32767 in Binary

The process of converting 32767 from decimal to binary involves dividing the number 32767 by 2. 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 32767 to binary. In the last step, the remainder is noted down bottom side up, and that becomes the converted value. For example, after dividing 32767 by 2 until getting 0 as the quotient, the remainders noted down are 111111111111111. 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 111111111111111. 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 111111111111111 in binary is indeed 32767 in the decimal number system.

32767 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 32767 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. 2⁰ = 1 2¹ = 2 ... 2¹⁴ = 16384 2¹⁵ = 32768 Since 32768 is greater than 32767, we stop at 2¹⁴ = 16384.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹⁴ = 16384. This is because, in this step, we have to identify the largest power of 2, less than or equal to the given number, 32767. Since 2¹⁴ is the number we are looking for, write 1 in the 2¹⁴ place. Now the value of 2¹⁴, which is 16384, is subtracted from 32767. 32767 - 16384 = 16383.

Step 3 - Continue identifying 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, 16383. Continue this process until the remainder is 0.

Step 4 - Identify the unused place values: Write 0s in any remaining places where powers of 2 are not used.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 32767 in binary. Therefore, 111111111111111 is 32767 in binary.

Grouping Method: In this method, we divide the number 32767 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 32767 by 2. 32767 / 2 = 16383. Here, 16383 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (16383) by 2. 16383 / 2 = 8191. Here, the quotient is 8191 and the remainder is 1.

Step 3 - Continue dividing each quotient by 2 until the quotient is 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 32767 (decimal) = 111111111111111 (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 32767. Since the answer is 2¹⁴, write 1 next to this power of 2. Subtract the value (16384) from 32767. So, 32767 - 16384 = 16383. Continue finding the largest powers of 2 for the remainder and place 1s appropriately. Final conversion will be 111111111111111.

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, 32767 is divided by 2 to get 16383 as the quotient and 1 as the remainder. Continue dividing each quotient by 2 until the quotient is 0. Now, write the remainders upside down to get the binary equivalent of 32767, 111111111111111.

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., 2¹⁴, 2¹³, 2¹²,..., 2⁰. Find the largest power that fits into 32767. 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 32767, we use 1s for the powers of 2 that sum up to 32767.

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 32767.

• Memorize to speed up conversions: We can memorize the binary forms for smaller numbers to aid in larger conversions.
   
• 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.

• 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.

• Power of 2: In binary systems, each digit represents a power of 2.