51 in Binary

The process of converting 51 from decimal to binary involves dividing the number 51 by 2. It is divided by 2 because the binary number system consists of 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 method is commonly used to convert 51 to binary. In the final step, the remainders are noted down in reverse order, which becomes the converted value.

For example, the remainders noted down after dividing 51 by 2 until getting 0 as the quotient is 110011. Remember, the remainders here are written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 110011. 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 110011 in binary is indeed 51 in the decimal number system.

51 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 51 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 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 Since 64 is greater than 51, we stop at 2^5 = 32.

Step 2 - Identify the largest power of 2: We stopped at 2^5 = 32. In this step, identify the largest power of 2 less than or equal to the given number, 51. Since 2^5 is the number we are looking for, write 1 in the 2^5 place. Now the value of 2^5, which is 32, is subtracted from 51. 51 - 32 = 19.

Step 3 - Identify the next largest power of 2: Find the largest power of 2 that fits into the result of the previous step, 19. The next largest power of 2 is 2^4, which is less than or equal to 19. Write 1 in the 2^4 place. Then subtract 16 from 19. 19 - 16 = 3.

Step 4 - Identify the next largest power of 2: Find the largest power of 2 that fits into 3. The next largest is 2^1, which is less than or equal to 3. Write 1 in the 2^1 place. Then subtract 2 from 3. 3 - 2 = 1.

Step 5 - Identify the next largest power of 2: Find the largest power of 2 that fits into 1, which is 2^0. Write 1 in the 2^0 place. Then subtract 1 from 1. 1 - 1 = 0.

Step 6 - Identify the unused place values: In steps 2, 3, 4, and 5, we wrote 1 in the 2^5, 2^4, 2^1, and 2^0 places. Write 0s in the remaining places, 2^3 and 2^2. Now, by substituting the values, we get, 1 in the 2^5 place 1 in the 2^4 place 0 in the 2^3 place 0 in the 2^2 place 1 in the 2^1 place 1 in the 2^0 place

Step 7 - Write the values in reverse order: Now, write the numbers to represent 51 in binary. Therefore, 110011 is 51 in binary.

Grouping Method: In this method, we divide the number 51 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 51 by 2. 51 / 2 = 25. Here, 25 is the quotient, and 1 is the remainder.

Step 2 - Divide the previous quotient (25) by 2. 25 / 2 = 12. Here, the quotient is 12, and the remainder is 1.

Step 3 - Repeat the previous step. 12 / 2 = 6. Now, the quotient is 6, and 0 is the remainder.

Step 4 - Repeat the previous step. 6 / 2 = 3. Now, the quotient is 3, and 0 is the remainder.

Step 5 - Repeat the previous step. 3 / 2 = 1. Now, the quotient is 1, and 1 is the remainder.

Step 6 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 7 - Write down the remainders from the bottom to the top. Therefore, 51 (decimal) = 110011 (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 51. Since the answer is 2^5, write 1 next to this power of 2. Subtract the value (32) from 51. So, 51 - 32 = 19. Find the largest power of 2 less than or equal to 19. The answer is 2^4. So, write 1 next to this power. Now, 19 - 16 = 3. Find the largest power of 2 less than or equal to 3. The answer is 2^1. So, write 1 next to this power. Now, 3 - 2 = 1. Find the largest power of 2 less than or equal to 1. The answer is 2^0. So, write 1 next to this power. Since there is no remainder, we can write 0 next to the remaining powers (2^3 and 2^2). Final conversion will be 110011.

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 a better understanding. First, 51 is divided by 2 to get 25 as the quotient and 1 as the remainder. Now, 25 is divided by 2. Here, we will get 12 as the quotient and 1 as the remainder. Dividing 12 by 2, we get 6 as the quotient and 0 as the remainder. Divide 6 by 2 to get 3 as the quotient and 0 as the remainder. Divide 3 by 2 to get 1 as the quotient and 1 as the remainder. Divide 1 by 2 to get 0 as the quotient and 1 as the remainder. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 51, 110011.

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., 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Find the largest power that fits into 51. 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 51, we use 1s for 2^5, 2^4, 2^1, and 2^0, and 0s for 2^3 and 2^2.

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 51.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 51. 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 example, 52 is even, and its binary form is 110100. Here, the binary of 52 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 51 (an odd number) is 110011. 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 102 (base 10), 1 occupies the hundreds place, 0 is in the tens place, and 2 is in the ones place.

• 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. In binary conversion, it is used to divide the number repeatedly by 2.