251 in Binary

The process of converting 251 from decimal to binary involves dividing the number 251 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 251 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 251 by 2 until getting 0 as the quotient is 11111011. 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 11111011. 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 11111011 in binary is indeed 251 in the decimal number system.

251 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 251 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 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 Since 256 is greater than 251, we stop at 2^7 = 128.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^7 = 128. 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, 251. Since 2^7 is the number we are looking for, write 1 in the 2^7 place. Now the value of 2^7, which is 128, is subtracted from 251. 251 - 128 = 123.

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, 123. So, the next largest power of 2 is 2^6, which is less than or equal to 123. Now, we have to write 1 in the 2^6 place. And then subtract 64 from 123. 123 - 64 = 59.

Step 4 - Repeat the above steps: Find the next largest power of 2 that fits into 59, which is 2^5. Write 1 next to 2^5 and subtract 32 from 59. 59 - 32 = 27. The next largest power that fits into 27 is 2^4. Write 1 next to 2^4 and subtract 16 from 27. 27 - 16 = 11. Now, 2^3 fits into 11. Write 1 next to 2^3 and subtract 8 from 11. 11 - 8 = 3. Continue with the next largest power, which is 2^1. Write 1 next to 2^1 and subtract 2 from 3. 3 - 2 = 1. Finally, 2^0 fits into 1. Write 1 next to 2^0.

Step 5 - Write the values: We now have the binary representation of 251 as 11111011.

Grouping Method: In this method, we divide the number 251 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 251 by 2. 251 / 2 = 125. Here, 125 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (125) by 2. 125 / 2 = 62. Here, the quotient is 62 and the remainder is 1.

Step 3 - Repeat the previous step. 62 / 2 = 31. Now, the quotient is 31, and 0 is the remainder.

Step 4 - Repeat the previous step. 31 / 2 = 15. Here, the remainder is 1.

Step 5 - Continue dividing until the quotient becomes 0. 15 / 2 = 7, remainder 1 7 / 2 = 3, remainder 1 3 / 2 = 1, remainder 1 1 / 2 = 0, remainder 1

Step 6 - Write down the remainders from bottom to top. Therefore, 251 (decimal) = 11111011 (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 251. Since the answer is 2^7, write 1 next to this power of 2. Subtract the value (128) from 251. So, 251 - 128 = 123. Find the largest power of 2 less than or equal to 123. The answer is 2^6. So, write 1 next to this power. Continue the process until the remainder is 0. Fill in zeros for the unused powers of 2. Final conversion will be 11111011.

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, 251 is divided by 2 to get 125 as the quotient and 1 as the remainder. Now, 125 is divided by 2. Here, we get 62 as the quotient and 1 as the remainder. Dividing 62 by 2, we get 31 as the quotient and 0 as the remainder. Continue the process until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 251, which is 11111011.

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., 2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Find the largest power that fits into 251. 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 251, we use 0s for unused 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 251.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers, which aids in quickly calculating larger 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 8 + 8 = 16 → 10000 16 + 16 = 32 → 100000…and so on. This is also called the double and add rule.

Even and odd rule: Whenever a number is even, its binary form will end in 0. For e.g., 16 is even and its binary form is 10000. Here, the binary of 16 ends in 0. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 19 (an odd number) is 10011. As you can see, the last digit here is 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 251 (base 10), 2 has occupied the hundreds place, 5 is in the tens place, and 1 is in the ones place.

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

• Remainder: It is the number left after division when the divisor does not divide the dividend exactly.