Decimal Number System

A number system is a method for representing numbers, defined by its base or radix. The decimal number system, also known as base-10, is the number system we use every day for tasks like counting people, tracking scores, or tallying votes.

It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

In this system, each digit’s value depends on its position in the number. The image below illustrates this with the number 1234.567:

Explanation: In the number 1234.567 :

• Starting from the left, 1 is in the thousands place (1 × 10³ = 1000),
• 2 is in the hundreds place (2 × 10² = 200),
• 3 is in the tens place (3 × 10¹ = 30),
• 4 is in the ones place (4 × 10⁰ = 4). After the decimal point, 5 is in the tenths place (5 × 10⁻¹ = 0.5),
• 6 is in the hundredths place (6 × 10⁻² = 0.06),
• 7 is in the thousandths place (7 × 10⁻³ = 0.007).

Adding these together (1000 + 200 + 30 + 4 + 0.5 + 0.06 + 0.007) gives the total value, 1234.567.

Note : Right most bit is the least significant bit (LSB), and leftmost bit is the most significant bit (MSB). The table below highlights these positions:

Conversion from Decimal to Other Number Systems

To convert decimal to another base, divide by the target base, record remainders from right to left until the quotient is 0, then specify the base for each section (binary: base-2, octal: base-8, hexadecimal: base-16 with A–F for 10–15).

Decimal to Binary Conversion 

To see how a decimal number is converted to binary, consider the following example:

Binary numbers use base-2. To convert a decimal number to binary, follow the steps given below:

• Step 1: Divide the Decimal Number with the base of the number system to be converted to. Here the conversion is to binary, hence the divisor will be 2. 
• Step 2: The remainder obtained from the division will become the least significant digit of the new number.
• Step 3: The quotient obtained from the division will become the next dividend and will be divided by the base i.e. 2.
• Step 4: The remainder obtained will become the second least significant digit i.e. it will be added to the left of the previously obtained digit.

Now, the steps 3 and 4 are repeated until the quotient obtained becomes 0, and the remainders obtained after each iteration are added to the left of the existing digits. 

After all the iterations are over, the last obtained remainder will be termed as the Most Significant digit.

Decimal to Octal Conversion

Let’s look at an example of converting a decimal number to octal:

Octal numbers use base-8. Here’s the process:

• Step 1: Divide the Decimal Number with the base of the number system to be converted to. Here the conversion is to octal, hence the divisor will be 8. 
• Step 2: The remainder obtained from the division will become the least significant digit of the new number.
• Step 3: The quotient obtained from the division will become the next dividend and will be divided by base i.e. 8.
• Step 4: The remainder obtained will become the second least significant digit i.e. it will be added to the left of the previously obtained digit.

Now, the steps 3 and 4 are repeated until the quotient obtained becomes 0, and the remainders obtained after each iteration are added to the left of the existing digits. 

Decimal to Hexadecimal Conversion

Here’s an example of converting a decimal number to hexadecimal:

Hexadecimal uses base-16, with digits 0–9 and A–F (where A = 10, B = 11, C = 12, D = 13, E = 14, F = 15). Follow these steps:

• Step 1: Divide the Decimal Number with the base of the number system to be converted to. Here the conversion is to Hex hence the divisor will be 16. 
• Step 2: The remainder obtained from the division will become the least significant digit of the new number.
• Step 3: The quotient obtained from the division will become the next dividend and will be divided by base i.e. 16.
• Step 4: The remainder obtained will become the second least significant digit i.e. it will be added to the left of the previously obtained digit.

Now, the steps 3 and 4 are repeated until the quotient obtained becomes 0, and the remainders obtained after each iteration are added to the left of the existing digits. 

NOTE: If the remainder is 10–15, use A–F (e.g., A = 10, B = 11, C = 12, D = 13, E = 14, F = 15).

Also Check:

1. Decimal to Binary Conversion and its implementation.
2. Decimal to Octal Conversion and its implementation.
3. Decimal to Hexadecimal Conversion and its implementation.

Beyond conversions, the decimal system also supports techniques like complements, used in subtraction and error detection.

9’s and 10’s Complement of Decimal (Base-10) Number

Complements are used in number systems to simplify subtraction and error detection in digital systems. In the decimal system, the 9’s complement and 10’s complement are commonly used.

Steps to Find 9’s Complement:

1. Write the given decimal number.
2. Subtract each digit from 9.

Example : 9’s Complement of 2020

 9 - 2 = 7
 9 - 0 = 9
 9 - 2 = 7
 9 - 0 = 9

Thus, the 9’s complement of 2020 is 7979.

Steps to Find 10’s Complement:

1. Find the 9’s complement of the number.
2. Add 1 to the least significant digit (LSB).

Example : 10’s Complement of 2020

• 9’s complement of 2020 = 7979
• Adding 1: 7979 + 1 = 7980

Thus, the 10’s complement of 2020 is 7980

Practice Question on Decimal Number System

Question 1: Convert the binary system 110₂ to its decimal equivalent.

Question 2: Find the 10's complement of 725,

Question 3: Convert the decimal number 29 to binary.

Question 4: Find the 9's complement of decimal number 486.

Question 5: Covert 423₁₀ to hexadecimal.

Answer key:

1. 6₁₀
2. 275
3. 11101₂
4. 513
5. 1A7₁₈