239 in Binary

The process of converting 239 from decimal to binary involves dividing the number 239 by 2. Here, 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 239 to binary. In the last step, the remainder is noted down from bottom to top, and that becomes the converted value. For example, the remainders noted down after dividing 239 by 2 until getting 0 as the quotient is 11101111. 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 11101111. 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 11101111 in binary is indeed 239 in the decimal number system.

239 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 239 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 239, 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, 239. 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 239. 239 - 128 = 111.

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, 111. So, the next largest power of 2 is 2⁶ = 64. Write 1 in the 2⁶ place and subtract 64 from 111. 111 - 64 = 47.

Step 4 - Continue the process: Repeat the process of finding the largest power of 2 for the remaining numbers. For 47, the largest power of 2 is 2⁵ = 32. Write 1 in the 2⁵ place. 47 - 32 = 15. For 15, the largest power of 2 is 2³ = 8. Write 1 in the 2³ place. 15 - 8 = 7. For 7, the largest power of 2 is 2² = 4. Write 1 in the 2² place. 7 - 4 = 3. For 3, the largest power of 2 is 2¹ = 2. Write 1 in the 2¹ place. 3 - 2 = 1. For 1, the largest power of 2 is 2⁰ = 1. Write 1 in the 2⁰ place. 1 - 1 = 0.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 239 in binary. Therefore, 11101111 is 239 in binary.

Grouping Method: In this method, we divide the number 239 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 239 by 2. 239 / 2 = 119. Here, 119 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (119) by 2. 119 / 2 = 59. Here, the quotient is 59 and the remainder is 1.

Step 3 - Repeat the previous step. 59 / 2 = 29. Now, the quotient is 29 and 1 is the remainder.

Step 4 - Repeat the previous step. 29 / 2 = 14. Here, the quotient is 14 and 1 is the remainder.

Step 5 - Continue the process: 14 / 2 = 7. Quotient is 7, remainder is 0. 7 / 2 = 3. Quotient is 3, remainder is 1. 3 / 2 = 1. Quotient is 1, remainder is 1. 1 / 2 = 0. Quotient is 0, remainder is 1.

Step 6 - Write down the remainders from bottom to top. Therefore, 239 (decimal) = 11101111 (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 239. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 239. So, 239 - 128 = 111. Find the largest power of 2 less than or equal to 111. The answer is 2⁶. So, write 1 next to this power. Continue the process until you get a remainder of 0.

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, 239 is divided by 2 to get 119 as the quotient and 1 as the remainder. Now, 119 is divided by 2. Here, we will get 59 as the quotient and 1 as the remainder. Dividing 59 by 2, we get 29 as the quotient and 1 as the remainder. Continue the division process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 239, 11101111.

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 239. 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.

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 239.

• Memorize to speed up conversions: We can memorize the binary forms for numbers in specific ranges, such as powers of 2.
   
• 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, 238 is even, and its binary form is 11101110. Here, the binary of 238 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 239 (an odd number) is 11101111. 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 239 (base 10), 2 is in the hundreds place, 3 is in the tens place, and 9 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, the quotient is used in subsequent division steps until it becomes zero.