455 in Binary

The process of converting 455 from decimal to binary involves dividing the number 455 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 455 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 455 by 2 until getting 0 as the quotient is 111000111. 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 111000111.

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 111000111 in binary is indeed 455 in the decimal number system.

455 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 455 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. 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 2^9 = 512 Since 512 is greater than 455, we stop at 2^8 = 256.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^8 = 256. 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, 455. Since 2^8 is the number we are looking for, write 1 in the 2^8 place. Now the value of 2^8, which is 256, is subtracted from 455. 455 - 256 = 199.

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, 199. The next largest power of 2 is 2^7, which is 128. Now, we have to write 1 in the 2^7 places. And then subtract 128 from 199. 199 - 128 = 71. Continue the process with the remaining result.

Step 4 - Continue identifying powers of 2 and subtracting: 71 - 64 (2^6) = 7. Write 1 in the 2^6 place. 7 - 4 (2^2) = 3. Write 1 in the 2^2 place. 3 - 2 (2^1) = 1. Write 1 in the 2^1 place. 1 - 1 (2^0) = 0. Write 1 in the 2^0 place.

Step 5 - Fill in the unused place values with 0s. Now, by substituting the values, we get: 0 in the 2^3 place 0 in the 2^4 place 0 in the 2^5 place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 455 in binary. Therefore, 111000111 is 455 in binary.

Grouping Method: In this method, we divide the number 455 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 455 by 2. 455 / 2 = 227. Here, 227 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (227) by 2. 227 / 2 = 113. Here, the quotient is 113 and the remainder is 1.

Step 3 - Repeat the previous step. 113 / 2 = 56. Now, the quotient is 56 and 1 is the remainder.

Step 4 - Continue the division: 56 / 2 = 28. The remainder is 0. 28 / 2 = 14. The remainder is 0. 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.

Step 5 - Write down the remainders from bottom to top. Therefore, 455 (decimal) = 111000111 (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 455. Since the answer is 2^8, write 1 next to this power of 2. Subtract the value (256) from 455. So, 455 - 256 = 199. Find the largest power of 2 less than or equal to 199. The answer is 2^7. So, write 1 next to this power. Continue this process until the remainder is 0, filling in 0s for unused powers of 2. Final conversion will be 111000111.

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, 455 is divided by 2 to get 227 as the quotient and 1 as the remainder. Now, 227 is divided by 2. Here, we will get 113 as the quotient and 1 as the remainder. Continue dividing until the quotient is 0, writing remainders upside down to get the binary equivalent of 455, which is 111000111.

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., 2^8, 2^7, 2^6, etc. Find the largest power that fits into 455. 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 455, we use 0s for 2^5, 2^4, and 2^3, and 1s for other necessary powers.

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 455.

Memorize to speed up conversions: Memorize the binary forms for small numbers to help with larger numbers.

Recognize the patterns: There is a pattern when converting numbers from decimal to binary. 1 → 1 2 → 10 4 → 100 8 → 1000 16 → 10000 32 → 100000 64 → 1000000 128 → 10000000 256 → 100000000

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 256 is even, and its binary form 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, cross-verify by converting the number back to the decimal form. This will eliminate any unforeseen errors in conversion.

Practice by using tables: Write the decimal numbers and their binary equivalents on a table to help remember 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 111000111 (binary), each digit has a specific power of 2 associated with it.

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

• Power of 2: In binary, each digit represents a power of 2, starting from 2^0 for the rightmost digit.