131072 in Binary

The process of converting 131072 from decimal to binary involves dividing the number 131072 by 2. This is done 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 131072 to binary. In the last step, the remainder is noted down from bottom to top, and that becomes the converted value.

For example, the remainders noted down after dividing 131072 by 2 until getting 0 as the quotient is 100000000000000000. 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 100000000000000000.

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 100000000000000000 in binary is indeed 131072 in the decimal number system.

131072 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 131072 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

2¹⁵ = 32768

2¹⁶ = 65536

2¹⁷ = 131072 Since 2¹⁷ is equal to 131072, we stop at 2¹⁷.

Step 2 - Identify the largest power of 2:

In the previous step, we stopped at 2¹⁷ = 131072. This is because, in this step, we have to identify the largest power of 2 that is equal to the given number, 131072. Since 2¹⁷ is the number we are looking for, write 1 in the 2¹⁷ place. There is no need to subtract since the number is exactly 131072.

Step 3 - Identify the unused place values: Since we wrote 1 in the 2¹⁷ place, we can just write 0s in the remaining places, which are from 2⁰ up to 2¹⁶. 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 0 in the 2⁵ place 0 in the 2⁶ place 0 in the 2⁷ place 0 in the 2⁸ place 0 in the 2⁹ place 0 in the 2¹⁰ place 0 in the 2¹¹ place 0 in the 2¹² place 0 in the 2¹³ place 0 in the 2¹⁴ place 0 in the 2¹⁵ place 0 in the 2¹⁶ place 1 in the 2¹⁷ place Therefore, 100000000000000000 is 131072 in binary.

Grouping Method: In this method, we divide the number 131072 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 131072 by 2. 131072 / 2 = 65536. Here, 65536 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (65536) by 2. 65536 / 2 = 32768. Here, the quotient is 32768 and the remainder is 0.

Step 3 - Repeat the previous step. Continue dividing by 2 and noting the quotient and remainders until the quotient is 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 131072 (decimal) = 100000000000000000 (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 that is less than or equal to 131072. Since the answer is 2¹⁷, write 1 next to this power of 2. There is no need to subtract since the number is exactly 131072. Now, we write 0 next to the remaining powers from 2⁰ to 2¹⁶. Final conversion will be 100000000000000000.

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, 131072 is divided by 2 to get 65536 as the quotient and 0 as the remainder. Now, 65536 is divided by 2. Here, we will get 32768 as the quotient and 0 as the remainder. Continue dividing by 2 until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 131072, 100000000000000000.

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., 2¹⁷, 2¹⁶, 2¹⁵, ..., 2⁰. Find the largest power that fits into 131072. 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 131072, we use 0s for 2⁰ to 2¹⁶ and 1 for 2¹⁷.

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 131072.

Memorize to speed up conversions: We can memorize the binary forms for numbers like 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, and 131072.

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 example, 131072 is even and its binary form is 100000000000000000. Here, the binary of 131072 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 131073 (an odd number) is 100000000000000001. 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 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: It refers to exponential values of 2, such as 2⁰, 2¹, 2², and so on, used in binary conversions.