Problem 1 - Discrete Mathematics with Applications

Question

Express each number in base \(10 .\) $$1101_{\text {two }}$$

Step-by-Step Solution

Answer

The decimal equivalent of \(1101_{\text{two}}\) is \(13_{\text{ten}}\).

Step 1: Express the number in base 10

In order to express the given base 2 (binary) number, \(1101_{\text{two}}\), in base 10 (decimal), we should follow these steps: 1. Write down the place values of the binary number. 2. Multiply each digit in the binary number by its corresponding place value. 3. Add up the products to get the decimal equivalent. Let's proceed with the steps.

Step 1: Write down place values

Write down the place values of each digit in the binary number. Place values are powers of 2, starting from the right. \(2^0, 2^1, 2^2, 2^3\)

Step 2: Multiply each digit with its place value

Now, multiply each digit of the binary number with its corresponding place value: \(1 \cdot 2^3, 1 \cdot 2^2, 0 \cdot 2^1, 1 \cdot 2^0\) These products represent: \(1\cdot 8, 1\cdot 4, 0\cdot 2, 1\cdot 1\)

Step 3: Add up the products

Finally, add up the products to get the decimal equivalent: \(8 + 4 + 0 + 1\) \(13\) So, the decimal equivalent of \(1101_{\text{two}}\) is \(13_{\text{ten}}\).

Key Concepts

Binary Numbers

Binary numbers are the backbone of computer systems, used to represent and process information. Unlike the familiar decimal system, which is base 10 and uses digits from 0 to 9, binary is a base 2 number system, employing only two digits: 0 and 1. This makes binary a highly efficient way for machines to operate, as electrical circuits can easily distinguish between the two states.

• Each digit in a binary number is called a 'bit'.
• The rightmost bit represents the smallest value, and is often referred to as the least significant bit (LSB).
• The leftmost bit is known as the most significant bit (MSB).

Binary numbers might seem complex at first, but they follow a straightforward logic. Once grasped, they become just another tool in the numerical toolkit.
Understanding how to convert between binary and other number systems, like decimal, is crucial in fields like computer science and digital electronics.

Decimal Numbers

Decimal numbers are what we typically use in daily life, also known as base 10. This system is believed to be so prevalent because we have ten fingers, providing a natural counting method. Decimal numbers use the digits 0 through 9.

• Each digit in a decimal number holds a specific place value, which dictates its true value.
• Moving from right to left, each digit represents a power of 10, starting from 10⁰.

The simplicity of decimal numbers lies in their straightforward representation and intuitive nature for most people. They are fundamental in mathematics and finance. Despite their simplicity, converting decimal numbers to other bases or understanding other base systems often requires additional insight and practice.

Place Value Method

The place value method is a process used to understand numbers by relating each digit to its position or 'place' in the number. This concept applies to any base system, whether binary, decimal, or others, by expressing the number in terms of powers of the base.

• For binary numbers (base 2), place values are powers of 2, such as 2⁰, 2¹, 2², etc.
• For decimal numbers (base 10), place values are powers of 10, like 10⁰, 10¹, 10², and so forth.

Using the Place Value Method

To convert a number from one base to another, one utilizes the place value method by multiplying each digit by its place value and then summing the results. For example:- In binary, '1101' is calculated as: \[1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 13_{\text{ten}}\]This meticulous approach ensures accurate conversions and deepens understanding of how different numerical systems are interconnected.