Decimal To Binary

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Decimal To Binary

Converting decimal numbers to binary is a fundamental concept in computer science and digital systems. The decimal number system (base-10) is the one we are most familiar with, where each digit position represents a power of 10. However, in digital systems, binary numbers (base-2) are used, where each digit represents a power of 2.

Please see the Examples below to convert a decimal number to a binary number:


Divide and Remainder Method

 Start by dividing the decimal number by 2 repeatedly and keep track of the remainders. Each remainder will be a digit in the binary representation, read from bottom to top. Continue this process until the quotient becomes 0.

Example: Let's convert the decimal number 25 to binary.

    1. 25 ÷ 2 = 12 remainder 1
    2. 12 ÷ 2 = 6 remainder 0
    3. 6 ÷ 2 = 3 remainder 0
    4. 3 ÷ 2 = 1 remainder 1
    5. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top, the binary representation of 25 is 11001.

Using Powers of 2

 Alternatively, you can use powers of 2 to convert a decimal number to binary. Write down the highest power of 2 that is less than or equal to the decimal number, and then subtract it. Continue this process with the remainder until the remainder becomes 0.

Example: Let's convert the decimal number 45 to binary.

    1. The largest power of 2 less than 45 is 2^5 (32).
    2. Subtracting 32 from 45 leaves a remainder of 13.
    3. The largest power of 2 less than 13 is 2^3 (8).
    4. Subtracting 8 from 13 leaves a remainder of 5.
    5. The largest power of 2 less than 5 is 2^2 (4).
    6. Subtracting 4 from 5 leaves a remainder of 1.
    7. The largest power of 2 less than 1 is 2^0 (1).

The binary representation of 45 is 101101.

Substitution of base 2 

In this method, the number is converted to binary by using powers of 2.

 Example: **Decimal: 23**

  1. Find the nearest and smallest power of 2 (2^4): 16 (1 * 2^4)
  2. Calculate the remainder: 23 - 16 = 7
  3. Find the nearest and smallest power of 2 (2^2): 4 (1 * 2^2)
  4. Calculate the remainder: 7 - 4 = 3
  5. Find the nearest and smallest power of 2 (2^1): 2 (1 * 2^1)
  6. Calculate the remainder: 3 - 2 = 1
  7. Multiply the remainder by 2^0: 1 * 2^0 = 1

As a result, the binary representation of the number is: 10111.

Converting decimal numbers to binary numbers is important for understanding how computers represent and process data. It is also the basis for other number systems used in computers, such as hexadecimal and octal.

Although this method requires more decomposition, each step ensures that the number is decomposed to a power of 2. This method can be particularly useful in the conversion of large numbers.

Remember that both methods will achieve the same result. Whichever method you use, practice will help you understand numbers in binary more easily.

Other methods for converting from decimal to binary:


Bit Array Method

 You can think of the number as an array of bits and convert each digit separately. For example, if you think of the number 23 as 8 bits, you get a sequence of 00010111.

Shift Method

 Each digit in the binary representation of the number can be seen as the remainders you get when you divide the number by 2. These remainders can form the binary representation with a shift operation.

Lookup Table Method

You can use a pre-built lookup table to convert numbers. This table contains the decimal and binary equivalents of the numbers and can be useful to speed up the conversion.

Using Python or a Programming Language

 Programming languages have built-in functions or algorithms for obtaining the binary representation. For example, in Python, you can obtain a number in binary using the bin() function.

Bit Manipulation Operations

 Some programming languages have bit manipulation operators. With these operators, it can be easier to convert or manipulate numbers into binary.

Each of these methods offers different levels of complexity and usefulness. The method you choose will depend on the purpose of the conversion, the use case and your personal preference. By experimenting and practicing each method, you can determine which method works best for you.


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