Base

Base is the count of the digits used to express any number in the number system. Generally, the word base is used to describe a starting point, foundation, or a fundamental idea. The building blocks that are used for number representation are known as bases. It is a key concept used to represent numbers using digits or letters. Each number system has a unique base.

 

The base in a number system is also known as “radix” and is the total number of digits used, starting from 0. Now, let us understand each number system and its base values. 

 

Number System	Base
Binary	2
Ternary	3
Quaternary	4
Quinary	5
Senary	6
Septenary	7
Octal	8
Nonary	9
Decimal	10
Undecimal	11
Duodecimal	12
Hexadecimal	16
Vigesimal	20
Sexagesimal	60

 

 Now, take a close look at the most common number systems and their base. 

 

 

• Binary number system  Base value is 2. 

 

• Octal number system  Base value is 8. 

 

• The Decimal number system  Base value is 10. 

 

• The hexadecimal number system  Base value is 16.
   

We can represent the base of a number by writing a small number as a subscript next to it. The subscript shows which number system is used to represent the numbers. The key elements that come into consideration when representing the base are: 

 

 

• The number. 
• The base.
• The position of the number and its base. 

 

We can represent the decimal number 436 along with its base as 43610.
It can be expressed as follows:
   43610 = (4 × 102) + (3 × 101) + (6 × 100)    
 

The most frequently used number systems are the binary, octal, decimal, and hexadecimal number systems. Let us now learn more about each of them.   

 

 

The decimal number system: In our daily lives, we use the decimal number system to count and calculate objects, money, measurements, and numbers. It is also known as a base-10 number system, which uses 10 digits from 0 to 9. The digits included in the decimal number system are:
       0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 
The base value of this number system is 10, and powers of 10, such as 100, 101, 102, decide the place value of every digit. For example, decimal numbers are:     
 54610, 32810, 2510 

 

 

The binary number system: The base of the binary number system is 2, which uses only two digits: 0 and 1. It is known as the base-2 number system. The powers of 2 (20, 21, 22) decide the place value of each binary digit. Computers and electronic devices mainly use this number system to store and process data and information.
For example, binary numbers are:
1012, 110012, 100012 

 

 

The octal number system: This number system is known as the base-8 number system. The 8 distinct digits in the octal number system include 0, 1, 2, 3, 4, 5, 6, and 7. The place values of all digits of this system are powers of 8, such as 80, 81, and 82. The octal number system simplifies binary numbers by grouping them into sets of three bits. For instance, numbers in the octal number system are 2548, 4328, 6548, etc.

 

 

The hexadecimal number system: The hexadecimal number system uses 16 symbols, like numbers and letters, to represent numbers. The sixteen digits are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
A, B, C, D, E, and F (6 letters)
It is also known as the base-16 number system, where powers of 16, such as 160, 161, or 162, decide the place value of each digit. Every 4-bit binary number is expressed as a single digit in hexadecimal.   

We can convert any number from one number system to another using its base value. A decimal number can be converted into binary, octal, and hexadecimal using their base values. Let’s discuss how to convert numbers from one system to another.

 

 

1. Decimal to binary number system

How to convert a decimal number to binary number are discussed below - 

Step 1: Divide binary number .

 

Step 2: Write the remainder.

 

Step 3: Continue the division until the quotient becomes 0. 

 

Step 4: Write the remainder in reverse order, from bottom to top. 

 

 

To convert a fractional part of a decimal to a binary number, we can follow the given steps: 

 

Step 1: Multiply the fractional part by 2. 

 

Step 2: Write the integer number of the product (0 or 1).

 

Step 3: Continue the multiplication with the remaining fractional part until the fraction becomes 0. 

 

Step 4: Write the obtained digits to form the binary number.   
By following these steps, we can get the binary number. 

Now, take an example to understand the conversion of a decimal to binary. 
For example, (10.25)10
For the integer part (10): 

 

Step 1: Divide 10 by 2:
      Quotient = 5
      Remainder = 0

 

Step 2: Divide 5 by 2:
       Quotient = 2
       Remainder = 1 

 

Step 3: Divide 2 by 2:
       Quotient = 1 
       Remainder = 0 

 

Step 4: Divide 1 by 2:
        Quotient = 0
        Remainder = 1 

 

Step 5: Write the remainder from bottom to top. The result is 1010. 

Next, we can convert the fractional part. 

Step 1: Multiply 0.25 by 2: 
     Product = 0.5 
     Integer part = 0 

 

Step 2: Multiply 0.5 by 2: 
     Product = 1.0 
     Integer part = 1 

 

Step 3: Write the obtained integers to get the binary number. 
Thus, the binary number of (10.25)10 is (1010.01)2.

 

 

2. Binary to decimal number system 

To convert the integer part of a binary number to a decimal, follow the given steps: 

Step 1: Write down the binary number. 
Step 2: Multiply each binary number by powers of 2. The first number on the left has the highest power, and it decreases to the right. The power starts at 0. For example, if the given binary number is (1010.01)2
It can be multiplied as follows:
(1 × 23) + (0 × 22) + (1 × 21) + (0 × 20) 
Here, the power is raised to its position, which starts at 0 for the last number on the right and increases by 1 to the left.  
Step 3: Add the results, and the sum is the decimal number. 

 

Next, we can convert the fractional part of the binary number.

Step 1: Note the binary fraction. 
Step 2: Multiply each number by powers of 2. Here, after the decimal point, the power starts from -1 and goes up by -2, -3, and so on.  
Step 3: Add the results, and the sum is the decimal fraction.    

 

For instance, we can convert a binary number (1010.01)2 to a decimal. 

Step 1: (1010.01)2
Step 2: (1 × 23) + (0 × 22) + (1 × 21) + (0 × 20) + (0 × 2-1) + (1 × 2-2) 
      = 8 + 0 + 2 + 0 + 0 + 0.25 = 10.25  
Thus, (1010.01)2 = (10.25)10 

 

 

3. Decimal to octal number system

How to convert any decimal number to octal one.
Step 1: Here, divide each number by the base number, 8.
Step 2: List the remainder (0 to 7).
Step 3: Continue the division until the quotient becomes 0. 
Step 4: Finally, write the remainder from bottom to top. 

 

 

To convert a fractional part of a decimal to an octal number, we can follow the given steps: 

Step 1: Multiply the fractional part by 8. 
Step 2: Write the integer number of the product (0 or 7).
Step 3: Continue the multiplication with the remaining fractional part until the fraction becomes 0. 
Step 4: Write the obtained integers to get the octal number.   
By following these steps, we can get the octal number. 

For example, the given decimal number is (10.25)10 can be converted to an octal number as follows: 
For the integer part (10): 
Step 1: Divide 10 by 8:
     Quotient = 1 
     Remainder = 2

Step 2: Divide 1 by 8: 
     Quotient = 0
      Remainder = 1 

Step 3: Write the remainder from bottom to top. Thus, the octal equivalent is 12.

For the fractional part (0.25): 
Step 1: Multiply 0.25 by 8:
      Product = 2.0 
      Integer part = 2 
Here, the multiplication ends, and the result is 0. Thus, (0.25) = 0.2
Therefore, (10.25)10 = (12.2)8

 

 

4. Octal to decimal number system 

To convert the integer part of an octal number to a decimal, perform these steps: 
Step 1: Write down the octal number. 
Step 2: Multiply each octal number by powers of 8. The first number on the left has the highest power, and it decreases when we move to the left. For example, if the given octal number is  (12.2)8
It can be multiplied as follows:
(1 × 81) + (2 × 80) 
Here, the power is raised to its position, which starts at 0 for the last number on the right and increases by 1 to the left.  
Step 3: Add the results, and the sum is the decimal number. 

Next, we can convert the fractional part of the octal number.
Step 1: Note the octal fraction. 
Step 2: Multiply each number by powers of 8. Here, after the decimal point, the power starts from -1 and goes up.  
Step 3: Add the results, and the sum is the octal fraction.  
For instance, we can convert an octal number (12.2)8 to a decimal. 

Step 1: (1 × 81) + (2 × 80) + (2 × 8 -1) = 8 + 2 + 0.25 = 10.25
Thus, (12.2)8 = (10.25)10 

 

 

5. Decimal to hexadecimal number system 

The given steps can be followed to convert the integer part of a hexadecimal number to a decimal: 
Step 1: Divide each number by the base number, 16.
Step 2: List the remainder (0 to 9 or A to F).
Step 3: Continue the division until the quotient becomes 0. 
Step 4: Write the remainder from bottom to top. 

 

To convert a fractional part of a decimal to a hexadecimal number, we can follow the given steps: 
Step 1: Multiply the fractional part by 16. 
Step 2: Write the integer number of the product (0 to 9 or A to F).
Step 3: Continue the multiplication with the remaining fractional part until the fraction becomes 0. 
Step 4: Write the obtained integers to get the hexadecimal number.   
By following these steps, we can get the hexadecimal number. 

For example, the given decimal number (10.25)10 can be converted to a hexadecimal number as follows: 
Step 1: Divide 10 by 16:
     Quotient = 0
     Remainder = A 
In hexadecimal, 10 is equal to A.

Next, we can convert the fractional part, 0.25.
Step 1: Multiply 0.25 by 16:
0.25 × 16 = 4
Thus, the hexadecimal value 0.25 is equivalent to 0.4
So, (10.25)10 = (A.4)16

 

 

6. Hexadecimal to decimal number system  

To convert the hexadecimal number to a decimal, follow the steps given below: 
Step 1: Write down the hexadecimal number. 
Step 2: Multiply each hexadecimal number by the powers of 16. Here, the power is raised to its position, which starts from 0. 
Step 3: Add the results, and the sum is the decimal number. 

 

Next, we can convert the fractional part of the hexadecimal number.
Step 1: Note the hexadecimal fraction. 
Step 2: Multiply each number by powers of 16. Here, after the decimal point, the power starts from -1 and goes up.  
Step 3: Add the results, and the sum is the hexadecimal fraction.  
For instance, we can convert a hexadecimal number (A.4)16 to a decimal. 
(A.4)16 = (A × 160) + (4 × 16 -1) = (10 × 1) + 4 × 0.0625) 
       = 10 + 0.25 = 10.25
So, (A.4)16 = 10.25

 

 

7. Hexadecimal to Binary number system

 
Remember that a 4-bit binary number is used to represent each hexadecimal number. To convert a hexadecimal number to a binary number, the given steps should be followed:
Step 1: Find the equivalent 4-bit binary number of each hexadecimal number 
Step 2: Write the numbers in a pattern to get the binary number. 
The given table shows the value of the decimal numbers that correspond to each hexadecimal number. 

  
 Let’s look at an example to understand how to convert a hexadecimal number to a binary number. For instance, (3A)16 is a hexadecimal number. 
Now, look at the table and write the equivalent decimal numbers.
(3)16  = (0011)2
(A)16 = (1010)2

Therefore, (3A)16 = (00111010)2

8. Binary to hexadecimal number system 
The given steps can be followed to convert a binary number to a hexadecimal number: 
Step 1: Break the given binary numbers into 4-bits of groups from the right side. 
Step 2: If needed, add zeros to the leftmost bit to make it 4 bits.
Step 3: Find the corresponding hexadecimal number for the binary group.
Step 4: Write them in a pattern to get the hexadecimal number. 

For example, (1111011011)2 is a binary number. 
Now, we can split the given number into groups of 4-bits. 
 
0011  1101  1011
Next, find the equivalent hexadecimal value from the table. 
0011 = 3 
1101 = D 
1011 = B
Therefore, the hexadecimal number equal to the binary number (1111011011)2 is (3DB)16. 
   (1111011011)2 = (3DB)16 
 
9. Binary to octal number system 
A binary number can be converted to a hexadecimal number by following these steps: 
Step 1: Split the given binary number into groups of 3-bit from the right side. 
Step 2: If needed, add zeros to the leftmost group to make them a 3-bit. 
Step 3: Identify the single octal number that equals the 3-bit binary set.  
Step 4: Write them in a pattern to get the octal number. 

 The given table represents the corresponding octal value of each binary group.

Octal
Binary Equivalent
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111

 
For example, convert the given binary number (111101101)2 to an octal number.
We must divide the given number into groups of 3-bits. 
 111   101   101
Find the corresponding octal number from the table. 
111 = 7 
101 = 5 
101 = 5 
Therefore, the octal number of (111101101)2 is (755)8.

10. Octal to binary number system 
There are corresponding 3-bit binary numbers for each single octal number. Therefore, split the octal number, find the binary numbers, and finally write them in order.
For example, convert (153)8 to a binary number. 
We can split the 153 as 1, 5, and 3. 
Now, replace each digit with a binary number. 
 1 = 001 
 5 = 101 
 3 = 011 
Next, write the binary numbers in a pattern. 
Hence, (153)8 = (001101011)2
 

The tips and tricks that should be kept in mind when working with bases of different number systems are: 

 

 

• Any number can be converted from one number system to another using its respective base values. 

 

• When we convert a binary, octal, or hexadecimal number to a decimal number, we multiply each digit by the powers of its base. The powers of base are raised to their position, starting from 0 on the right and increasing as you move to the left. Then, add the results to get the decimal number.  

 

• When we convert a decimal number to a binary, octal, or hexadecimal number, we must divide the number by the base (2, 8, or 16) and remember to list the remainder. Continue the division until the quotient becomes 0, and write the remainder from bottom to top. 
   

In our daily lives, we use the base value of different number systems to convert a number from one system to another. The endless real-world applications of base are: 

 

 

• Electronic devices and computers use a binary number system to process data and information and give instructions and commands. The data and inputs are stored in a binary format in hard drives and memory cards. 

 

• Octal number system and its base are useful for some computing processes and navigation systems. The data in the form of binary is simplified using an octal system, and it helps to execute and generate codes and instructions for aviation and navigation. 

 

• In our everyday lives, we use the decimal number system to count and calculate measurements, money, temperature, and numbers. In finance and baking, the decimal number system is used to represent financial transactions and capital. 

 

• Computers and websites use the hexadecimal number system to identify colors for designing and graphic editing. Technology professionals use the hexadecimal number system to locate the memory addresses and decode the errors of a computer.