In the world of mathematics and computer science, number systems play a fundamental role in representing and manipulating numbers. One such conversion is from binary to decimal. Binary is a base-2 numeral system, while decimal is a base-10 numeral system. Understanding how to convert binary numbers to decimal is essential for various applications in computer programming and electronic systems.
What is a Binary Number System?
The binary number system is a mathematical notation that uses only two digits, 0 and 1. It is also known as the base-2 numeral system because it represents numeric values using two distinct symbols. In binary, each digit represents a power of 2, with the rightmost digit being the least significant bit (LSB) and the leftmost digit being the most significant bit (MSB).
What is a Decimal Number System?
The decimal number system, also known as the base-10 numeral system, is the most commonly used number system in our daily lives. It uses ten digits from 0 to 9 to represent numeric values. In the decimal system, the position of each digit to the left of the decimal point represents units, tens, hundreds, thousands, and so on. The base of the decimal number system is 10.
What is Binary to Decimal Conversion?
Binary to decimal conversion is the process of converting a number given in the binary system to its equivalent in the decimal system. This conversion is necessary to interpret binary numbers, which are represented as a sequence of 0s and 1s, into decimal numbers that are more easily understood by humans.
How to Convert Binary to Decimal Numbers?
There are two main methods for converting binary to decimal numbers: the positional notation method and the doubling method. Let’s explore each method in detail.
Binary to Decimal Conversion Using Positional Notation Method
The positional notation method involves multiplying each digit of the binary number by the corresponding power of 2 based on its position. Here are the steps to convert binary to decimal using this method:
- Write down the binary number and count the powers of 2 from right to left, starting from 0.
- Multiply each binary digit from right to left with the corresponding power of 2.
- Add all the products obtained in the previous step.
- The sum of the products is the decimal equivalent of the binary number.
Binary to Decimal Conversion Using Doubling Method
The doubling method is another approach to convert binary to decimal. It involves doubling the previous number and adding the current digit for each digit in the binary number, starting from the leftmost digit. Here are the steps to convert binary to decimal using this method:
- Write the binary number and start from the leftmost digit.
- Double the previous number (or consider it as 0 for the leftmost digit) and add the current digit.
- Repeat the above step for each digit in the binary number.
- The final result is the decimal equivalent of the binary number.
Binary to Decimal Formula
The formula for converting a binary number to a decimal number is as follows:
Decimal Number = (b0 × 20) + (b1 × 21) + (b2 × 22) + … + (bn × 2n)
Here, b0, b1, b2, …, bn are the binary digits, and n is the total number of digits in the binary number.
Binary to Decimal Table
The conversion chart below shows the binary to decimal conversion for commonly used binary numbers:
| Binary | Decimal |
|---|---|
| 0000 | 0 |
| 0001 | 1 |
| 0010 | 2 |
| 0011 | 3 |
| 0100 | 4 |
| 0101 | 5 |
| 0110 | 6 |
| 0111 | 7 |
| 1000 | 8 |
| 1001 | 9 |
| 1010 | 10 |
| 1011 | 11 |
| 1100 | 12 |
| 1101 | 13 |
| 1110 | 14 |
| 1111 | 15 |
How to use Binary to Decimal Calculator?
To convert binary to decimal, you can use a binary to decimal converter tool. Here’s how you can use it:
- Enter the binary number in the input field.
- Click on the convert button to initiate the conversion process.
- The result will be displayed, showing the decimal equivalent of the binary number.
Using a binary to decimal calculator makes the conversion process quick and accurate, saving you time and effort.
How to Read a Binary Number?
Reading a binary number involves understanding the positional value of each digit in the number. Each digit in a binary number represents a power of 2, starting from the rightmost digit. The rightmost digit is the least significant bit (LSB), and the leftmost digit is the most significant bit (MSB). To read a binary number, multiply each digit by the corresponding power of 2 and sum up the results.
For example, to read the binary number 1010, we multiply the leftmost digit (1) by 2^3, the next digit (0) by 2^2, the next digit (1) by 2^1, and the rightmost digit (0) by 2^0. The sum of these products gives us the decimal equivalent of the binary number.
How to Read a Decimal Number?
Reading a decimal number is straightforward as it follows the base-10 system that we use in our daily lives. Each digit in a decimal number represents a power of 10, starting from the rightmost digit. The rightmost digit is the ones place, the next digit to the left is the tens place, the next is the hundreds place, and so on. To read a decimal number, multiply each digit by the corresponding power of 10 and sum up the results.
For example, to read the decimal number 456, we multiply the rightmost digit (6) by 10^0, the next digit (5) by 10^1, and the leftmost digit (4) by 10^2. The sum of these products gives us the value of the decimal number.
Solved Examples on Binary to Decimal Conversion
Let’s solve a few examples to demonstrate the process of converting binary numbers to decimal numbers:
Example 1: Convert the binary number (1101) to a decimal number.
Solution: Given binary number = (1101) To convert it to the decimal number, we multiply each digit from the left to right by the powers of 2. (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0) = 8 + 4 + 0 + 1 = 13 Therefore, the decimal equivalent of the binary number (1101) is 13.
Example 2: Convert the binary number (101010) to a decimal number.
Solution: Given binary number = (101010) To convert it to the decimal number, we multiply each digit from the left to right by the powers of 2. (1 × 2^5) + (0 × 2^4) + (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0) = 32 + 0 + 8 + 0 + 2 + 0 = 42 Therefore, the decimal equivalent of the binary number (101010) is 42.
Example 3: Convert the binary number (111011) to a decimal number.
Solution: Given binary number = (111011) To convert it to the decimal number, we multiply each digit from the left to right by the powers of 2. (1 × 2^5) + (1 × 2^4) + (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0) = 32 + 16 + 8 + 0 + 2 + 1 = 59 Therefore, the decimal equivalent of the binary number (111011) is 59.
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