Problem 2 - Discrete Mathematics with Applications

Question

Express each number in base \(10 .\) $$11011_{\text {two }}$$

Step-by-Step Solution

Answer

The decimal equivalent of the binary number \(11011_2\) is \(27_{10}\).

Step 1: Identify the powers of 2 for each digit

Start from the rightmost digit (the least significant digit) and follow through to the leftmost digit (the most significant digit), and assign the corresponding power of 2 to each digit. We'll have: $$1 \times 2^0 \\ 1 \times 2^1 \\ 0 \times 2^2 \\ 1 \times 2^3 \\ 1 \times 2^4$$

Step 2: Multiply each digit by the corresponding power of 2

Now, multiply each digit by the power of 2 assigned to it: $$1 \times 2^4 = 1 \times 16 = 16 \\ 1 \times 2^3 = 1 \times 8 = 8 \\ 0 \times 2^2 = 0 \times 4 = 0 \\ 1 \times 2^1 = 1 \times 2 = 2 \\ 1 \times 2^0 = 1 \times 1 = 1$$

Step 3: Add the results of each multiplication

Finally, add the results obtained in Step 2 to get the decimal equivalent of the binary number: $$16 + 8 + 0 + 2 + 1 = 27$$ As a result, the decimal equivalent of the binary number $$11011_2$$ is $$27_{10}$$.

Key Concepts

Base Number Systems

Base number systems are a way of representing numbers using a specific set of digits and a base, which determines the number of unique digits, including zero, that a number system uses. The most commonly known base system is the decimal system, or base 10, which uses ten digits from 0 to 9. There are various base systems used in mathematics and computer science:

• Binary system (base 2): Uses digits 0 and 1, and is fundamental in digital electronics and computing.
• Octal system (base 8): Uses digits 0 to 7, historically used in computing.
• Hexadecimal system (base 16): Uses digits 0 to 9 and letters A to F, widely used in programming and computer systems.

When we convert numbers from one base to another, such as from binary to decimal, we apply the concept of using each digit's position, similar to how we interpret digits in the decimal system, to determine its value based on its position in the number.

Powers of Two

The concept of powers of two is essential in understanding binary numbers, as each binary digit represents a power of two. A power of two is a number of the form \(2^n\), where \(n\) is a non-negative integer. In binary numbers, the rightmost digit (known as the least significant bit) corresponds to \(2^0\), which is equal to 1, and each digit to the left is a higher power of 2. For example, in the binary number \(11011_2\):

• The rightmost 1 is at position \(2^0\), contributing 1 to the overall value.
• The next 1 is at position \(2^1\), contributing 2.
• The 0 is at position \(2^2\), contributing 0.
• The next 1 is at position \(2^3\), contributing 8.
• The leftmost 1 is at position \(2^4\), contributing 16.

This pattern continues for binary numbers, enabling easy calculation of their value in base 10 by summing these contributions.

Decimal System

The decimal system, also known as base 10, is the numeric system most commonly used in everyday life. It utilizes ten digits, from 0 through 9, with each position in a number representing a power of 10. For example, the number 273 in decimal is expanded as:

• The digit 3 is in the \(10^0\) position, which equals 1, thus contributing 3.
• The digit 7 is in the \(10^1\) position, which equals 10, thus contributing 70.
• The digit 2 is in the \(10^2\) position, which equals 100, thus contributing 200.

The decimal value is obtained by summing these contributions: 200 + 70 + 3 = 273.In the context of converting a binary number to the decimal system, we reassign each binary digit's corresponding power of two (as seen in binary systems) and sum their values to achieve the decimal equivalent. This method allows for efficient translation between number systems, which is crucial in computer science for data processing and manipulation.