215 in Binary

The process of converting 215 from decimal to binary involves dividing the number 215 by 2. 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 215 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 215 by 2 until getting 0 as the quotient is 11010111. 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 11010111.

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 11010111 in binary is indeed 215 in the decimal number system.

215 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 215 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

2⁷ = 128

2⁸ = 256

Since 256 is greater than 215, we stop at 2⁷ = 128.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁷ = 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, 215. Since 2⁷ is the number we are looking for, write 1 in the 2⁷ place. Now the value of 2⁷, which is 128, is subtracted from 215. 215 - 128 = 87.

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, 87. So, the next largest power of 2 is 2⁶, which is less than or equal to 87. Now, we have to write 1 in the 2⁶ places. And then subtract 64 from 87. 87 - 64 = 23.

Step 4 - Repeat the process for the remaining value: Continue finding the largest suitable power of 2 for the remainder 23, which is 2⁴ = 16. Write 1 in the 2⁴ place and subtract 16 from 23.

23 - 16 = 7.

For 7, the largest power of 2 is 2² = 4.

Write 1 in the 2² place and subtract 4 from 7.

7 - 4 = 3. Finally, for 3, the largest power of 2 is 2¹ = 2.

Write 1 in the 2¹ place and subtract 2 from 3.

3 - 2 = 1.

Finally, 1 is 2⁰.

Step 5 - Identify the unused place values:

In step 2 to 4, we wrote 1 in the 2⁷, 2⁶, 2⁴, 2², 2¹, and 2⁰ places.

Now, we can just write 0s in the remaining places, which are 2⁵ and 2³.

Now, by substituting the values, we get, 1 in the 2⁷ place 1 in the 2⁶ place 0 in the 2⁵ place 1 in the 2⁴ place 0 in the 2³ place 1 in the 2² place 1 in the 2¹ place 1 in the 2⁰ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 215 in binary.

Therefore, 11010111 is 215 in binary.

Grouping Method: In this method, we divide the number 215 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 215 by 2. 215 / 2 = 107. Here, 107 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (107) by 2. 107 / 2 = 53. Here, the quotient is 53 and the remainder is 1.

Step 3 - Repeat the previous step. 53 / 2 = 26. Now, the quotient is 26, and 1 is the remainder.

Step 4 - Repeat the previous step. 26 / 2 = 13. Here, the remainder is 0.

Step 5 - Repeat the previous step. 13 / 2 = 6. Here, the remainder is 1.

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

Step 7 - Repeat the previous step. 3 / 2 = 1. Here, the remainder is 1.

Step 8 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 9 - Write down the remainders from bottom to top.

Therefore, 215 (decimal) = 11010111 (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 215.

Since the answer is 2⁷, write 1 next to this power of 2.

Subtract the value (128) from 215. So, 215 - 128 = 87.

Find the largest power of 2 less than or equal to 87.

The answer is 2⁶.

So, write 1 next to this power. Now, 87 - 64 = 23.

Repeat the process until the remainder is 0.

Final conversion will be 11010111.

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, 215 is divided by 2 to get 107 as the quotient and 1 as the remainder.

Now, 107 is divided by 2. Here, we will get 53 as the quotient and 1 as the remainder.

Dividing 53 by 2, we get 26 as the quotient and 1 as the remainder.

Divide 26 by 2 to get 13 as the quotient and 0 as the remainder.

Divide 13 by 2 to 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 215, 11010111.

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., 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰.

Find the largest power that fits into 215.

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 215, we use 0s for 2⁵ and 2³, and 1s for 2⁷, 2⁶, 2⁴, 2², 2¹, 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 215.

Memorize to speed up conversions: We can memorize the binary forms for numbers with specific patterns.

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, 214 is even, and its binary form is 11010110.

If the number is odd, then its binary equivalent will end in 1. For example, the binary of 215 (an odd number) is 11010111. 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 215 (base 10), 2 occupies the hundreds place, 1 is in the tens place, and 5 is in the ones place.

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

• Quotient: In division, the quotient is the result obtained when one number is divided by another. In binary conversion, the quotient is used as the dividend for the next step in the division process.