208 in Binary

The process of converting 208 from decimal to binary involves dividing the number 208 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 208 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 208 by 2 until getting 0 as the quotient is 11010000. 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 11010000. 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 11010000 in binary is indeed 208 in the decimal number system.

208 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 208 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 208, 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 we have to identify the largest power of 2, which is less than or equal to the given number, 208. 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 208. 208 - 128 = 80.

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, 80. So, the next largest power of 2 is 2⁶ = 64. Now, we have to write 1 in the 2⁶ place. And then subtract 64 from 80. 80 - 64 = 16.

Step 4 - Continue identifying the next largest power of 2: Now, find the largest power of 2 less than or equal to 16, which is 2⁴ = 16. Write 1 in the 2⁴ place. Subtract 16 from 16. 16 - 16 = 0. We need to stop the process here since the remainder is 0.

Step 5 - Identify the unused place values: In steps 2, 3, and 4, we wrote 1 in the 2⁷, 2⁶, and 2⁴ places. Now, we can just write 0s in the remaining places, which are 2⁵, 2³, 2², 2¹, and 2⁰. Now, by substituting the values, we get: 0 in the 2⁰ place 0 in the 2¹ place 0 in the 2² place 0 in the 2³ place 0 in the 2⁵ place 1 in the 2⁴ 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 208 in binary. Therefore, 11010000 is 208 in binary.

Grouping Method: In this method, we divide the number 208 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 208 by 2. 208 / 2 = 104. Here, 104 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (104) by 2. 104 / 2 = 52. Here, the quotient is 52 and the remainder is 0.

Step 3 - Repeat the previous step. 52 / 2 = 26. Now, the quotient is 26, and 0 is the remainder.

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

Step 5 - Continue dividing. 13 / 2 = 6. The quotient is 6, and the remainder is 1.

Step 6 - Continue dividing. 6 / 2 = 3. The quotient is 3, and the remainder is 0.

Step 7 - Continue dividing. 3 / 2 = 1. The quotient is 1, and the remainder is 1.

Step 8 - Continue dividing. 1 / 2 = 0. The quotient is 0, and the remainder is 1. We stop the division here because the quotient is 0.

Step 9 - Write down the remainders from bottom to top. Therefore, 208 (decimal) = 11010000 (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 208. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 208. So, 208 - 128 = 80. Find the largest power of 2 less than or equal to 80. The answer is 2⁶. So, write 1 next to this power. Subtract 64 from 80. Now, 80 - 64 = 16. Find the largest power of 2 less than or equal to 16. The answer is 2⁴. So, write 1 next to this power. Now, 16 - 16 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2⁰, 2¹, 2², 2³, and 2⁵). Final conversion will be 11010000.

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, 208 is divided by 2 to get 104 as the quotient and 0 as the remainder. Now, 104 is divided by 2. Here, we will get 52 as the quotient and 0 as the remainder. Dividing 52 by 2, we get 26 as the quotient and 0 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 208, 11010000.

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 208. 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 208, we use 0s for 2⁰, 2¹, 2², 2³, and 2⁵ and 1s for 2⁷, 2⁶, and 2⁴.

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 208.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 208.
   
• Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary.
   
• Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 208 is even, and its binary form is 11010000. Here, the binary of 208 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 209 (an odd number) is 11010001. 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 that 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 11010000 (binary), the leftmost 1 is in the 2⁷ place, equivalent to 128 in decimal.

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

• Power of 2: In the binary system, each digit's position represents a power of 2, such as 2⁰, 2¹, 2², etc.