Problem 1 - Discrete Mathematics with Applications

Question

Express each number in base 10 . $$ 1101_{\mathrm{two}} $$

Step-by-Step Solution

Answer

The base 10 equivalent of \(1101_{\mathrm{two}}\) is \(13_{\mathrm{ten}}\).

Step 1: Identify each digit and its position

First, let's write down the given binary number and identify each digit (bit) and its position: \(1101_{\mathrm{two}}\) Here, we have four binary digits at the following positions: \(1 \times 2^3\) (This is the highest position, which corresponds to the digit 1 in the 2^3 place) \(1 \times 2^2\) (This digit, also 1, is in the 2^2 place) \(0 \times 2^1\) (This digit is 0, and it's in the 2^1 place) \(1 \times 2^0\) (This is the lowest position, which corresponds to the digit 1 in the 2^0 place)

Step 2: Calculate the decimal value for each position

Now, let's calculate the decimal value for each position: \(1 \times 2^3 = 1 \times 8 = 8\) \(1 \times 2^2 = 1 \times 4 = 4\) \(0 \times 2^1 = 0 \times 2 = 0\) \(1 \times 2^0 = 1 \times 1 = 1\)

Step 3: Sum the decimal values

Lastly, we'll sum up the decimal values we calculated for each position: \(8 + 4 + 0 + 1 = 13\)

Step 4: Conclusion

Therefore, the base 10 equivalent of \(1101_{\mathrm{two}}\) is \(13_{\mathrm{ten}}\).

Key Concepts

Binary Numbers

Binary numbers are part of a number system that uses only two digits: 0 and 1. This system is known as "base-2." Unlike the "decimal system" which relies on ten digits (0 through 9), binary numbers are essential in computer operations because they align perfectly with computer architecture. Computers operate using transistors, which can only be in one of the two states - on or off. Consequently, the binary system is ideal for representing these states.
For example, the binary number \( 1101 \) consists solely of the binary digits 1 and 0. Understanding how to read and interpret these numbers is key to performing binary to decimal base conversions.

Place Value

Place value is a fundamental concept not only in the decimal system but also in binary numbers. Each digit in a binary number has a specific place value, determined by its position in the sequence. The rightmost digit represents the \(2^0\) place, the next one represents \(2^1\), the following represents \(2^2\), and so forth.
In the given binary number \(1101\), each digit has a different place value. The leftmost digit is in the \(2^3\) position, followed by the \(2^2\), \(2^1\), and \(2^0\) positions respectively. By assigning the correct place value to each digit, we can accurately calculate its decimal equivalent.

Base Conversions

Base conversions involve changing numbers from one numeral system to another. This is a common necessity as different systems are better suited to specific types of calculations. To convert from binary (base 2) to decimal (base 10), we multiply each binary digit by \(2^n\), where \(n\) is the position of the digit starting from 0 at the far right.
The conversion process of the binary number \(1101\) involves multiplying each bit by its place value and then summing the results:

• \(1 \times 2^3 = 8\)
• \(1 \times 2^2 = 4\)
• \(0 \times 2^1 = 0\)
• \(1 \times 2^0 = 1\)

Adding these values gives us \(13_{\mathrm{ten}}\), which is the decimal equivalent.

Decimal System

The decimal system, or base-10, is the most widely used numeral system in the world. It employs ten base digits: 0 through 9. Each digit's place value is a power of ten.
For example, in the decimal number 257:

• The 7 is in the \(10^0\) place, representing 7.
• The 5 is in the \(10^1\) place, representing 50.
• The 2 is in the \(10^2\) place, representing 200.

By understanding the decimal system, we can better comprehend the binary system and how base conversions work. This knowledge allows us to visualize the place value of digits across different numeral systems, facilitating conversions and arithmetic operations effortlessly.