214 in Binary

The process of converting 214 from decimal to binary involves dividing the number 214 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 214 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 214 by 2 until getting 0 as the quotient is 11010110. 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 214. 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 11010110 in binary is indeed 214 in the decimal number system.

214 can be converted easily froaluesm 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 214 using the expansion method.

Step 1 - Figure out the place v: 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 214, 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, 214. 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 214. 214 - 128 = 86.

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, 86. So, the next largest power of 2 is 2⁶, which is less than or equal to 86. Now, we have to write 1 in the 2⁶ place. And then subtract 64 from 86. 86 - 64 = 22.

Step 4 - Continue the process: Repeat the steps for the remaining result. 22 - 16 = 6. (Write 1 in the 2⁴ place) 6 - 4 = 2. (Write 1 in the 2² place) 2 - 2 = 0. (Write 1 in the 2¹ place)

Step 5 - Identify the unused place values: In the steps above, we wrote 1 in the 2⁷, 2⁶, 2⁴, 2², and 2¹ places. Now, we can just write 0s in the remaining places, which are 2⁵, 2³, and 2⁰. Now, by substituting the values, we get, 0 in the 2⁰ place 1 in the 2¹ place 0 in the 2² place 1 in the 2³ place 0 in the 2⁴ place 1 in the 25 place 1 in the 2⁶ place 1 in the 2⁷ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 214 in binary. Therefore, 11010110 is 214 in binary.

Grouping Method: In this method, we divide the number 214 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 214 by 2. 214 / 2 = 107. Here, 107 is the quotient and 0 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, 13 is the quotient and 0 is the remainder.

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

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

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

Step 8 - Repeat the previous step. 1 / 2 = 0. Here, 0 is the quotient and 1 is the remainder.

Step 9 - Write down the remainders from bottom to top. Therefore, 214 (decimal) = 11010110 (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 214. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 214. So, 214 - 128 = 86. Find the largest power of 2 less than or equal to 86. The answer is 2^6. So, write 1 next to this power. Now, 86 - 64 = 22. Repeat this process for the remaining numbers. Final conversion will be 11010110.

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, 214 is divided by 2 to get 107 as the quotient and 0 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. Continue this process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 214, 11010110.

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., 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 214. 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 214, we use 0s for 2⁵, 2³, and 2⁰ and 1s for 2⁷, 2⁶, 2⁴, 2², and 2¹.

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 214.

• Memorize to speed up conversions: We can memorize the binary forms for smaller numbers and use them as building blocks for 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., 214 is even and its binary form is 11010110. Here, the binary of 214 ends 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.

• Quotient: The result of division, showing how many times one number is contained within another.