How to Convert from Decimal to Binary

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Decimal and binary are two different number systems commonly used in various fields, such as computer science and mathematics. Decimal, also known as base-10, is the numerical system most familiar to us, where numbers are represented using ten different symbols (0-9). On the other hand, binary, often referred to as base-2, is a numerical system that uses only two symbols (0 and 1). Converting a decimal number to its binary equivalent can be a useful skill when working with binary codes or understanding how computers process information. In this guide, we will explore the step-by-step process of converting a decimal number to binary, providing techniques and examples to help you understand and master this conversion method. Whether you are a computer science student, a programmer, or simply interested in expanding your knowledge, learning how to convert from decimal to binary is a valuable skill that can enhance your understanding of number systems and their applications. So, let’s dive into the world of binary and unravel the process of converting decimal numbers to their binary counterparts.

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The decimal system (base ten) has ten values (0,1,2,3,4,5,6,7,8, or 9) for each value. In contrast, the binary system (base two) has two representations of 0 and 1 for each value. ^{[1] X Research Source} Since binary is an intrinsic language used in electronic computers, computer programmers should understand how to convert from decimal to binary. Follow these simple steps to see how to convert.

Table of Contents

Steps

Short division by Two with Remainder

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Problem solving. For this example, we will convert the decimal number 156 ₁₀ to binary. Write the decimal as the number to be divided inside the long division notation. Write down the coefficient of the objective system (in our case, write the number “2” for binary) as the divisor outside the curve of the long division symbol.

• This method is easier to understand when described on paper, and much easier for beginners, because it relies only on division by two.
• To avoid confusion before and after conversion, write the number of the base system you are working on below each number. In this case, the decimal number will have a subscript of 10 and the binary equivalent will have a subscript of 2.

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Division. Enter the quotient below the long division symbol, and write the remainder (0 or 1) to the right of the divisor. ^{[2] X Research Source}

• Since we divide by 2, when the number being divided is even the binary remainder will be 0, and when the number being divided is odd the binary remainder will be 1.

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Continue the division until the division by 2 is zero. Continue the division down below, divide the new quotient by two and write the remainder to the right of the division. Stops when the quotient is 0.

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Write a new binary number. Starting from the balance at the bottom, write the order of the balance from bottom to top. As in this example, you will get 10011100. This is the binary equivalent of the decimal number 156. Or it can be written as subscript: 156 ₁₀ = 10011100 ₂

• This method can be adapted to convert from decimal to ‘any’ system. The divisor is 2 because the system you want to convert is 2 (binary). If the conversion system is another system, replace the divisor by 2 in the calculation with the system you want to convert. For example, if the system you want to convert is a 9, replace the divisor 2 with the number 9. The final result will be according to the system you want to convert.

Decreasing Powers and Subtraction

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Start by making a table. List powers of two in a “table of coefficients 2” from right to left. Starting at 2 ⁰ , whose value is “1”. Increase the exponent by one for each power. Build a series of powers until you’ve come to a number close to the decimal system you’re starting from. In this example, we will convert the decimal number 156 ₁₀ to binary.

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Find the greatest power of 2. Choose the largest number that matches the number you are converting. 128 is the largest power of 2 that matches 156, write the number 1 below this cell in your worksheet as the binary number on the leftmost side. Then subtract 128 from the original number. You get 28.

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Move to the next power of two smaller. Using the new number (28), move down the worksheet to mark the power of 2 that can match the divisor. 64 is greater than 28, write a 0 below it as the next binary number to the right. Continue until you find the number that “could” include in 28.

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Subtract the next number that might match, and mark it with 1. 16 might match 28, so you’d put 1 under that box and subtract 28 from 16. You’d get 12. 8 matches 12, so put 1 below 8 and subtract 12 from 8. You get 4.

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Continue until you finish your workbook. Mark a 1 below the number included in the new number, and put a 0 under the boxes that are greater than the new number.

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Write the resulting binary number. The primary binary numbers are the 1s and 0s below the worksheet from left to right. You will get the binary number 10011100. This is the binary corresponding to the decimal number 156. Or it can be written as subscript: 156 ₁₀ = 10011100 ₂ .

• Repeating this method will help memorize powers of 2, allowing you to skip step 1.

Advice

• The computer installed in your operating system can do this conversion for you, but as a programmer you should have a clear understanding of how the conversion works. You can view your computer’s conversion options by opening the “View” section of the toolbar and selecting “Programmer”.
• Converting backwards, from binary to decimal, is usually easier to learn first.
• Practice. Try converting decimals 178 ₁₀ , 63 ₁₀ , and 8 ₁₀ . The corresponding binary numbers are 10110010 ₂ , 111111 ₂ , and 1000 ₂ . Try converting 209 ₁₀ , 25 ₁₀ , and 241 ₁₀ to the binary numbers 11010001 ₂ , 11001 ₂ , and 11110001 ₂ respectively.

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X

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 94 people, some of whom are anonymous, have edited and improved the article over time.

This article has been viewed 207,215 times.

The decimal system (base ten) has ten values (0,1,2,3,4,5,6,7,8, or 9) for each value. In contrast, the binary system (base two) has two representations of 0 and 1 for each value. ^{[1] X Research Source} Since binary is an intrinsic language used in electronic computers, computer programmers should understand how to convert from decimal to binary. Follow these simple steps to see how to convert.

In conclusion, converting from decimal to binary is a relatively simple process that involves dividing the decimal number by 2 repeatedly until the quotient becomes zero. The remainders obtained at each step are then written in reverse order to obtain the binary representation of the decimal number. This method allows us to represent decimal numbers in a binary format, which is essential in various computational tasks, such as computer programming and digital data storage. By understanding the steps involved in this conversion process, individuals can effectively convert decimal numbers to binary and make use of binary representations in various applications.

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