224 in Binary

The process of converting 224 from decimal to binary involves dividing the number 224 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 224 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 224 by 2 until getting 0 as the quotient is 11100000. 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 11100000.

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 11100000 in binary is indeed 224 in the decimal number system.

224 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 224 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 224, 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, 224.

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 224. 224 - 128 = 96

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, 96.

So, the next largest power of 2 is 2⁶, which is 64. Now, we have to write 1 in the 2⁶ places. And then subtract 64 from 96. 96 - 64 = 32

Step 4 - Identify the next largest power of 2: Now, we need to find the power of 2 that fits into 32. That would be 2⁵. Write 1 in the 2⁵ places, and subtract 32 from 32. 32 - 32 = 0

Step 5 - Identify the unused place values: In step 2, step 3, and step 4, we wrote 1 in the 2⁷, 2⁶, and 2⁵ places. Now, we can just write 0s in the remaining places, which are 2⁴, 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 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 upside down to represent 224 in binary. Therefore, 11100000 is 224 in binary. Grouping Method: In this method, we divide the number 224 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 224 by 2. 224 / 2 = 112. Here, 112 is the quotient and 0 is the remainder. Step 2 - Divide the previous quotient (112) by 2. 112 / 2 = 56.

Here, the quotient is 56 and the remainder is 0. Step 3 - Repeat the previous step. 56 / 2 = 28. Now, the quotient is 28, and 0 is the remainder.

Step 4 - Repeat the previous step. 28 / 2 = 14. Here, the quotient is 14, and the remainder is 0. Step 5 - Repeat the previous step. 14 / 2 = 7.

Here, the quotient is 7, and the remainder is 0.

Step 6 - Repeat the previous step. 7 / 2 = 3. Here, the quotient is 3, and the remainder is 1. Step 7 - Repeat the previous step. 3 / 2 = 1.

Here, the quotient is 1, and the remainder is 1.

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, 224 (decimal) = 11100000 (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 224. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 224. So, 224 - 128 = 96.

Find the largest power of 2 less than or equal to 96. The answer is 2^6. So, write 1 next to this power. Now, 96 - 64 = 32. Find the largest power of 2 less than or equal to 32. The answer is 2^5. So, write 1 next to this power. Now, 32 - 32 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2^4, 2^3, 2^2, 2^1, and 2^0). Final conversion will be 11100000.

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, 224 is divided by 2 to get 112 as the quotient and 0 as the remainder.

Now, 112 is divided by 2. Here, we will get 56 as the quotient and 0 as the remainder. Dividing 56 by 2, we get 0 as the remainder and 28 as the quotient. Divide 28 by 2 to get 0 as the remainder and 14 as the quotient. Divide 14 by 2 to get 0 as the remainder and 7 as the quotient. Divide 7 by 2 to get 1 as the remainder and 3 as the quotient. Divide 3 by 2 to get 1 as the remainder and 1 as the quotient. Divide 1 by 2 to get 1 as the remainder and 0 as the quotient. We stop the division once the quotient becomes 0.

Now, we write the remainders upside down to get the binary equivalent of 224, 11100000.

Rule 3: Representation Method This rule also involves breaking of the number into powers of 2. Identify the powers of 2 and write it down in decreasing order i.e., 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰.

Find the largest power that fits into 224. 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 224, we use 0s for 2⁴, 2³, 2², 2¹, and 2⁰ and 1s for 2⁷, 2⁶, and 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 224. Memorize to speed up conversions:

We can memorize the binary forms for powers of 2.

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., 224 is even and its binary form is 11100000. Here, the binary of 224 ends in 0. If the number is odd, then its binary equivalent will end in 1.

For e.g., the binary of 33 (an odd number) is 100001. 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 102 (base 10), 1 has occupied the hundreds place, 0 is in the tens place, and 2 is in the ones place.

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

• Power of 2: It refers to numbers that can be expressed as 2 raised to an exponent, such as 2⁰, 2¹, 2², etc.