53 in Binary

The process of converting 53 from decimal to binary involves dividing the number 53 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 53 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 53 by 2 until getting 0 as the quotient is 110101. 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 110101.

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 110101 in binary is indeed 53 in the decimal number system.

53 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 53 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

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

Since 64 is greater than 53, we stop at 2⁵ = 32.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁵ = 32. 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, 53. Since 2⁵ is the number we are looking for, write 1 in the 2⁵ place. Now the value of 2⁵, which is 32, is subtracted from 53. 53 - 32 = 21.

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, 21. So, the next largest power of 2 is 2⁴, which is 16. Now, we have to write 1 in the 2⁴ places. And then subtract 16 from 21. 21 - 16 = 5.

Step 4 - Identify the next largest power of 2: Now, we find the largest power of 2 that fits into 5, which is 2² = 4. Write 1 in the 2² place and subtract 4 from 5. 5 - 4 = 1.

Step 5 - Identify the next largest power of 2: Now, we find the largest power of 2 that fits into 1, which is 2⁰ = 1. Write 1 in the 2⁰ place and subtract 1 from 1. 1 - 1 = 0. We need to stop the process here since the remainder is 0.

Step 6 - Identify the unused place values: In previous steps, we wrote 1 in the 2⁵, 2⁴, 2², and 2⁰ places. Now, we can just write 0s in the remaining places, which are 2³ and 2¹. Now, by substituting the values, we get, 1 in the 2⁵ place 1 in the 2⁴ place 0 in the 2³ place 1 in the 2² place 0 in the 2¹ place 1 in the 2⁰ place

Step 7 - Write the values in reverse order: We now write the numbers upside down to represent 53 in binary. Therefore, 110101 is 53 in binary.

Grouping Method: In this method, we divide the number 53 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 53 by 2. 53 / 2 = 26. Here, 26 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (26) by 2. 26 / 2 = 13. Here, the quotient is 13 and the remainder is 0.

Step 3 - Repeat the previous step. 13 / 2 = 6. Now, the quotient is 6, and 1 is the remainder.

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

Step 5 - Repeat the previous step. 3 / 2 = 1. Here, 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 bottom to top. Therefore, 53 (decimal) = 110101 (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 53. Since the answer is 2⁵, write 1 next to this power of 2. Subtract the value (32) from 53. So, 53 - 32 = 21. Find the largest power of 2 less than or equal to 21. The answer is 2⁴. So, write 1 next to this power. Subtract 16 from 21 to get 5. Find the largest power of 2 less than or equal to 5. The answer is 2². So, write 1 next to this power. Subtract 4 from 5 to get 1. Find the largest power of 2 less than or equal to 1. The answer is 2⁰. So, write 1 next to this power. Now, 1 - 1 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2³ and 2¹). Final conversion will be 110101.

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, 53 is divided by 2 to get 26 as the quotient and 1 as the remainder. Now, 26 is divided by 2. Here, we will get 13 as the quotient and 0 as the remainder. Dividing 13 by 2, we get 6 as the quotient and 1 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 53, 110101.

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., 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 53. 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 53, we use 0s for 2³ and 2¹ and 1s for 2⁵, 2⁴, 2², and 2⁰.

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 53.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 53.

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 53 (an odd number) is 110101. 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 has occupied 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.

• Powers of 2: These are the values obtained by raising 2 to various exponents (e.g., 2⁰, 2¹, 2²). They are used to determine place values in the binary system.