131071 in Binary

The process of converting 131071 from decimal to binary involves dividing the number by 2. This division is necessary because the binary number system uses only two digits (0 and 1).

The quotient becomes the dividend in the next step, and the process continues until the quotient becomes 0. This is a commonly used method to convert 131071 to binary. In the last step, the remainder is noted down bottom side up, and that becomes the converted value.

For example, the remainders noted down after dividing 131071 by 2 until getting 0 as the quotient is 11111111111111111. Remember, the remainders here have been written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 11111111111111111.

The second column represents the place values of each digit, and the third column is the value calculation, where the binary digits are multiplied by their corresponding place values.

The results of the third column can be added to cross-check if 11111111111111111 in binary is indeed 131071 in the decimal number system.

131071 can be converted easily from decimal to binary. The methods mentioned below will help us convert the number. Let’s see how it is done.

Expansion Method: Let us see the step-by-step process of converting 131071 using the expansion method.

Step 1 - Figure out the place values: In the binary system, each place value is a power of 2. Therefore, in the first step, we will ascertain the powers of 2. 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 ...(continue this pattern)... 2^16 = 65536 2^17 = 131072 Since 131072 is greater than 131071, we stop at 2^16 = 65536.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^16 = 65536. This is because we have to identify the largest power of 2, which is less than or equal to the given number, 131071. Since 2^16 is the number we are looking for, write 1 in the 2^16 place. Now the value of 2^16, which is 65536, is subtracted from 131071. 131071 - 65536 = 65535.

Step 3 - Identify the next largest power of 2: In this step, we need to continue this process by finding the largest power of 2 that fits into the result of the previous step, 65535, and so on, until the remainder is zero. Now, by substituting the values, we get a series of 1s for each power of 2 from 2^0 to 2^16.

Step 4 - Write the values in reverse order: We now write the numbers upside down to represent 131071 in binary. Therefore, 11111111111111111 is 131071 in binary.

There are certain rules to follow when converting any number to binary. Some of them are mentioned below:

Rule 1: Place Value Method

This is one of the most commonly used rules to convert any number to binary. The place value method is the same as the expansion method, where we need to find the largest power of 2. Let’s see a brief step-by-step explanation to understand the first rule. Find the largest power of 2 less than or equal to 131071. Since the answer is 2^16, write 1 next to this power of 2. Subtract the value (65536) from 131071. So, 131071 - 65536 = 65535. Continue this process for all powers of 2 until the remainder is 0.

Rule 2: Division by 2 Method

The division by 2 method is the same as the grouping method. A brief step-by-step explanation is given below for better understanding. First, 131071 is divided by 2 to get a quotient and a remainder. Now, the quotient is divided by 2, repeating this process until the quotient becomes 0. Write the remainders upside down to get the binary equivalent of 131071, which is 11111111111111111.

Rule 3: Representation Method

This rule also involves breaking the number into powers of 2. Identify the powers of 2 and write them down in decreasing order. Find the largest power that fits into 131071. Repeat the process and allocate 1s to all suitable powers of 2. Combine the digits (0 and 1) to get the binary result.

Rule 4: Limitation Rule

The limitation of the binary system is that only 0s and 1s can be used to represent numbers. The system doesn’t use any other digits other than 0 and 1. This is a base 2 number system, where the binary places represent powers of 2. So, every digit is either a 0 or a 1. To convert 131071, we use 1s for all powers of 2 from 2^0 to 2^16.

Learning a few tips and tricks is a great way to solve any mathematical problems easily. Let us take a look at some tips and tricks for binary numbers up to 131071.

Memorize to speed up conversions: We can memorize the binary forms for powers of 2.

Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary.

Even and odd rule: Whenever a number is even, its binary form will end in 0. If the number is odd, then its binary equivalent will end in 1.

Cross-verify the answers: Once the conversion is done, we can cross-verify the answers by converting the number back to the decimal form. This will eliminate any unforeseen errors in conversion.

Practice by using tables: Writing the decimal numbers and their binary equivalents on a table will help us remember the conversions.

• Decimal: It is the base 10 number system that uses digits from 0 to 9.

• Binary: This number system uses only 0 and 1. It is also called the base 2 number system.

• Octal: It is the number system with a base of 8. It uses digits from 0 to 7.

• Place value: Every digit has a value based on its position in a given number. For example, in 102 (base 10), 1 has occupied the hundreds place, 0 is in the tens place, and 2 is in the ones place.

• Power of 2: In the binary system, each digit's position is expressed as a power of 2, determining its value in the sequence.