Number System Conversion

As we know, the number system is a form of expressing the numbers. In number system conversion, we will study to convert a number of one base, to a number of another base. There are a variety of number systems such as binary numbers, decimal numbers, hexadecimal numbers, octal numbers, which can be exercised.

In this article, you will learn the conversion of one base number to another base number considering all the base numbers such as decimal, binary, octal and hexadecimal with the help of examples. Here, the following number system conversion methods are explained.

• Binary to Decimal Number System
• Decimal to Binary Number System
• Octal to Binary Number System
• Binary to Octal Number System
• Binary to Hexadecimal Number System
• Hexadecimal to Binary Number System

Get the pdf of number system with a brief description in it. The general representation of number systems are;

Decimal Number – Base 10 – N10

Binary Number – Base 2 – N2

Octal Number – Base 8 – N8

Hexadecimal Number – Base 16 – N16

Number System Conversion Table

Binary Numbers	Octal Numbers	Decimal Numbers	Hexadecimal Numbers
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	10	8	8
1001	11	9	9
1010	12	10	A
1011	13	11	B
1100	14	12	C
1101	15	13	D
1110	16	14	E
1111	17	15	F

Number System Conversion Methods

Number system conversions deal with the operations to change the base of the numbers. For example, to change a decimal number with base 10 to binary number with base 2. We can also perform the arithmetic operations like addition, subtraction, multiplication on the number system. Here, we will learn the methods to convert the number of one base to the number of another base starting with the decimal number system. The representation of number system base conversion in general form for any base number is;

(Number)b = dn-1 dn-2—–.d1 d0 . d-1 d-2 —- d-m

In the above expression, dn-1 dn-2—–.d1 d0 represents the value of integer part and d-1 d-2 —- d-m represents the fractional part.

Also, dn-1 is the Most significant bit (MSB) and d-m is the Least significant bit (LSB).

Now let us learn, conversion from one base to another.

Decimal to Other Bases

Converting a decimal number to other base numbers is easy. We have to divide the decimal number by the converted value of the new base.

Decimal to Binary Number:

Suppose if we have to convert decimal to binary, then divide the decimal number by 2.

Example 1. Convert (25)10 to binary number.

Solution: Let us create a table based on this question.

Operation	Output	Remainder
25 ÷ 2	12	1(MSB)
12 ÷ 2`	6	0
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1(LSB)

Therefore, from the above table, we can write,

(25)10 = (11001)2

Decimal to Octal Number:

To convert decimal to octal number we have to divide the given original number by 8 such that base 10 changes to base 8. Let us understand with the help of an example.

Example 2: Convert 12810 to octal number.

Solution: Let us represent the conversion in tabular form.

Operation	Output	Remainder
128÷8	16	0(MSB)
16÷8	2	0
2÷8	0	2(LSB)

Therefore, the equivalent octal number = 2008

Decimal to Hexadecimal:

Again in decimal to hex conversion, we have to divide the given decimal number by 16.

Example 3: Convert  12810 to hex.

Solution: As per the method, we can create a table;

Operation	Output	Remainder
128÷16	8	0(MSB)
8÷16	0	8(LSB)

Therefore, the equivalent hexadecimal number is 8016

Here MSB stands for a Most significant bit and LSB stands for a least significant bit.

Other Base System to Decimal Conversion

Binary to Decimal:

In this conversion, binary number to a decimal number, we use multiplication method, in such a way that, if a number with base n has to be converted into a number with base 10, then each digit of the given number is multiplied from MSB to LSB with reducing the power of the base. Let us understand this conversion with the help of an example.

Example 1. Convert (1101)2 into a decimal number.

Solution: Given a binary number (1101)2.

Now, multiplying each digit from MSB to LSB with reducing the power of the base number 2.

1 × 23 + 1 × 22 + 0 × 21 + 1 × 20

= 8 + 4 + 0 + 1

= 13

Therefore, (1101)2 = (13)10

Octal to Decimal:

To convert octal to decimal, we multiply the digits of octal number with decreasing power of the base number 8, starting from MSB to LSB and then add them all together.

Example 2: Convert 228 to decimal number.

Solution: Given, 228

2 x 81 + 2 x 80

= 16 + 2

= 18

Therefore, 228 = 1810

Hexadecimal to Decimal:

Example 3: Convert 12116 to decimal number.

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

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

= 289

Therefore, 12116 = 28910

Hexadecimal to Binary Shortcut Method

To convert hexadecimal numbers to binary and vice versa is easy, you just have to memorize the table given below.

Hexadecimal Number	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
A	1010
B	1011
C	1100
D	1101
E	1110
F	1111

You can easily solve the problems based on hexadecimal and binary conversions with the help of this table. Let us take an example.

Example: Convert (89)16 into a binary number.

Solution: From the table, we can get the binary value of 8 and 9, hexadecimal base numbers.

8 = 1000 and 9 = 1001

Therefore, (89)16 = (10001001)2

Octal to Binary Shortcut Method

To convert octal to binary number, we can simply use the table. Just like having a table for hexadecimal and its equivalent binary, in the same way, we have a table for octal and its equivalent binary number.

Octal Number	Binary
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

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

Solution: From the table, we know,

2 → 010

1 → 001

4 → 100

Therefore,(214)8 = (010001100)2

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