16383 in Binary

The process of converting 16383 from decimal to binary involves repeatedly dividing the number 16383 by 2. This is done 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 16383 to binary. In the last step, the remainder is noted down bottom-side-up, and that becomes the converted value.

The remainders noted down after dividing 16383 by 2 until getting 0 as the quotient are 11111111111111. 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 11111111111111.

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 11111111111111 in binary is indeed 16383 in the decimal number system.

16383 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 16383 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 2^4 = 16 ... 2^13 = 8192 2^14 = 16384 Since 16384 is greater than 16383, we stop at 2^13 = 8192.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^13 = 8192. 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, 16383. Since 2^13 is the number we are looking for, write 1 in the 2^13 place. Now the value of 2^13, which is 8192, is subtracted from 16383. 16383 - 8192 = 8191.

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, 8191. So, the next largest power of 2 is 2^12, which is less than or equal to 8191. Now, we have to write 1 in the 2^12 places. And then subtract 4096 from 8191. 8191 - 4096 = 4095. Continue this process until you reach 0.

Step 4 - Identify the unused place values: In step 2 and step 3, we wrote 1 in the places. Now, we can just write 0s in the remaining places, which are not used. Now, by substituting the values, we get, 1 in the 2^0 place 1 in the 2^1 place 1 in the 2^2 place ... 1 in the 2^13 place

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 16383 in binary. Therefore, 11111111111111 is 16383 in binary.

Grouping Method: In this method, we divide the number 16383 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 16383 by 2. 16383 / 2 = 8191. Here, 8191 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (8191) by 2. 8191 / 2 = 4095. Here, the quotient is 4095 and the remainder is 1.

Step 3 - Repeat the previous step. 4095 / 2 = 2047. Now, the quotient is 2047, and 1 is the remainder. Continue in this manner until the quotient is 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 16383 (decimal) = 11111111111111 (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 16383. Since the answer is 2^13, write 1 next to this power of 2. Subtract the value (8192) from 16383. So, 16383 - 8192 = 8191. Find the largest power of 2 less than or equal to 8191. The answer is 2^12. So, write 1 next to this power. Continue this process until the remainder is 0. Final conversion will be 11111111111111.

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, 16383 is divided by 2 to get 8191 as the quotient and 1 as the remainder. Now, 8191 is divided by 2. Here, we will get 4095 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 16383, 11111111111111.

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., 2^13, 2^12, ..., 2^0. Find the largest power that fits into 16383. 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 16383, we use 1s for all powers up to 2^13.

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 16383.

Memorize to speed up conversions: We can memorize the binary forms for numbers to quickly convert them.

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. For example, in 11111111111111 (binary), each 1 has a place value corresponding to the power of 2.

• Octal: It is the number system with a base of 8. It uses digits from 0 to 7 and can represent numbers using fewer digits than binary.

• Hexadecimal: This number system has 16 digits. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here, A to F represent the values 10 to 15.