A Guide to Converting Binary to Decimal (and Back Again)

Over the past 5,500 years, humanity has developed more than 100 distinct methods for representing numbers. While Roman numerals have their place, the modern decimal system is currently the most widely used. This system allows users to represent any whole number using just ten digits: 0 through 9.

On the other hand, computers operate differently. Devices like laptops and smartphones depend on binary code, a numerical language that sends precise instructions. It enables your computer to interpret a podcaster's voice, display accurate colors in videos, and count the characters in your boss's latest email.

Binary code truly embodies its name. In contrast to the decimal system, it operates with just two digits, known as "bits." These are typically "0" and "1," and nothing more. Fortunately, we’ll guide you through transforming binary numbers into the more intuitive decimal format. Then, like a skilled magician, we’ll reverse the process, converting decimals back into binary.

Know Your Exponents

Grasping positional notation is essential for navigating both number systems and their conversions. Each digit contributes to the calculation, from the highest to the lowest value. While binary numbers are written using only 0 and 1, interpreting them requires understanding a third element: the number 2.

Let’s illustrate this with an example. The number 138 is represented in binary as "10001010." How does your computer interpret this seemingly random sequence as "138"? Programming plays a role here. Your device has been programmed to recognize that this binary sequence represents a number, not a word or sentence—decoding the latter involves a different approach.

Once this fundamental principle is clear, the code functions by assigning a unique power of 2 to each bit (every 0 and 1). An exponent signifies a number multiplied by itself a specific number of times. For instance, 2 to the third power, written as 2, equals 2 x 2 x 2, which is 8.

Take a moment to explore this handy list of powers of 2. It’s worth your attention!

2 to the power of 0 equals 1

2 to the power of 1 equals 2

2 to the power of 2 equals 4

2 to the power of 3 equals 8

2 to the power of 4 equals 16

2 to the power of 5 equals 32

2 to the power of 6 equals 64

2 to the power of 7 equals 128

2 to the power of 8 equals 256

2 to the power of 9 equals 512

2 to the power of 10 equals 1024

Understanding the Binary Number System

Let’s revisit our initial binary number: 10001010. If you’re a native English speaker, prepare to challenge your instincts. Unlike reading English from left to right, we’ll analyze this binary number by moving from right to left.

In binary numbers, the rightmost bit is multiplied by 2, followed by the bit to its left, and so on, moving leftward. This pattern uses ascending powers of 2, progressing from right to left.

Our task is to continue this process until each bit in the binary number, whether it's a 0 or 1, is paired with its corresponding power of 2. We'll conclude once the leftmost bit has been multiplied by its appropriate exponent of 2.

A practical method to ensure accuracy is to align the exponents directly above their respective binary bits on a piece of paper. The setup should resemble the following:

Converting Binary Numbers to Decimal Value

Great. Now, let’s dive back into converting binary to decimal. Since 10001010 consists of 8 bits, we’ll tackle 8 distinct multiplication tasks. Starting with the rightmost 0, what’s 0 multiplied by 2? The answer is, of course, 0.

One step completed, seven more to go. Shift one place to the left. Notice the "1"? Well, 1 multiplied by 2 equals 2. Move another space left, and you’ll find 0 x 2, which results in 0. By following this method, moving from the rightmost to the leftmost digit, here’s what you’ll discover:

0 multiplied by 2 equals 0

1 multiplied by 2 equals 2

0 multiplied by 2 equals 0

1 multiplied by 2 results in 8

0 multiplied by 2 equals 0

0 multiplied by 2 yields 0

0 multiplied by 2 gives 0

1 multiplied by 2 equals 128

Determining the Decimal Equivalent

Hold on, we’re nearly done! Sum up the results of all those multiplication tasks. Remember, it’s addition, not multiplication. Got it? So, what’s 0 + 2 + 0 + 8 + 0 + 0 + 0 + 128?

Before we solve that, let’s eliminate the zeros. They’re unnecessary in addition. The real task is solving this: 2 + 8 + 128 = ? And the answer? It’s 138. Well done! You’ve completed the cycle. Celebrate your success!

Keep in mind that 138 is an integer. For numbers with fractional parts, such as 0.25 or 3.14, there’s a method to convert them to binary. However, it’s quite complex. If you’re curious and want to explore further, the Institute of Electrical and Electronics Engineers (IEEE) provides a standardized approach.

Decimal Conversion Formula

After converting "10001010" to "138," let’s reverse the process. Imagine starting with 138 and needing to turn it back into binary. Exponents are once again the foundation of this transformation.

Revisit the "powers of 2" list. Identify the value closest to 138 without going over. A quick check reveals 138 falls between 256 (2^8) and 128 (2^7). Subtract 128 from 138: 138 - 128 = 10.

Now, take the result, 10, and refer back to the exponent list. The closest power of 2 to 10 is 8 (2^3). Subtract 8 from 10: 10 - 8 = 2. Interestingly, 2 equals 2^1. This process highlights three key numbers: 128, 8, and 2. Add them together: 128 + 8 + 2 = 138.

Interpreting the Decimal Value

Grab a sheet of paper if you haven’t already. List the values of each power of 2, starting with 128 (2^7) and ending with 1 (2^0). Arrange them in descending order from left to right, leaving space between each number.

Your notes should resemble this sequence: 128, 64, 32, 16, 8, 4, 2, 1. Notice there are eight distinct values. Draw a downward arrow (↓) beneath each one. Then, refer to the addition problem we noted earlier: 128 + 8 + 2 = 138.

Do you spot a "128" in that equation? If yes, place a "1" under its arrow. Is there a "64" in the problem? No! So, write a "0" under that arrow. Follow this pattern, and you’ll end up with:

Does this look familiar? The result is 10001010, which, as we’ve confirmed, translates to "138." In no time, you’ve mastered the binary system, its decimal counterpart, and the conversion process. You’ve also reversed the process, returning to just two digits. Like a magician, you’ve made the rabbit vanish and reappear. Celebrate your success!