65535 in Binary

The process of converting 65535 from decimal to binary involves dividing the number by 2. Here, it is getting 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 65535 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 65535 by 2 until getting 0 as the quotient is 1111111111111111. 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 1111111111111111. 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 1111111111111111 in binary is indeed 65535 in the decimal number system.

65535 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 65535 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 2² = 4 ... 2¹⁵ = 32768 2¹⁶ = 65536 Since 65536 is greater than 65535, we stop at 2¹⁵ = 32768.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹⁵ = 32768. This is because, in this step, we have to identify the largest power of 2, which is less than or equal to the given number, 65535. Since 2¹⁵ is the number we are looking for, write 1 in the 2¹⁵ place. Now the value of 2¹⁵, which is 32768, is subtracted from 65535. 65535 - 32768 = 32767.

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, 32767. So, the next largest power of 2 is 2¹⁴, which is 16384. Now, we have to write 1 in the 2¹⁴ place. And then subtract 16384 from 32767. 32767 - 16384 = 16383. The process continues until all place values are filled. Finally, by substituting the values, we get, 1111111111111111.

Grouping Method: In this method, we divide the number 65535 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 65535 by 2. 65535 / 2 = 32767. Here, 32767 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (32767) by 2. 32767 / 2 = 16383. Here, the quotient is 16383 and the remainder is 1.

Step 3 - Repeat the previous step for all subsequent quotients until the quotient is 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 65535 (decimal) = 1111111111111111 (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 65535. Since the answer is 2¹⁵, write 1 next to this power of 2. Subtract the value (32768) from 65535. So, 65535 - 32768 = 32767. Continue this process until all place values are filled. Final conversion will be the binary equivalent of 65535.

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, 65535 is divided by 2 to get 32767 as the quotient and 1 as the remainder. Now, 32767 is divided by 2. Here, we will get 16383 as the quotient and 1 as the remainder. Continue this process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 65535.

Rule 3: Representation Method

This rule also involves breaking the number into powers of 2. Identify the powers of 2 and write it down in decreasing order i.e., 2¹⁵, 2¹⁴, ..., 2⁰. Find the largest power that fits into 65535. Repeat the process and allocate 1s to all 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. For 65535, all positions up to 2¹⁵ have a 1.

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 65535.

• Memorize to speed up conversions: We can memorize the binary forms for key numbers.
   
• Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary. 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 ... 32768 + 32768 = 65536 → 10000000000000000
   
• Even and odd rule: Whenever a number is even, its binary form will end in 0. For e.g., 65534 is even and its binary form is 1111111111111110. 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.

• Quotient: The result obtained by dividing one number by another.