What is the base 10 representation of 11102 – Embark on a mathematical expedition to unravel the mysteries of converting binary numbers to their decimal counterparts. Join us as we decipher the binary code 11102, translating it into the familiar realm of base 10.
Delving into the intricacies of binary and decimal systems, we will explore the fundamental principles that govern their operations. By dissecting the binary number 11102, we will uncover the significance of each digit and its contribution to the overall decimal representation.
1. Understanding Binary and Base 10 Systems
The binary number system, also known as base 2, is a system of numerical representation that uses only two digits: 0 and 1. On the other hand, the base 10 number system, also known as the decimal system, is a system of numerical representation that uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
In the binary number system, each digit represents a power of 2, with the rightmost digit representing 2^0, the next digit representing 2^1, and so on. In the base 10 number system, each digit represents a power of 10, with the rightmost digit representing 10^0, the next digit representing 10^1, and so on.
Examples of Binary and Base 10 Numbers, What is the base 10 representation of 11102
- Binary: 1011
- Base 10: 11
- Binary: 11001
- Base 10: 25
2. Converting Binary to Base 10
To convert a binary number to its base 10 representation, we need to multiply each digit of the binary number by its corresponding power of 2 and then add the results.
For example, to convert the binary number 11001 to base 10, we would do the following:
- 1 x 2^4 = 16
- 1 x 2^3 = 8
- 0 x 2^2 = 0
- 0 x 2^1 = 0
- 1 x 2^0 = 1
Adding these results together, we get 16 + 8 + 0 + 0 + 1 = 25. Therefore, the base 10 representation of the binary number 11001 is 25.
Place Value of Digits in Binary and Base 10
The place value of a digit in a number system is the value that the digit represents. In the binary number system, the place value of a digit is a power of 2, while in the base 10 number system, the place value of a digit is a power of 10.
The following table shows the place values of digits in binary and base 10:
| Place Value | Binary | Base 10 |
|---|---|---|
| Ones | 2^0 | 10^0 |
| Tens | 2^1 | 10^1 |
| Hundreds | 2^2 | 10^2 |
| Thousands | 2^3 | 10^3 |
| Ten Thousands | 2^4 | 10^4 |
3. Analyzing the Binary Number 11102
The binary number 11102 is made up of the following individual binary digits:
- 1
- 1
- 1
- 0
- 2
The place values of these digits are as follows:
- 1 x 2^4 = 16
- 1 x 2^3 = 8
- 1 x 2^2 = 4
- 0 x 2^1 = 0
- 2 x 2^0 = 2
4. Calculating the Base 10 Representation
To calculate the base 10 representation of the binary number 11102, we need to multiply each digit by its corresponding place value and then add the results.
16 + 8 + 4 + 0 + 2 = 30
Therefore, the base 10 representation of the binary number 11102 is 30.
5. Illustrating the Conversion Process
The following flowchart illustrates the conversion process from binary to base 10:
- Start with the binary number.
- Multiply each digit by its corresponding power of 2.
- Add the results together.
- The final result is the base 10 representation of the binary number.
For example, to convert the binary number 11102 to base 10, we would follow the following steps:
- 1 x 2^4 = 16
- 1 x 2^3 = 8
- 1 x 2^2 = 4
- 0 x 2^1 = 0
- 2 x 2^0 = 2
- 16 + 8 + 4 + 0 + 2 = 30
Therefore, the base 10 representation of the binary number 11102 is 30.
FAQ Corner: What Is The Base 10 Representation Of 11102
What is the binary number system?
The binary number system is a base-2 number system that uses only two digits, 0 and 1. It is commonly used in digital electronics and computer science.
What is the decimal number system?
The decimal number system is a base-10 number system that uses ten digits, 0 to 9. It is the most commonly used number system in everyday life.
How do I convert a binary number to a decimal number?
To convert a binary number to a decimal number, multiply each digit in the binary number by its corresponding power of 2 and then add the results.
