Decimal Result: 1101101₂ - 4A₁₆ Conversion & Subtraction

Oct 28, 2025 by SLV Team 57 views

Hey guys! Let's break down this math problem where we need to find the decimal result of subtracting a hexadecimal number from a binary number. It might seem a bit tricky at first, but don’t worry, we'll go through it step by step. The key here is understanding number base conversions before we even think about subtracting. So, grab your thinking caps, and let’s dive in!

Understanding Number Base Conversions

Before we can perform the subtraction, we need to make sure both numbers are in the same base. Since the question asks for the result in decimal, our goal is to convert both the binary number (1101101₂) and the hexadecimal number (4A₁₆) into their decimal equivalents. Let's explore how to do that:

Binary to Decimal Conversion

Binary numbers, as you probably know, use base-2, meaning they only have two digits: 0 and 1. Each digit's position represents a power of 2, starting from the rightmost digit as 2⁰, then 2¹, 2², and so on. To convert a binary number to decimal, we multiply each digit by its corresponding power of 2 and then sum the results. Let’s take the binary number 1101101₂ as an example.

To convert 1101101₂ to decimal, we can break it down like this:

(1 x 2⁶) + (1 x 2⁵) + (0 x 2⁴) + (1 x 2³) + (1 x 2²) + (0 x 2¹) + (1 x 2⁰)
= (1 x 64) + (1 x 32) + (0 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1)
= 64 + 32 + 0 + 8 + 4 + 0 + 1
= 109

So, the decimal equivalent of 1101101₂ is 109. See? Not so scary when you break it down! We're essentially figuring out the value each digit contributes based on its position and then adding those values together. This method works for any binary number, no matter how long it is.

Hexadecimal to Decimal Conversion

Hexadecimal numbers, on the other hand, use base-16. This means they have 16 digits: 0-9 and A-F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. Similar to binary, each digit's position in a hexadecimal number represents a power of 16. To convert from hexadecimal to decimal, we multiply each digit by its corresponding power of 16 and sum the results.

Let's convert the hexadecimal number 4A₁₆ to decimal. Here's how:

(4 x 16¹) + (A x 16⁰)
Since A represents 10, we can rewrite this as:
(4 x 16¹) + (10 x 16⁰)
= (4 x 16) + (10 x 1)
= 64 + 10
= 74

Therefore, the decimal equivalent of 4A₁₆ is 74. The process is quite similar to binary conversion, but instead of powers of 2, we're dealing with powers of 16. Remembering that A-F represent 10-15 is crucial for these conversions.

Performing the Subtraction

Now that we've converted both numbers to decimal, we can finally perform the subtraction. We found that 1101101₂ is equal to 109 in decimal, and 4A₁₆ is equal to 74 in decimal. So, the operation becomes:

109 - 74 = 35

Wait a minute! Did I make a mistake? The result 35 is nowhere to be found in the options A) 123, B) 110, C) 85, D) 95. Let's double-check our binary conversion:

(1 x 2⁶) + (1 x 2⁵) + (0 x 2⁴) + (1 x 2³) + (1 x 2²) + (0 x 2¹) + (1 x 2⁰)
= (1 x 64) + (1 x 32) + (0 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (1 x 1)
= 64 + 32 + 0 + 8 + 4 + 0 + 1
= 109

And our hexadecimal conversion:

(4 x 16¹) + (A x 16⁰)
= (4 x 16) + (10 x 1)
= 64 + 10
= 74

Okay, the conversions are correct! So, 109 - 74 still equals 35. It seems there might be an issue with the provided answer choices. The correct answer, based on our calculations, should be 35.

Why Understanding Number Base Conversion is Important

You might be wondering, “Why all this fuss about converting between bases?” Well, in the world of computers and digital systems, different number bases are used for different purposes. Binary is the fundamental language of computers, as it directly corresponds to the on/off states of transistors. Hexadecimal is often used as a shorthand for binary, making it easier for humans to read and write long binary sequences. Decimal, of course, is the base we use in our everyday lives.

Being able to convert between these bases is crucial for anyone working with computers, programming, or digital electronics. It allows you to understand how data is represented and manipulated at a low level. Plus, it's a great exercise for your brain!

Common Mistakes to Avoid

When converting between number bases, there are a few common mistakes you should watch out for:

Forgetting the place values: Always remember that each digit's position represents a power of the base. It’s super important to get this right.
Incorrectly substituting hexadecimal digits: Don't forget that A-F represent 10-15 in hexadecimal. Mixing this up is a common error.
Math errors: Double-check your multiplication and addition. Simple arithmetic mistakes can throw off your entire calculation.
Not double-checking your work: It's always a good idea to review your steps to make sure you haven't made any mistakes. Especially during exams!

Practice Makes Perfect

The best way to master number base conversions is to practice! Try converting different numbers between binary, hexadecimal, and decimal. You can find plenty of online resources and practice problems to help you along the way. The more you practice, the more comfortable you'll become with the process. You'll start to see patterns and develop a sense of how numbers translate between different bases.

Conclusion

So, in conclusion, while the answer choices provided in the original question don't seem to include the correct result (which is 35), we've successfully walked through the process of converting binary and hexadecimal numbers to decimal and performing the subtraction. Remember, the key is to understand the place values and the rules for each number base. Keep practicing, and you'll become a number base conversion pro in no time! And if you ever encounter a similar problem, you’ll know exactly how to tackle it. Keep up the great work, guys!