December 16, 2022

The decimal and binary number systems are the world’s most frequently used number systems today.


The decimal system, also known as the base-10 system, is the system we use in our everyday lives. It uses ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to illustrate numbers. On the other hand, the binary system, also called the base-2 system, utilizes only two digits (0 and 1) to portray numbers.


Learning how to transform from and to the decimal and binary systems are important for multiple reasons. For instance, computers use the binary system to represent data, so computer engineers must be proficient in converting between the two systems.


Additionally, learning how to change between the two systems can help solve mathematical problems including enormous numbers.


This blog article will cover the formula for converting decimal to binary, offer a conversion table, and give examples of decimal to binary conversion.

Formula for Changing Decimal to Binary

The method of converting a decimal number to a binary number is performed manually utilizing the ensuing steps:


  1. Divide the decimal number by 2, and account the quotient and the remainder.

  2. Divide the quotient (only) collect in the previous step by 2, and record the quotient and the remainder.

  3. Reiterate the last steps until the quotient is equal to 0.

  4. The binary equal of the decimal number is acquired by inverting the sequence of the remainders received in the previous steps.


This might sound confusing, so here is an example to illustrate this process:


Let’s change the decimal number 75 to binary.


  1. 75 / 2 = 37 R 1

  2. 37 / 2 = 18 R 1

  3. 18 / 2 = 9 R 0

  4. 9 / 2 = 4 R 1

  5. 4 / 2 = 2 R 0

  6. 2 / 2 = 1 R 0

  7. 1 / 2 = 0 R 1


The binary equivalent of 75 is 1001011, which is obtained by reversing the sequence of remainders (1, 0, 0, 1, 0, 1, 1).

Conversion Table

Here is a conversion chart portraying the decimal and binary equivalents of common numbers:


Decimal

Binary

0

0

1

1

2

10

3

11

4

100

5

101

6

110

7

111

8

1000

9

1001

10

1010


Examples of Decimal to Binary Conversion

Here are some examples of decimal to binary transformation employing the steps discussed priorly:


Example 1: Change the decimal number 25 to binary.


  1. 25 / 2 = 12 R 1

  2. 12 / 2 = 6 R 0

  3. 6 / 2 = 3 R 0

  4. 3 / 2 = 1 R 1

  5. 1 / 2 = 0 R 1


The binary equivalent of 25 is 11001, which is gained by inverting the sequence of remainders (1, 1, 0, 0, 1).


Example 2: Change the decimal number 128 to binary.


  1. 128 / 2 = 64 R 0

  2. 64 / 2 = 32 R 0

  3. 32 / 2 = 16 R 0

  4. 16 / 2 = 8 R 0

  5. 8 / 2 = 4 R 0

  6. 4 / 2 = 2 R 0

  7. 2 / 2 = 1 R 0

  1. 1 / 2 = 0 R 1


The binary equivalent of 128 is 10000000, that is achieved by inverting the invert of remainders (1, 0, 0, 0, 0, 0, 0, 0).


Although the steps defined above provide a method to manually change decimal to binary, it can be tedious and error-prone for big numbers. Fortunately, other ways can be used to quickly and effortlessly convert decimals to binary.


For example, you could employ the built-in functions in a spreadsheet or a calculator application to change decimals to binary. You could also use web-based tools for instance binary converters, that enables you to type a decimal number, and the converter will automatically generate the respective binary number.


It is worth noting that the binary system has some limitations contrast to the decimal system.

For example, the binary system fails to portray fractions, so it is solely suitable for representing whole numbers.


The binary system also requires more digits to represent a number than the decimal system. For instance, the decimal number 100 can be represented by the binary number 1100100, which has six digits. The extended string of 0s and 1s can be prone to typos and reading errors.

Concluding Thoughts on Decimal to Binary

In spite of these limitations, the binary system has several merits with the decimal system. For instance, the binary system is lot easier than the decimal system, as it just utilizes two digits. This simplicity makes it easier to carry out mathematical functions in the binary system, for example addition, subtraction, multiplication, and division.


The binary system is more suited to depict information in digital systems, such as computers, as it can simply be portrayed utilizing electrical signals. As a consequence, understanding how to change among the decimal and binary systems is important for computer programmers and for unraveling mathematical questions concerning large numbers.


Even though the process of changing decimal to binary can be time-consuming and vulnerable to errors when done manually, there are applications that can quickly convert between the two systems.

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