234 in Binary

The process of converting 234 from decimal to binary involves dividing the number 234 by 2. This is done 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 234 to binary. In the last step, the remainder is noted down in reverse order, and that becomes the converted value. For example, the remainders noted down after dividing 234 by 2 until getting 0 as the quotient are 11101010. Remember, the remainders here have been written in reverse order.

In the table shown below, the first column shows the binary digits (1 and 0) as 11101010. 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 11101010 in binary is indeed 234 in the decimal number system.

234 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 234 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 234, 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, 234. 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 234. 234 - 128 = 106.

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, 106. So, the next largest power of 2 is 2⁶, which is 64. Now, we have to write 1 in the 2⁶ place. And then subtract 64 from 106. 106 - 64 = 42.

Step 4 - Continue the process: Identify the next largest power of 2 that fits into 42, which is 2⁵ = 32. Write 1 in the 2⁵ place and subtract 32 from 42. 42 - 32 = 10. The next largest power of 2 that fits into 10 is 2³ = 8. Write 1 in the 2³ place. 10 - 8 = 2. Finally, the largest power of 2 that fits into 2 is 2¹ = 2. Write 1 in the 2¹ place. 2 - 2 = 0. We need to stop the process here since the remainder is 0.

Step 5 - Identify the unused place values: In the previous steps, we wrote 1 in the 2⁷, 2⁶, 2⁵, 2³, and 2¹ places. Now, we can just write 0s in the remaining places, which are 2⁴, 2², 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 from top to bottom to represent 234 in binary. Therefore, 11101010 is 234 in binary.

Grouping Method: In this method, we divide the number 234 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 234 by 2. 234 / 2 = 117. Here, 117 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (117) by 2. 117 / 2 = 58. Here, the quotient is 58 and the remainder is 1.

Step 3 - Repeat the previous step. 58 / 2 = 29. Now, the quotient is 29, and 0 is the remainder.

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

Step 5 - Continue the division. 14 / 2 = 7. The remainder is 0. 7 / 2 = 3. The remainder is 1. 3 / 2 = 1. The remainder is 1. 1 / 2 = 0. The remainder is 1. And we stop the division here because the quotient is 0.

Step 6 - Write down the remainders from bottom to top. Therefore, 234 (decimal) = 11101010 (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 234. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 234. So, 234 - 128 = 106. Find the largest power of 2 less than or equal to 106. The answer is 2⁶. So, write 1 next to this power. Now, 106 - 64 = 42. Continue finding powers of 2 and subtracting until you reach 0. Final conversion will be 11101010.

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, 234 is divided by 2 to get 117 as the quotient and 0 as the remainder. Now, 117 is divided by 2. Here, we will get 58 as the quotient and 1 as the remainder. Dividing 58 by 2, we get 29 as the quotient and 0 as the remainder. Continue dividing down to 1. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 234, 11101010.

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, and so on. Find the largest power that fits into 234. 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 234, we use 0s and 1s in the respective places to represent the number.

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 234.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 15 to help with 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., 234 is even, and its binary form is 11101010. Here, the binary of 234 ends in 0. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 235 (an odd number) is 11101011. 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 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 e.g., in 234 (base 10), the place values are 2 hundreds, 3 tens, and 4 ones.

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

• Quotient: The result of a division process, used especially in converting numbers to binary.