11100 in Binary

The process of converting 28 from decimal to binary involves dividing the number 28 by 2. This is because the binary number system is base 2, using only the digits 0 and 1.

The quotient becomes the dividend in the next step, and the process continues until the quotient becomes 0. This method is commonly used to convert numbers to binary. For instance, the remainders noted down after dividing 28 by 2 until getting 0 as the quotient are 11100. Remember, the remainders are written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 11100. 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 11100 in binary is indeed 28 in the decimal number system.

Converting 28 from decimal to binary can be done easily using the methods mentioned below.

Expansion Method: Let us see the step-by-step process of converting 28 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

Since 32 is greater than 28, we stop at 2⁴ = 16.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁴ = 16. 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, 28. Since 2⁴ is the number we are looking for, write 1 in the 2⁴ place. Now the value of 2⁴, which is 16, is subtracted from 28. 28 - 16 = 12.

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, 12. So, the next largest power of 2 is 2³, which is less than or equal to 12. Now, we have to write 1 in the 2³ place. And then subtract 8 from 12. 12 - 8 = 4.

Step 4 - Identify the next largest power of 2: Now, we need to find the largest power of 2 that fits into 4. The next largest power of 2 is 2^2. Now, we have to write 1 in the 2² place. And then subtract 4 from 4. 4 - 4 = 0. We stop the process here since the remainder is 0.

Step 5 - Identify the unused place values: In step 2, step 3, and step 4, we wrote 1 in the 2⁴, 2³, and 2² places. Now, we can just write 0s in the remaining places, which are 2¹ and 2⁰. Now, by substituting the values, we get, 0 in the 2⁰ place 0 in the 2¹ place 1 in the 2² place 1 in the 2³ place 1 in the 2⁴ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 28 in binary. Therefore, 11100 is 28 in binary.

Grouping Method: In this method, we divide the number 28 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 28 by 2. 28 / 2 = 14. Here, 14 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (14) by 2. 14 / 2 = 7. Here, the quotient is 7 and the remainder is 0.

Step 3 - Divide the previous quotient (7) by 2. 7 / 2 = 3. Now, the quotient is 3, and 1 is the remainder.

Step 4 - Divide the previous quotient (3) by 2. 3 / 2 = 1. Here, the quotient is 1, and 1 is the remainder.

Step 5 - Divide the previous quotient (1) by 2. 1 / 2 = 0. Here, 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, 28 (decimal) = 11100 (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 28. Since the answer is 2⁴, write 1 next to this power of 2. Subtract the value (16) from 28. So, 28 - 16 = 12. Find the largest power of 2 less than or equal to 12. The answer is 2³. So, write 1 next to this power. Now, 12 - 8 = 4. Find the largest power of 2 less than or equal to 4. The answer is 2². So, write 1 next to this power. Now, 4 - 4 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2¹ and 2⁰). Final conversion will be 11100.

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, 28 is divided by 2 to get 14 as the quotient and 0 as the remainder. Now, 14 is divided by 2. Here, we will get 7 as the quotient and 0 as the remainder. Dividing 7 by 2, we get 3 as the quotient and 1 as the remainder. Dividing 3 by 2, we get 1 as the quotient and 1 as the remainder. 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 28, 11100.

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., 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 28. 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 28, we use 0s for 2¹ and 2⁰ and 1s for 2⁴, 2³, and 2².

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 28.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 28.
   
• 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., 28 is even and its binary form is 11100. Here, the binary of 28 ends in 0. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 29 (an odd number) is 11101. 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 known as the base 2 number system.

• Place value: Every digit has a value based on its position in a given number. For example, in 11100 (binary), 1 is in the 2⁴, 2³, and 2² places.

• 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, which is fundamental to understanding binary conversion.