131077 in Binary

The process of converting 131077 from decimal to binary involves dividing the number by 2 repeatedly. Here, it is divided by 2 because the binary number system uses only 2 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 decimal numbers to binary. In the last step, the remainder is noted down bottom side up, which becomes the converted value.

For example, the remainders noted down after dividing 131077 by 2 until getting 0 as the quotient is 1111111111110101. 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 1111111111110101.

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 1111111111110101 in binary is indeed 131077 in the decimal number system.

131077 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 131077 using the expansion method.

Step 1 - Figure out the place values: In the binary system, each place value is a power of 2. 2^0 = 1 2^1 = 2 2^2 = 4 ... 2^16 = 65536 2^17 = 131072 2^18 = 262144 Since 262144 is greater than 131077, we stop at 2^17 = 131072.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^17 = 131072. This is because we need to identify the largest power of 2, which is less than or equal to the given number, 131077. Since 2^17 is the number we are looking for, write 1 in the 2^17 place. Now, subtract 131072 from 131077. 131077 - 131072 = 5.

Step 3 - Identify the next largest power of 2: In this step, we need to find the largest power of 2 that fits into the result of the previous step, 5. So, the next largest power of 2 is 2^2, which is equal to 4. Now, we have to write 1 in the 2^2 places. And then subtract 4 from 5. 5 - 4 = 1.

Step 4 - Identify the unused place values: In step 2 and step 3, we wrote 1 in the 2^17, 2^2, and 2^0 places. Now, we can just write 0s in the remaining places.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 131077 in binary. Therefore, 1111111111110101 is 131077 in binary.

Grouping Method: In this method, we divide the number 131077 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 131077 by 2. 131077 / 2 = 65538. Here, 65538 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (65538) by 2. 65538 / 2 = 32769. Here, the quotient is 32769 and the remainder is 0.

Step 3 - Repeat the previous step. 32769 / 2 = 16384. Now, the quotient is 16384, and 1 is the remainder.

Continue this process until the quotient becomes 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 131077 (decimal) = 1111111111110101 (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 131077. Since the answer is 2^17, write 1 next to this power of 2. Subtract the value (131072) from 131077. So, 131077 - 131072 = 5. Find the largest power of 2 less than or equal to 5. The answer is 2^2. So, write 1 next to this power. Now, 5 - 4 = 1. Since there is no remainder, we can write 0 next to the remaining powers except 2^0, where we write 1. Final conversion will be 1111111111110101.

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, 131077 is divided by 2 to get 65538 as the quotient and 1 as the remainder. Now, 65538 is divided by 2. Here, we will get 32769 as the quotient and 0 as the remainder. Continue dividing each quotient by 2 and noting down the remainders until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 131077, 1111111111110101.

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 i.e., 2^17, 2^16, 2^15, and so on. Find the largest power that fits into 131077. Repeat the process and allocate 1s and 0s to the 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 131077, we use 1s for powers of 2 that add up to 131077 and 0s for those that do not.

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.

Memorize to speed up conversions: We can memorize the binary forms for numbers that frequently occur.

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 which uses digits from 0 to 9.

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

• Place value: Every digit has a value based on its position in a given number.

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

• Division by 2 Method: A method of converting decimal numbers to binary by dividing the number by 2 and writing down the remainders.