Number System Conversion

As we all know, the number system is a way of expressing numbers in a structured manner. In number system conversion, we will look at how to convert a number from one base to another base using a different base system. There are many different number systems that can be used, including binary numbers, decimal numbers, hexadecimal numbers, and octal numbers, all of which can be tested.

With the help of examples, you will learn how to convert one base number to another base number while taking into consideration all of the base numbers available, including decimal, binary, octal, and hexadecimal. The following number system conversion methods are discussed in detail in this section.

• System for Converting Binary Numbers to Decimal Numbers
• System of converting between decimal and binary numbers
• Octal Number System to Binary Number System
• Changing from a Binary to an Octal Number System
• System of Binary to Hexadecimal Number Conversion
• The Hexadecimal to Binary Number System is a system that converts numbers from hexadecimal to binary.

Methods for Converting Number Systems

Number system conversions are concerned with the operations that are performed in order to change the base of the numbers. Changing a decimal number with base 10 to a binary number with base 2 is an example of this transformation. On the number system, we can also perform arithmetic operations such as addition, subtraction, and multiplication, among others. In this section, we will learn how to convert numbers from one base to another base, starting with the decimal number system and progressing to the binary number system. When dealing with any base number, the representation of number system base conversion in general form is;

(Number)b = dn-1 dn-2- -.d1d0. d-1 d-2—- d-m 

In the preceding expression, dn-1 dn-2—– is used.

The value of the integer part is represented by d1d0, and the value of the fractional part is represented by d-1 d-2—- d-m.

Aside from that, dn-1 is the Most Significant Bit (MSB) and d-m is the Least Significant Bit (LSB) .

Let us now look at the process of converting from one base to another.

Decimal Conversion to Other Bases

Converting a decimal number to another base number is a straightforward process. We must divide the decimal number by the converted value of the new base in order to get the new base value.

Converting a decimal number to a binary number:

If we need to convert a decimal number to a binary number, we would divide the decimal number by two.

Example 1: Convert the decimal number (25)10 to a binary number.

Solution: Let’s create a table based on the information in this question:

As a result of the information in the preceding table, we can write,

(25)10 = (11001)2

Converting a Decimal Number to an Octal Number

In order to convert a decimal number to an octal number, we must divide the given original number by 8 in such a way that base 10 becomes base 8. Allow us to better understand by way of an illustration.

2nd example: Convert the number 12810 to an octal number.

Method of representation: Let’s represent the conversion in tabular form instead.

As a result, the corresponding octal number is = 2008.

Decimal to Hexadecimal Conversion

Again, in order to convert from decimal to hexadecimal, we must divide the given decimal number by 16.

Example 3: Convert the number 12810 to hexadecimal.

Solution: We can create a table using the method described above.

As a result, the hexadecimal equivalent of 8016 is obtained.

The letters MSB and LSB stand for the most significant bit and the least significant bit, respectively.

Other Base System to Decimal

Conversions Binary to Decimal

During this binary number to decimal number conversion, we employ the multiplication method in such a way that, if a number with base n must be converted into a number with base 10, each digit of the given number is multiplied from MSB to LSB while decreasing the power of the base. Let’s look at an example to better understand how this conversion works.

Consider the first example, which is to convert (1101)2 into a decimal number.

Solution: Given a binary number (1101)2, the solution is

Now, multiplying each digit from the MSB to the LSB while decreasing the power of the base number 2 will result in the answer.

1 × 23 + 1 × 22 + 0 × 21 + 1 × 20

= 8 + 4 + 0 + 1

= 13

Therefore, (1101)2 = (13)10

Octal to Decimal Conversion

As an example, we can multiply the digits of an octal number with decreasing powers of base 8, starting from the MSB to the LSB, and then add them all together to get a decimal representation of the number.

Example 2: Convert the number 228 to a decimal value.

228 is the solution that has been provided.

the sum of 2 x 81 and 2 x 80

= 16 + 2

=18

As a result, 228 equals 1810.

Converting Hexadecimal to Decimal

Example 3: Convert the binary number 12116 to a decimal number.

Solution: 1 x 162 + 2 x 161 + 1 x 160 

= 16 x 16 + 2 x 16 + 1 x 1

= 289

As a result, 12116 = 28910 is obtained.

Method for Converting Hexadecimal to Binary in a Shortcut

It is simple to convert hexadecimal numbers to binary numbers and vice versa; all you need to do is memorise the table provided below.

With the assistance of this table, you can quickly and easily solve problems involving hexadecimal and binary conversions. Take, for instance, the following scenario.

Using the example above, convert (89)16 to a binary number.

Using the table, we can determine the binary value of numbers 8 and 9, which are hexadecimal base numbers.

8 equals 1000, and 9 equals 1001.

As a result, (89)16= (10001001)2 is obtained.

Method for Converting Octal to Binary in a Shortcut

We can simply use the table to convert an octal number to a binary number. In the same way that we have a table for hexadecimal and its equivalent binary, we also have a table for octal and its equivalent binary number in the same way.

Example: Convert the hexadecimal number (214)8 into a binary number.

Solution: Based on what we’ve learned at the table,

2 → 010

1 → 001

4 → 100

As a result, (214)8= (010001100)2 is obtained.

Conclusion

In number system conversion, we will look at how to convert a number from one base to another base using a different base system. There are many different number systems that can be used, including binary numbers, decimal numbers, hexadecimal numbers, and octal numbers, all of which can be tested.Number system conversions are concerned with the operations that are performed in order to change the base of the numbers. Changing a decimal number with base 10 to a binary number with base 2 is an example of this transformation. On the number system, we can also perform arithmetic operations such as addition, subtraction, and multiplication, among others.