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Decimal into Binary Conversion

Written by Prerit Jain

Updated on: 15 Aug 2023

Decimal into Binary Conversion

Decimal into Binary Conversion

Decimal into Binary

In Decimal to Binary, we transform a base 10 number into a base 10 number by Decimal to Binary conversion method. Decimal into binary conversion method is simple to transform the given decimal number into binary number. In this decimal into binary conversion we discuss the definition of decimal number system and binary number system with comparison and also discuss the difference between decimal number system and binary number system. In this we explain the decimal number to binary number conversion method 

Decimal Number System Definition

Definition of decimal number system, the number system which involves base 10 that uses a total of 10 numbers such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In the decimal number system the number is defined by base 10. Decimal number system as well called base-10 number system. In mathematics, the decimal number system is also known Hindu-Arabic number system or Arabic number system.

    \[{\left( {11} \right)_{10}}\]

,

    \[{\left( {201} \right)_{10}}\]

are some examples of decimal number systems.

Binary Number system Definition

Definition of binary number system, the number system which involves base 2 that uses only 2 numbers such as 0, 1.  In binary number a one only digit is called bit. A number takes many bits.

    \[{\left( {111} \right)_2}\]

,

    \[{\left( {11011} \right)_2}\]

are some examples of binary number systems.

Comparison of the two number system and their differences

Decimal Number SystemBinary Number System
The Decimal Number System contains 10 numbers such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The Binary Number System contains 2 only two numbers 0, 1.
The base 10 is used for the decimal number system 10.The base 2 is use for binary number system is 2
For example:

    \[{\left( {45} \right)_{10}}\]

,

    \[{\left( {56} \right)_2}\]

 
For example:

    \[{\left( {1110} \right)_2}\]

,

    \[{\left( {10101} \right)_2}\]

 

Conversion Process

In this, we discuss how to convert decimal numbers into binary numbers.

For example: convert number 19 in binary.

In this 19 is defined as a decimal number system. In which number 19 is involving number such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 

Now let us explain how to convert 19 in the binary number system.

These are the following rules for converting a decimal number system into a binary number system.

Rule: 1 for convert decimal number into binary number first divided the number,

IMG_1065.jpg

 In this, here according to the rule 1 decimal number 19 divides by the 2. 

Rule: 2. Now again divide the number by 2. Continue divide the number by 2 until we get the 0 in reminders. 

Rule: 3. MSB is the most important Bit and LSB is the least significant Bit. Write the result/reminders in reverse from or bottom to top from.

Like this  

Hence by the decimal into binary convert method decimal number 19 in binary number

    \[{\left( {10011} \right)_2}\]

Converting fractional decimal numbers to binary

In this, there will be one more conversion of the numbers system. So everything is fine if there is no fractional part. It is very simple to convert a decimal value into a binary number. What we do if given the number in fraction from how it converts into a binary number. 

These are the following steps for converting a fractional decimal number into binary.

Step: 1. Firstly separate the integral part and fractional part.

Step: 2.Now divide the integral part by 2. 

Step: 3. Convert the integral part into binary number by the decimal to binary convert method.

Step: 4.Now, take the fractional part, and multiple by 2 and again repeat the same process until 0. 

Fraction parts become 0. So we have stopped the calculation. 

This will not happen in all cases. Sometimes we do not get the result 0 multiple by 2. It may be that the fractional part is the same as a previous result. So if the fractional part is the same as the previous result that automatically will continue in the process. So in such cases you will stop the calculations.

We will explain with both the examples.

For a fraction, it becomes 0.

Example: 1.

Convert

    \[34.25\]

into binary

Separate the integral part and fractional part in 34.25

    \[\begin{array}{l}34 - {\mathop{\rm int}} egral\\0.25 - fractional\end{array}\]

 

Now, divide the integral part by 2

IMG_1066.jpg

By the decimal to binary convert method decimal number 34 in binary is 

    \[{\left( {100010} \right)_2}\]

 

Now, take the fraction part 0.25 and multiple by 2

We get the result 0.5, and separate the integral part and fraction part. 

And the binary of

    \[0.25\]

is

    \[{\left( {01} \right)_2}\]

This one is arranged from top to bottom.

IMG-1067.jpg

So 

    \[34.25 = {\left( {100010.01} \right)_2}\]

 

Now we will take one more example:

That is 

    \[34.45\]

In this same process of 34 

We get 34 in binary is

    \[{\left( {100010} \right)_2}\]

 

For 0.45

IMG-1069.jpg

In the last case, see the repetition of multiple. So this is the termination in such cases we will not get the 0. 

    \[0.45 = {\left( {011100} \right)_2}\]

 

 34.45 in binary number is  

    \[{\left( {100010.011100} \right)_2}\]

 

Explanation of the descending power of two and subtraction method for conversion

In this method, these are the following steps for converting decimal into binary number.

Convert 201 into binary number

Step: 1. First we need to create a table of power of 2 such as

1286432168421
2726252423222120

Step: 1. See the nearest value of 201, here 128 is nearest value of 201. 

Step: 2. subtract the 128 from 201 reminder is 73 and we write the bit of this is 1. 

Step: 3. Again subtract the 64 from 73 we get the remainder is 9 and write the bit of this is also 1.

Step: 4. Here we will see reminder 9 is smaller than 32. In this case we write 0 bits of this.

IMG-1070.jpg

1286432168421
2726252423222120
11001001

So, 201 in binary number is 

    \[{\left( {11001001} \right)_2}\]

 

Real-world Applications

Human languages can’t be understood by the computer. As a result, we must communicate with the computer to solve our problems. Mostly we use the conversion of decimal number to binary number in computer technology in programming languages such as java, c ++ and so on.

Conclusion

In this chapter, we discuss the method of conversion of decimal numbers into binary numbers. Also we discuss the uses of decimal numbers and binary numbers. 

Solved examples

  1.  

        \[{\left( {202} \right)_{10}}\]

    Convert to a binary number?

    \[{\left( {11001010} \right)_2}\]

 

  1. Convert

        \[{\left( {0.375} \right)_{10}}\]

    into binary numbers?

    \[{\left( {0.011} \right)_2}\]

 

  1. Write

        \[{\left( {11.75} \right)_{10}}\]

    in binary numbers?

The value of 11.75 into binary is 1011.110

  1. Convert

        \[{\left( {201} \right)_{10}}\]

    in binary number by descending power of two and subtraction method?
1286432168421
2726252423222120
1286432168421
2726252423222120
11001001

    \[{\left( {201} \right)_{10}}\]

=

    \[{\left( {11001001} \right)_2}\]

 

  1. What is the value of

        \[15\]

    in binary numbers?

    \[{\left( {1111} \right)_2}\]

FAQs

1. How many numbers consist of a decimal number system?

Ans: The decimal number system contains 9 numbers such as 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

2. How many numbers consist of a binary number system?

Ans: A binary number system consists of only 2 numbers such as 0 and 1.

3. Explain what are bits.

Ans: In the binary number system, one only digit is called a bit.

4. What are the steps for converting a decimal number into a binary number?

Ans: These are the following steps to convert decimal numbers to binary numbers:

Step 1: First divide the decimal number by 2, which we want to convert into a binary number.

Step 2: Continue to divide the decimal number by 2, until we get the 0 in reminder.

Step 3: Write the reminder bottom to top.

5. What is the use of a binary number system?

Ans: The most common use of binary number systems is in programming languages.

6. What is the value of 25 in the binary number system?

Ans: 25 in binary number is

    \[{\left( {11010} \right)_2}\]

 

References

https://www.techtarget.com/whatis/definition/decimal

Written by

Prerit Jain

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