223 in Binary

The process of converting 223 from decimal to binary involves dividing the number 223 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 223 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 223 by 2 until getting 0 as the quotient is 11011111. 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 223.

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 11011111 in binary is indeed 223 in the decimal number system.

223 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 223 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 223, 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, 223. 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 223. 223 - 128 = 95.

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, 95. The next largest power of 2 is 2⁶, which is equal to 64. Now, we have to write 1 in the 2⁶ place. And then subtract 64 from 95. 95 - 64 = 31.

Step 4 - Continue the process: Repeat the steps to find the largest power of 2 that fits into the current difference. Continue this until the remainder is 0. 31 - 16 = 15 15 - 8 = 7 7 - 4 = 3 3 - 2 = 1 1 - 1 = 0

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 223 in binary. Therefore, 11011111 is 223 in binary.

Grouping Method: In this method, we divide the number 223 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 223 by 2. 223 / 2 = 111. Here, 111 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (111) by 2. 111 / 2 = 55. Here, the quotient is 55 and the remainder is 1.

Step 3 - Repeat the previous step. 55 / 2 = 27. Now, the quotient is 27, and 1 is the remainder.

Step 4 - Repeat the previous step. 27 / 2 = 13. Now, the quotient is 13, and 1 is the remainder.

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

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

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

Step 8 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 9 - Write down the remainders from bottom to top. Therefore, 223 (decimal) = 11011111 (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 223. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 223. So, 223 - 128 = 95. Find the largest power of 2 less than or equal to 95. The answer is 2⁶. So, write 1 next to this power. Now, 95 - 64 = 31. Continue this process until the remainder is 0. Final conversion will be 11011111.

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, 223 is divided by 2 to get 111 as the quotient and 1 as the remainder. Now, 111 is divided by 2. Here, we will get 55 as the quotient and 1 as the remainder. Dividing 55 by 2, we get 27 as the quotient and 1 as the remainder. Divide 27 by 2 to get 13 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 223, 11011111.

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, and so on. Find the largest power that fits into 223. 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 223, we use 1s for the powers of 2 (2⁷, 2⁶, 2⁴, 2³, 2², 2¹, 2⁰) and 0 for 2⁵.

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 223.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 223.

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 e.g., 222 is even and its binary form is 11011110. Here, the binary of 222 ends in 0. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 223 (an odd number) is 11011111. 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 223 (base 10), 2 has occupied the hundreds place, 2 is in the tens place, and 3 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 of division in which a number is divided by another number.