients of the divisions b) The remainders of the divisions c) The sum of the divisions d) The difference between the divisions POST-TEST
  • 144. 5. What happens when the divisor becomes 0? a) The process is complete b) You start over with a new decimal number POST-TEST
  • 145. 5. What happens when the divisor becomes 0? c) You add 8 to the octal number d) You multiply the octal number by 8 POST-TEST
  • 146. 6. In what order should the digits of the octal number be written? a) In the order they were generated b) In reverse order c) From largest to smallest d) From smallest to large POST-TEST
  • 147. 7. How many unique digits are used in the octal number system? a) 2 b) 8 c) 10 d) 16 8.What is the decimal equivalent of the binary number 1010? a) 5 b) 8 c) 10 d) 15 POST-TEST
  • 148. 9. Which of the following is the hexadecimal representation of the decimal number 27? a) 1B b) 1F c) 1A d) 1C 10.What is the octal equivalent of the binary number 101011? a) 43 b) 53 c) 63 d) 73 POST-TEST