Number System

A Number System is a writing system used to express numbers.

It is a mathematical notation for representing numbers using digits or symbols in a consistent manner.

For a number system having base or radix r, the digits range from:

0 to r − 1

The total number of digits available in that number system is r.

Different values of r produce different number systems.

The most commonly used number systems are:

1. Decimal Number System
2. Binary Number System
3. Octal Number System
4. Hexadecimal Number System

1. Decimal Number System

The Decimal Number System is also known as the Base-10 Number System.

It uses 10 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

The position of each digit determines its value.

For the integer part, the positional weights are:

10⁰, 10¹, 10², 10³, …

For the fractional part, the positional weights are:

10⁻¹, 10⁻², 10⁻³, …

The decimal point separates the:

• Integer part
• Fractional part

Example

Let:

N = 34.5

Here:

• Integer part = 34
• Fractional part = 0.5

The digits have the following weights:

• 3 × 10¹
• 4 × 10⁰
• 5 × 10⁻¹

Therefore:

(34.5)₁₀ = 3 × 10¹ + 4 × 10⁰ + 5 × 10⁻¹

2. Binary Number System

The Binary Number System is the foundation of digital electronics and computer science.

It uses only two digits:

0 and 1

Therefore, its base or radix is:

2

The two binary digits represent two different states, such as:

• ON
• OFF

Binary digits are commonly called bits.

Digital circuits use binary because electronic components can easily represent two states.

A binary number can contain:

• Integer part
• Fractional part

The portion to the left of the binary point is called the integer part.

The portion to the right of the binary point is called the fractional part.

3. Octal Number System

The Octal Number System is a Base-8 Number System.

It uses eight digits:

0, 1, 2, 3, 4, 5, 6, 7

According to the provided notes, one advantage of the octal system is that it may require fewer digits in some calculations.

4. Hexadecimal Number System

The Hexadecimal Number System has a base or radix of:

16

It is also commonly called Hex.

Hexadecimal uses 16 symbols:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The letters represent decimal values:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Hexadecimal numbers are widely used in digital systems and computer applications.

Converting Decimal to Other Number Systems

Decimal numbers can be converted into:

1. Binary
2. Hexadecimal
3. Octal

1. Decimal to Binary

To convert a decimal number into binary:

Step 1: Divide the decimal number by 2 and note the remainder.

Step 2: Divide the quotient again by 2 and note the remainder.

Step 3: Continue dividing by 2 until the quotient becomes 0.

Step 4: Write the remainders in reverse order.

Step 5: The first remainder obtained is the Least Significant Bit (LSB), while the final remainder is the Most Significant Bit (MSB).

Step 6: Read the remainders from bottom to top.

The resulting number is the binary equivalent.

Example: Add Decimal to Binary conversion example here.

2. Decimal to Hexadecimal

To convert a decimal number into hexadecimal:

Step 1: Divide the decimal number by 16.

Step 2: Note the remainder.

Step 3: Divide the quotient again by 16.

Step 4: Continue until the quotient becomes 0.

Step 5: Write the remainders in reverse order.

If the remainder is greater than 9, use:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Example: Add Decimal to Hexadecimal conversion example here.

3. Decimal to Octal

To convert a decimal number into octal:

Step 1: Write the given decimal number.

Step 2: If the decimal number is less than 8, the octal number is the same.

Step 3: If the decimal number is greater than 7, divide it by 8.

Step 4: Note the remainder.

Step 5: Repeat the division using the quotient until the quotient becomes less than 8.

Step 6: Write the remainders in reverse order from bottom to top.

Example: Add Decimal to Octal conversion example here.

Converting Binary to Other Number Systems

Binary numbers can be converted into:

1. Decimal
2. Octal
3. Hexadecimal

1. Binary to Decimal

To convert binary into decimal:

Step 1: Write the binary number.

Step 2: Assign powers of 2 from right to left, starting from 0.

Step 3: Multiply each binary digit by its corresponding power of 2.

Step 4: Add all the products.

The result is the decimal equivalent.

Example: Add Binary to Decimal conversion example here.

2. Binary to Octal

According to the provided notes, the following method can be used:

Step 1: Take the given binary number.

Step 2: Multiply each digit by its corresponding power of 2.

Step 3: Add the results to obtain the equivalent decimal number.

Step 4: Divide the decimal number by 8.

Step 5: Note the remainder.

Step 6: Continue dividing the quotient by 8 until it becomes 0.

Step 7: Write the remainders in reverse order.

The result is the octal equivalent.

Example: Add Binary to Octal conversion example here.

3. Binary to Hexadecimal

To convert binary into hexadecimal:

Step 1: Write the required binary number.

Step 2: Divide the binary number into groups of four bits.

Step 3: Find the decimal value of each group.

Step 4: Convert each group into its hexadecimal equivalent.

Step 5: Combine the hexadecimal digits.

Example: Add Binary to Hexadecimal conversion example here.

Converting Octal to Other Number Systems

Octal numbers can be converted into:

1. Decimal
2. Binary
3. Hexadecimal

1. Octal to Decimal

To convert octal into decimal:

Step 1: Write the octal digits with their corresponding powers of 8.

Step 2: Evaluate the powers of 8.

For example:

• 8⁰ = 1
• 8¹ = 8
• 8² = 64

Step 3: Multiply each octal digit by its corresponding power of 8.

Step 4: Add all the products.

The result is the decimal equivalent.

Example: Add Octal to Decimal conversion example here.

2. Octal to Binary

To convert octal into binary:

Step 1: Write the octal number.

Step 2: Convert each octal digit into its 3-bit binary equivalent.

Step 3: Combine all the binary groups.

The resulting number is the binary equivalent.

Example: Add Octal to Binary conversion example here.

3. Octal to Hexadecimal

To convert octal into hexadecimal:

Step 1: Convert each octal digit into its 3-bit binary equivalent.

Step 2: Join all the binary groups.

Step 3: Divide the binary number into groups of four bits.

Step 4: Convert each 4-bit group into its hexadecimal equivalent.

Step 5: Combine the hexadecimal digits.

Example: Add Octal to Hexadecimal conversion example here.

According to the provided example, the binary equivalent obtained during one conversion is:

111101010

Converting Hexadecimal to Other Number Systems

Hexadecimal numbers can be converted into:

1. Binary
2. Decimal
3. Octal

1. Hexadecimal to Binary

According to the provided notes, the following method can be used:

Step 1: Take the given hexadecimal number.

Step 2: Determine the number of digits.

Step 3: Multiply each digit by the corresponding power of 16.

Step 4: Add all the products.

Step 5: The result gives the decimal equivalent.

Step 6: Divide the decimal number by 2.

Step 7: Note the remainder.

Step 8: Continue dividing by 2 until the quotient becomes 0.

Step 9: Write the remainders in reverse order.

Step 10: The result is the required binary number.

Example: Add Hexadecimal to Binary conversion example here.

2. Hexadecimal to Decimal

To convert hexadecimal into decimal:

Step 1: Identify each hexadecimal digit.

Step 2: Assign its decimal value.

Remember:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Step 3: Multiply each digit by the corresponding power of 16.

Step 4: Add all the products.

The result is the decimal equivalent.

Example: Add Hexadecimal to Decimal conversion example here.

3. Hexadecimal to Octal

To convert hexadecimal into octal:

Step 1: Convert the hexadecimal number into binary.

Step 2: Combine the binary bits.

Step 3: Split the binary number into groups of three bits.

Step 4: Convert each 3-bit group into its octal equivalent.

Step 5: Combine the octal digits.

Example: Add Hexadecimal to Octal conversion example here.

1’s Complement Form

The 1’s Complement of a binary number is obtained by inverting each bit.

This means:

• 0 becomes 1
• 1 becomes 0

According to the provided notes, the 1’s complement of a positive number represents the corresponding negative number, and vice versa.

Example

Suppose the binary number is:

1010

Its 1’s complement is:

0101

2’s Complement Form

The 2’s Complement of a binary number is obtained by adding 1 to the 1’s complement of that number.

In simple terms:

2’s Complement = 1’s Complement + 1

Example

Suppose the binary number is:

1010

First, find its 1’s complement:

0101

Then add 1:

0101 + 1 = 0110

Therefore:

2’s Complement = 0110

Subtraction Using 2’s Complement

Binary subtraction can be performed using the 2’s complement method.

Subtracting a Smaller Number

According to the provided notes:

Step 1: Determine the 2’s complement of the smaller number.

Step 2: Add this value to the larger number.

Step 3: Omit the final carry.

According to the notes, there is a carry in this case.

Example: Add subtraction using 2’s complement example here.

Subtracting a Larger Number

According to the provided notes:

Step 1: Determine the 2’s complement as required for the subtraction.

Step 2: Add it to the other binary number.

Step 3: If there is no carry, the result is in 2’s complement form and represents a negative number.

Step 4: To obtain the answer in true form, take the 2’s complement of the result and change its sign.

Example: Add subtraction of larger number using 2’s complement here.

Advantages of 2’s Complement Over 1’s Complement

The provided notes list the following advantages of 2’s complement.

1. Simplicity of Addition

In 2’s complement, signed-number addition can use the same basic hardware as unsigned-number addition.

This simplifies arithmetic operations.

2. Unique Representation of Zero

In 2’s complement, there is only one representation of zero.

In 1’s complement, there are two representations:

• Positive zero
• Negative zero

Having only one zero representation simplifies arithmetic operations.

3. No Separate Sign Bit Handling

According to the provided notes, the Most Significant Bit (MSB) serves as the sign bit in 2’s complement representation.

This can simplify comparisons and logical operations.

4. Easier Hardware Implementation

2’s complement arithmetic can be implemented efficiently in hardware.

It is particularly useful for subtraction because of its simpler carry logic.

5. Range of Representable Numbers

According to the provided notes, 2’s complement provides a convenient range of signed values and simplifies the handling and manipulation of signed binary numbers.