12475 in Binary

The process of converting 12475 from decimal to binary involves dividing the number 12475 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 12475 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 12475 by 2 until getting 0 as the quotient is 11000010010111. 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 11000010010111.

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 11000010010111 in binary is indeed 12475 in the decimal number system.

12475 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 12475 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 ... 2^13 = 8192 2^14 = 16384 Since 16384 is greater than 12475, 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, 12475. 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 12475. 12475 - 8192 = 4283.

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, 4283. So, the next largest power of 2 is 2^12, which is less than or equal to 4283. Now, we have to write 1 in the 2^12 places. And then subtract 4096 from 4283. 4283 - 4096 = 187. We need to continue the process until the remainder is 0.

Step 4 - Continue the steps: Repeat steps to find the next largest power of 2 and write 1 or 0 in each position until we reach 0. 187 - 128 = 59 (next largest power is 2^7) 59 - 32 = 27 (next largest power is 2^5) 27 - 16 = 11 (next largest power is 2^4) 11 - 8 = 3 (next largest power is 2^3) 3 - 2 = 1 (next largest power is 2^1) 1 - 1 = 0 (next largest power is 2^0) Now, by substituting the values, we get, 1 in the 2^13 place 1 in the 2^12 place 0 in the 2^11 place 0 in the 2^10 place 0 in the 2^9 place 0 in the 2^8 place 1 in the 2^7 place 0 in the 2^6 place 0 in the 2^5 place 1 in the 2^4 place 1 in the 2^3 place 1 in the 2^2 place 1 in the 2^1 place 1 in the 2^0 place

Step 5 - Combine the values to write the binary: Therefore, 11000010010111 is 12475 in binary.

Grouping Method: In this method, we divide the number 12475 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 12475 by 2. 12475 / 2 = 6237 with a remainder of 1.

Step 2 - Divide the previous quotient (6237) by 2. 6237 / 2 = 3118 with a remainder of 1.

Step 3 - Repeat the previous step. 3118 / 2 = 1559 with a remainder of 0.

Step 4 - Repeat the previous step. 1559 / 2 = 779 with a remainder of 1.

Step 5 - Continue the process until you reach quotient 0. 779 / 2 = 389 with a remainder of 1. 389 / 2 = 194 with a remainder of 0. 194 / 2 = 97 with a remainder of 0. 97 / 2 = 48 with a remainder of 1. 48 / 2 = 24 with a remainder of 0. 24 / 2 = 12 with a remainder of 0. 12 / 2 = 6 with a remainder of 0. 6 / 2 = 3 with a remainder of 0. 3 / 2 = 1 with a remainder of 1. 1 / 2 = 0 with a remainder of 1.

Step 6 - Write down the remainders from bottom to top. Therefore, 12475 (decimal) = 11000010010111 (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 12475. Since the answer is 2^13, write 1 next to this power of 2. Subtract the value (8192) from 12475. So, 12475 - 8192 = 4283. Find the largest power of 2 less than or equal to 4283. The answer is 2^12. So, write 1 next to this power. Continue this process until the remainder reaches 0. Final conversion will be 11000010010111.

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, 12475 is divided by 2 to get 6237 as the quotient and 1 as the remainder. Now, 6237 is divided by 2. Here, we will get 3118 as the quotient and 1 as the remainder. Continue dividing the quotient by 2 until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 12475, 11000010010111.

Rule 3: Representation Method

This rule also involves breaking down the number into powers of 2. Identify the powers of 2 and write them down in decreasing order i.e., 2^13, 2^12, 2^11, etc. Find the largest power that fits into 12475. 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 12475, we use 1s and 0s according to the powers of 2.

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 12475.

Memorize to speed up conversions: We can memorize the binary forms for numbers, especially smaller ones, to speed up conversion processes.

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.

• Power of 2: Each digit's place value in the binary system is based on powers of 2.