Lesson 4 Number System ( English medium)

1. What do you understand by number system? Explain decimal and binary number system.

Ans:- A number system is a way to represent numbers using a set of symbols or digits. It defines how numbers are written and understood. Different number systems exist, depending on the base they use.

Decimal Number System:

1. It is the most common number system, which we use in daily life.
2. It is based on base 10, meaning it has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
3. Each digit in a number has a place value depending on its position, and the place values are powers of 10. For example, in the number 345, the place values are 3 × 100, 4 × 10, and 5 × 1.

Binary Number System:

1. This system is used in computers and digital devices.
2. It is based on base 2, meaning it has only two digits: 0 and 1.
3. Each digit in a binary number also has a place value, but the place values are powers of 2. For example, in the binary number 101, the place values are 1 × 4, 0 × 2, and 1 × 1, which equals 5 in decimal.

These number systems help in different fields, from everyday calculations to computer processing.

2. Explain what do you mean by digital and analog operations.

Ans:-Digital operations and analog operations refer to different ways of processing and representing data.

Analog Operations: Analog operations deal with continuous data, where values can vary smoothly over a range. This means there are no fixed steps between values.

Analog signals are often used in devices like radios or older TVs, where the signal varies in amplitude or frequency.

These operations are prone to noise or distortion because they work with continuous signals.

For example, a thermometer shows temperature with a smooth rise or fall, and a clock with hands shows time in a continuous flow.

Digital Operations: Digital operations work with discrete, fixed values, usually represented by binary digits (0s and 1s).

Computers, calculators, and most modern electronic devices use digital operations to process information.

Unlike analog signals, digital signals are less affected by noise because they work with clear, fixed values.

For example, a digital clock shows specific times , without a smooth transition between seconds.

3. Write the steps to convert a decimal base to another base. Describe it with an example.

Ans:- To convert a decimal (base 10) number to another base (e.g., binary, base 2), follow these steps:

Steps for Conversion:

1. Divide the decimal number by the new base.
2. Record the remainder (this will be a digit in the new base).
3. Divide the quotient by the new base again.
4. Repeat steps 2 and 3 until the quotient becomes zero.
5. Read the remainders in reverse order to get the final result in the new base.
6. Example: Convert 45 (decimal) to binary (base 2):
7. Divide 45 by 2: quotient = 22, remainder = 1.
8. Divide 22 by 2: quotient = 11, remainder = 0.
9. Divide 11 by 2: quotient = 5, remainder = 1.
10. Divide 5 by 2: quotient = 2, remainder = 1.
11. Divide 2 by 2: quotient = 1, remainder = 0.
12. Divide 1 by 2: quotient = 0, remainder = 1 (stop here since quotient is 0).

Now, read the remainders from bottom to top: 101101. Thus, 45 in decimal = 101101 in binary.

4. Write the steps to convert a hexadecimal to binary.

Ans:-  To convert a hexadecimal (base 16) number to binary (base 2), follow these simple steps:

Steps for Conversion:

1. Write down the hexadecimal number.
2. Convert each hexadecimal digit into its 4-bit binary equivalent (since 16 = 2⁴, each hex digit can be represented by 4 binary digits).
3. Combine all the 4-bit binary groups to form the final binary number.
4. Hexadecimal to Binary Table:

Hexadecimal Binary Equivalent

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A (10) 1010

B (11) 1011

C (12) 1100

D (13) 1101

E (14) 1110

F (15) 1111

Example: Convert 3A7 (hex) to binary.

3 = 0011

A = 1010

7 = 0111

Now, combine the binary groups: 0011 1010 0111.

Thus, 3A7 (hex) = 001110100111 (binary).

5. Write the steps to convert a octal to decimal.

Ans:- To convert an octal (base 8) number to decimal (base 10), follow these steps:

Steps for Conversion:

Write down the octal number.

1. Label the position of each digit, starting from the right (the rightmost digit is at position 0, the next is 1, and so on).
2. Multiply each octal digit by 8 raised to the power of its position.
3. Add all the products together to get the decimal equivalent.

Example: Convert 235 (octal) to decimal.

Write down the octal number: 235.

Label the positions:

2 is at position 2,

3 is at position 1,

5 is at position 0.

Multiply each digit by 8 raised to its position:

2 × 8² = 2 × 64 = 128,

3 × 8¹ = 3 × 8 = 24,

5 × 8⁰ = 5 × 1 = 5.

Add the results: 128 + 24 + 5 = 157.

Thus, 235 (octal) = 157 (decimal).

6. What do you understand by logic gates?

Ans:- Logic gates are the basic building blocks of digital circuits. They are electronic devices that perform logical operations on one or more binary inputs to produce a single binary output. Logic gates work with two states: 0 (low/off) and 1 (high/on), and they are used in computers, calculators, and other digital systems to process data.

There are several basic types of logic gates, each performing a specific function:

1. AND Gate: Outputs 1 only if all inputs are 1. Otherwise, it outputs 0.
2. OR Gate: Outputs 1 if at least one input is 1. Outputs 0 if all inputs are 0.
3. NOT Gate: It is also called an inverter. It has one input and reverses the input—if the input is 1, it outputs 0, and vice versa.
4. NAND Gate: Outputs 0 only if all inputs are 1. Otherwise, it outputs 1 (opposite of AND gate).
5. NOR Gate: Outputs 0 if at least one input is 1. Otherwise, it outputs 1 (opposite of OR gate).
6. XOR Gate: Outputs 1 if exactly one input is 1, but outputs 0 if both inputs are the same.
7. XNOR Gate: Outputs 1 if both inputs are the same, 0 if they differ (opposite of XOR gate).

Logic gates are essential in performing arithmetic operations, decision-making processes, and controlling digital systems.

7. Explain NAND and NOR logic gate.

Ans:- NAND and NOR are two important types of logic gates, both of which are combinations of basic logic operations.

NAND Gate: The NAND gate stands for NOT AND. It is the opposite (negation) of the AND gate.

It takes two or more inputs and outputs 1 unless all inputs are 1. If all inputs are 1, the output is 0.

Truth table for a two-input NAND gate:

Input A Input B Output (A NAND B)

0 0 1

0 1 1

1 0 1

1 1 0

NAND gates are versatile and can be used to create any other logic gate, making them essential in digital circuits.

NOR Gate: The NOR gate stands for NOT OR. It is the opposite (negation) of the OR gate.

It takes two or more inputs and outputs 1 only if all inputs are 0. If any input is 1, the output is 0.

Truth table for a two-input NOR gate:

Input A Input B Output (A NOR B)

0 0 1

0 1 0

1 0 0

1 1 0

Like NAND, NOR gates can also be used to create other logic gates, and they are fundamental in digital design.

Both NAND and NOR gates are key to constructing complex digital systems

Very Short Answer Qusetions.

1. Difference between decimal and hexadecimal number system.

Ans:- The decimal system is based on base 10, using digits 0 to 9. The hexadecimal system is based on base 16, using digits 0 to 9 and letters A to F (where A = 10, B = 11, up to F = 15).

2. Write down the binary addition and subtraction process.

Ans:- Binary addition and subtraction:

Binary addition  rules:

0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10 (carry 1).

Binary subtraction rules:

0 - 0 = 0, 1 - 0 = 1, 1 - 1 = 0, 0 - 1 = 1 (borrow 1).

3. Give brief information about the octal number system.

Ans:-The octal system is a base 8 number system. It uses digits 0 to 7. Octal is commonly used in computing as a shorthand for binary since each octal digit corresponds to three binary digits.

4. What is the importance of 0 and 1 in the binary number system?

Ans:- The binary system uses 0 and 1 to represent information. These two states (off/on) are crucial in digital electronics and computing because they directly correspond to the logic of electronic circuits.

5. What is hexadecimal number system?

Ans:- The hexadecimal system is a base 16 number system. It uses digits 0-9 and letters A-F (A = 10, B = 11, ..., F = 15). Hexadecimal is often used in computing to represent large binary numbers compactly.

6. Write the first 15 binary and hexadecimal numbers by table.

Ans:- First 15 binary and hexadecimal numbers:

Decimal Binary Hexadecimal

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 9

10 1010 A

11 1011 B

12 1100 C

13 1101 D 

14 1110 E

15 1111 F

7. How binary numbers are converted to decimal numbers?

Ans:- To convert a binary number to decimal, multiply each binary digit by 2 raised to the power of its position (starting from 0 on the right), then sum the results.

8. Convert the following binary numbers into decimal:

(a) (10l) 2

(b) (100001),

(c) (1100l) 2

(d) (1111) 2

Ans:-   

(a) (101)₂: = 1 × 2² + 0 × 2¹ + 1 × 2⁰ = 5 (decimal).

(b) (100001)₂: = 1 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰ = 33 (decimal).

(c) (11001)₂: = 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰ = 25 (decimal).

(d) (1111)₂: = 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰ = 15 (decimal).