Binary To Decimal Conversion: Find Natural Number X

Hey guys! Let's dive into an exciting math problem where we need to convert binary numbers to decimal and then find the value of a natural number, x. This type of problem is super common in mathematics and computer science, so understanding the process is really beneficial. We'll break it down step by step, making it easy to follow along. So, grab your thinking caps, and let’s get started!

Understanding the Problem

Our mission, should we choose to accept it (and we do!), is to determine the natural number x given the equation: 1011(2) + 110111(2) = x(10). What this means is that we have two binary numbers (numbers in base 2) that we need to add together. The result of this addition will then be equal to x, which is represented in base 10, or the decimal system – the one we use every day.

To solve this, we need to perform two main steps:

1. Convert each binary number into its decimal equivalent. This means understanding how binary digits (bits) translate to decimal values.
2. Add the decimal equivalents together. This will give us the value of x in decimal form.

Before we jump into the nitty-gritty, let's quickly recap what binary and decimal systems are all about. This will help solidify our understanding and make the conversion process much smoother.

Diving into Number Systems: Binary and Decimal

The Decimal System (Base 10)

We’re super familiar with the decimal system because it's what we use every day. It's based on 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). Each position in a decimal number represents a power of 10. For example, in the number 123, the '1' represents 1 hundred (10^2), the '2' represents 2 tens (10^1), and the '3' represents 3 ones (10^0).

So, we can break down the number 123 like this:

(1 * 10^2) + (2 * 10^1) + (3 * 10^0) = (1 * 100) + (2 * 10) + (3 * 1) = 100 + 20 + 3 = 123

The Binary System (Base 2)

The binary system is the language of computers! It's a base-2 system, meaning it only uses two digits: 0 and 1. Each position in a binary number represents a power of 2. Just like in the decimal system, the position matters. For instance, in the binary number 101, the '1' on the left represents 1 four (2^2), the '0' represents 0 twos (2^1), and the '1' on the right represents 1 one (2^0).

Let's break down the binary number 101:

(1 * 2^2) + (0 * 2^1) + (1 * 2^0) = (1 * 4) + (0 * 2) + (1 * 1) = 4 + 0 + 1 = 5

So, the binary number 101 is equivalent to the decimal number 5. This understanding is crucial for converting binary numbers to decimal, which is exactly what we need to do to solve our original problem!

Now that we've refreshed our understanding of binary and decimal systems, let's get back to the main task: converting the binary numbers in our equation.

Converting Binary Numbers to Decimal

Okay, guys, time to put our knowledge into action! We need to convert the binary numbers 1011(2) and 110111(2) into their decimal equivalents. We'll use the method we just discussed, where we multiply each digit by the corresponding power of 2 and then add them all up.

Converting 1011(2) to Decimal

Let's break down the binary number 1011:

• The rightmost '1' is in the 2^0 (ones) place.
• The next '1' is in the 2^1 (twos) place.
• The '0' is in the 2^2 (fours) place.
• The leftmost '1' is in the 2^3 (eights) place.

So, we can calculate the decimal equivalent as follows:

(1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = (1 * 8) + (0 * 4) + (1 * 2) + (1 * 1) = 8 + 0 + 2 + 1 = 11

Therefore, the binary number 1011(2) is equal to the decimal number 11.

Converting 110111(2) to Decimal

Now, let's tackle the binary number 110111:

• The rightmost '1' is in the 2^0 (ones) place.
• The next '1' is in the 2^1 (twos) place.
• The next '1' is in the 2^2 (fours) place.
• The '0' is in the 2^3 (eights) place.
• The next '1' is in the 2^4 (sixteens) place.
• The leftmost '1' is in the 2^5 (thirty-twos) place.

Let's calculate the decimal equivalent:

(1 * 2^5) + (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (1 * 2^1) + (1 * 2^0) = (1 * 32) + (1 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (1 * 1) = 32 + 16 + 0 + 4 + 2 + 1 = 55

So, the binary number 110111(2) is equal to the decimal number 55.

We've successfully converted both binary numbers to their decimal equivalents! Now we're just one step away from finding x. Let's move on to the final calculation.

Adding the Decimal Equivalents

We've done the hard work of converting binary numbers to decimal. Now comes the easy part: adding the decimal equivalents together. We found that:

• 1011(2) = 11
• 110111(2) = 55

Our original equation was: 1011(2) + 110111(2) = x(10)

Now we can rewrite it using the decimal values:

11 + 55 = x

Adding 11 and 55, we get:

66 = x

So, the value of x is 66. We've successfully solved the problem!

Conclusion: We Found X!

Alright, guys, we did it! We successfully determined the natural number x by converting binary numbers to decimal and then adding them together. The answer is x = 66. This exercise highlights the importance of understanding different number systems and how to convert between them. It’s a fundamental concept in both mathematics and computer science.

Remember, the key to converting binary to decimal is to recognize that each digit in a binary number represents a power of 2. By multiplying each digit by its corresponding power of 2 and summing the results, we can easily find the decimal equivalent. And, as we've seen, once we have the decimal equivalents, the rest is simple addition!

I hope you found this explanation helpful and that you now feel more confident in tackling similar problems. Keep practicing, and you'll become a pro at number system conversions in no time! If you have any questions or want to explore more examples, feel free to ask. Keep up the great work!