How to Convert from Binary to Decimal

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Converting numbers from one numeral system to another is a fundamental skill in mathematics and computer science. In the realm of digital systems, the binary (base-2) numeral system is of utmost importance. We commonly use decimal (base-10) numbers in our daily lives, but computers and other digital devices primarily work with binary numbers. Therefore, knowing how to convert a binary number to its decimal equivalent is a crucial skill for understanding how computers operate and manipulating data in a digital environment. In this article, we will explore the step-by-step process of converting binary numbers to decimal numbers, providing clarity on the conversion mechanism and offering practical examples to illustrate the concept further. Whether you are a student exploring the fundamentals of computer science or simply curious about the inner workings of computers, learning how to convert from binary to decimal will expand your comprehension of the fundamental building blocks of digital systems.

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The binary system is the internal language of electronic computers. As a real programmer, you need to know how to convert from binary to decimal . In this article, wikiHow will show you how.

Table of Contents

Steps

Use position notation

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Write binary numbers and lists of powers of 2 from right to left. Let’s say with binary number 10011011 ₂ . First, write this number. Next, write the power of 2 from right to left. Starting at 2 ⁰ , gives the value “1”. Increment the exponent over each power value. Stops when the number of elements in the list is equal to the number of digits present in the binary number. 10011011 has eight digits so our list has eight elements, which are: 128, 64, 32, 16, 8, 4, 2, 1.

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Write the digits of the binary number below its corresponding element in the list of powers of 2. In the example problem, we just need to write 10011011 under the numbers 128, 64, 32, 16, 8 respectively. , 4, 2, and 1. The digit “1” at the end of the binary number corresponds to the rightmost “1” of the power of two. You can also write the digits of the binary number on top if you like. It is important that they correspond to having an element in the power sequence of 2.

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Match the digit in the binary number to the corresponding power of 2. From the right, draw a line connecting each digit of the binary number to a power of 2 just above it. The first is the first digit of the binary number with 2 to the power of 1. Next, the second digit to the 2 to the power of 2. Continue to the end. With this, you can see the relationship between two sets of numbers.

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Write down the final value. For the digit 1, write its corresponding power of 2 just below the dashed line. If it’s a 0, write 0 just below the horizontal line.

• Since “1” corresponds to “1”, our final value will be “1”. “2” corresponds to “1” so the final value will be “2”. “4” corresponds to “0” so the final value will be “0”. “8” corresponds to “1” so the final value is “8” and “16” corresponds to “1” so we have “16”. “32” corresponds to “0” and returns “0”. “64” corresponds to “0” so the final value is “0” while “128” corresponds to “1” so we have 128.

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Add the final values. Now, add the numbers written under the dashed line. We have: 128 + 0 + 0 + 16 + 8 + 0 + 2 + 1 = 155. This is the decimal corresponding to the binary number 10011011.

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Write the sum found with its base. In the example problem, that would be 155 ₁₀ , indicating that this is the answer in decimal. The more you get used to converting from binary to decimal, the easier it will be to remember powers of 2 and the quicker the conversion will become.

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Use this method to convert binary numbers with commas to decimal. You can use this method for binary numbers like 1,1 ₂ . Just remember that the number to the left of the comma is in the units part, as usual, and the number to the right of the comma is in the “half”, or 1 x (1/2) part.

• “1” to the left of the comma is equivalent to 2 ⁰ , or 1. 1 to the right of the comma is equivalent to 2 ^-1 , or ,5. Adding 1 to ,5 gives 1.5, which is 1.1 ₂ when expressed in decimal notation.

Use the double method

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Write binary numbers. With this method, we do not use exponentiation. Therefore, it is easier to do mental calculations on large numbers: for now, you only need to pay attention to the subtotal. First, write down on paper the binary number you plan to convert using this doubling method. Take for example the number 1011001 ₂ . We will write this number down on paper.

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Starting from the left, double the previous sum and add the current digit. For 1011001 ₂ , the leftmost digit is 1. The previous sum is 0 because we haven’t started doing anything before. You will have to double the previous sum, 0, and add 1, the digit in question. 0 x 2 + 1 = 1, so our new sum is 1.

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Double the current sum and add the next digit. The current sum is 1 and the current digit is 0. So multiplying 1 and adding 0, we get: 1 x 2 + 0 = 2. The new sum is 2.

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Repeat the above step. Just continue like that. Double the current sum and add 1, the next digit. 2 x 2 + 1 = 5. The new total is 5.

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Repeat the above step. Double the current sum, 5, and add 1, the next digit. 5 x 2 + 1 = 11. Your new total is 11.

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Repeat the above step. Double the current sum, 11, and add 0, the next digit. 2 x 11 + 0 = 22.

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Repeat next step. Double the current sum, 22, and add 0, the next digit. 22 x 2 + 0 = 44.

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Keep doubling the current sum and adding the next digit until the end. Now we have only the last number left and we’re almost done! All we have to do is take the current sum, 44, double and add 1, the last digit. 2 x 44 + 1 = 89. Done! We converted 10011011 ₂ to 89, its decimal form.

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Write the answer with the same base. Write the answer as 89 ₁₀ to show that here we are working with a decimal number with base 10.

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Use this method to convert from any base to decimal. Here, we’re doubling because the given number has a base 2. For another base, we just need to replace 2 with that base. For example, for a number with base 37, you would replace “x 2” with “x 37”. The result is always decimal (base 10).

Advice

• Practice. Try swapping the binary numbers 11010001 ₂ , 11001 ₂ , and 11110001 ₂ . They correspond to 209 ₁₀ , 25 ₁₀ , and 241 ₁₀ , respectively .
• A PC pre-installed with Microsoft Windows can do the conversion for you, but as a programmer you should understand how this conversion works. You can display the conversion options on your computer by opening the “View” menu and choosing “Scientific” or “Programmer”. On Linux, you can use a personal computer.
• Note: this article is ONLY about the calculation and not about the ASCII encoding.

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Warning

• This article uses unsigned binary numbers instead of signed numbers, static floating point real numbers, or floating point real numbers.

X

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 120 people, some of whom are anonymous, have edited and improved the article over time.

This article has been viewed 339,533 times.

The binary system is the internal language of electronic computers. As a real programmer, you need to know how to convert from binary to decimal . In this article, wikiHow will show you how.

In conclusion, converting from binary to decimal may seem daunting at first, but with the right understanding and practice, it becomes a straightforward process. By using the place value system and understanding the relationship between each digit in binary and its corresponding decimal value, one can successfully convert any binary number to decimal. The step-by-step method explained earlier provides a reliable approach to tackle this conversion. Confidence and familiarity with the binary and decimal number systems will enable individuals to perform these conversions quickly and accurately. Understanding how to convert between binary and decimal is not only helpful in computer science and digital technologies but also enhances our overall numerical literacy. With practice, one can master this process and expand their understanding of numbers beyond the traditional decimal system.

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