11000 in Binary

The process of converting 11000 from decimal to binary involves dividing the number 11000 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 11000 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 11000 by 2 until getting 0 as the quotient is 1010101101000. 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 11000.

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 1010101101000 in binary is indeed 11000 in the decimal number system.

11000 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 11000 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

2⁹ = 512

2¹⁰ = 1024

2¹¹ = 2048

2¹² = 4096

2¹³ = 8192

2¹⁴ = 16384

Since 16384 is greater than 11000, we stop at 2^13 = 8192.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹³ = 8192. 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, 11000. Since 2¹³ is the number we are looking for, write 1 in the 2¹³ place. Now the value of 2¹³, which is 8192, is subtracted from 11000. 11000 - 8192 = 2808.

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, 2808. So, the next largest power of 2 is 2¹¹, which is less than or equal to 2808. Now, we have to write 1 in the 2¹¹ places. And then subtract 2048 from 2808. 2808 - 2048 = 760.

Step 4 - Identify the next largest power of 2 for 760: The next largest power of 2 is 2⁹, which is less than or equal to 760. Write 1 in the 2⁹ place and subtract 512 from 760. 760 - 512 = 248.

Step 5 - Identify the next largest power of 2 for 248: The next largest power of 2 is 2⁷, which is less than or equal to 248. Write 1 in the 2⁷ place and subtract 128 from 248. 248 - 128 = 120.

Step 6 - Identify the next largest power of 2 for 120: The next largest power of 2 is 2⁶, which is less than or equal to 120. Write 1 in the 2⁶ place and subtract 64 from 120. 120 - 64 = 56.

Step 7 - Identify the next largest power of 2 for 56: The next largest power of 2 is 2⁵, which is less than or equal to 56. Write 1 in the 2⁵ place and subtract 32 from 56. 56 - 32 = 24.

Step 8 - Identify the next largest power of 2 for 24: The next largest power of 2 is 2⁴, which is less than or equal to 24. Write 1 in the 2⁴ place and subtract 16 from 24. 24 - 16 = 8.

Step 9 - Identify the next largest power of 2 for 8: The next largest power of 2 is 2³, which is equal to 8. Write 1 in the 2³ place and subtract 8 from 8. 8 - 8 = 0. We need to stop the process here since the remainder is 0.

Step 10 - Identify the unused place values: In the previous steps, we wrote 1 in the 2¹³, 2¹¹, 2⁹, 2⁷, 2⁶, 2⁵, 2⁴, and 2³ places. Now, we can just write 0s in the remaining places, which are 2¹², 2¹⁰, 2⁸, 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 1 in the 2³ place 1 in the 2^4 place 1 in the 2⁵ place 1 in the 2⁶ place 1 in the 2⁷ place 0 in the 2^8 place 1 in the 2⁹ place 0 in the 2¹⁰ place 1 in the 2¹¹ place 0 in the 2¹² place 1 in the 2¹³ place

Step 11 - Write the values in reverse order: We now write the numbers upside down to represent 11000 in binary. Therefore, 11000 is 1010101101000 in binary.

Grouping Method: In this method, we divide the number 11000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 11000 by 2. 11000 / 2 = 5500. Here, 5500 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (5500) by 2. 5500 / 2 = 2750. Here, the quotient is 2750 and the remainder is 0.

Step 3 - Repeat the previous step. 2750 / 2 = 1375. Now, the quotient is 1375, and 0 is the remainder.

Step 4 - Repeat the previous step. 1375 / 2 = 687. Here, the remainder is 1.

Step 5 - Repeat the previous step. 687 / 2 = 343. Here, the remainder is 1.

Step 6 - Repeat the previous step. 343 / 2 = 171. Here, the remainder is 1.

Step 7 - Repeat the previous step. 171 / 2 = 85. Here, the remainder is 1.

Step 8 - Repeat the previous step. 85 / 2 = 42. Here, the remainder is 1.

Step 9 - Repeat the previous step. 42 / 2 = 21. Here, the remainder is 0.

Step 10 - Repeat the previous step. 21 / 2 = 10. Here, the remainder is 1.

Step 11 - Repeat the previous step. 10 / 2 = 5. Here, the remainder is 0.

Step 12 - Repeat the previous step. 5 / 2 = 2. Here, the remainder is 1.

Step 13 - Repeat the previous step. 2 / 2 = 1. Here, the remainder is 0.

Step 14 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 15 - Write down the remainders from bottom to top. Therefore, 11000 (decimal) = 1010101101000 (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 11000. Since the answer is 2¹³, write 1 next to this power of 2. Subtract the value (8192) from 11000. So, 11000 - 8192 = 2808. Find the largest power of 2 less than or equal to 2808. The answer is 2¹¹. So, write 1 next to this power. Now, 2808 - 2048 = 760. Continue this process until the remainder becomes 0. Final conversion will be 1010101101000.

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, 11000 is divided by 2 to get 5500 as the quotient and 0 as the remainder. Now, 5500 is divided by 2. Here, we will get 2750 as the quotient and 0 as the remainder. Dividing 2750 by 2, we get 1375 as the quotient and 0 as the remainder. Divide 1375 by 2 to get 687 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 11000, 1010101101000.

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., 2¹³, 2¹², 2¹¹, ..., 2⁰. Find the largest power that fits into 11000. 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 11000, we use 0s for 2¹², 2¹⁰, 2⁸, 2², 2¹, and 2⁰ and 1s for 2¹³, 2¹¹, 2⁹, 2⁷, 2⁶, 2⁵, 2⁴, and 2³.

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 11000.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers to help with larger conversions.

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. If the number is odd, then its binary equivalent will end in 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 that 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 occupies 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: A power of 2 represents 2 raised to an exponent, crucial in binary conversion.