2147483647 in Binary

The process of converting 2147483647 from decimal to binary involves dividing the number 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 2147483647 to binary. In the last step, the remainder is noted down bottom side up, and that becomes the converted value. For example, after dividing 2147483647 by 2 until getting 0 as the quotient, the remainders noted down are 1111111111111111111111111111111. 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 1111111111111111111111111111111. 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 1111111111111111111111111111111 in binary is indeed 2147483647 in the decimal number system.

2147483647 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 2147483647 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 ...

Since 2³⁰ = 1073741824 and 2³¹ = 2147483648, we stop at 2³⁰ because 2147483648 is greater than 2147483647.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2³⁰ = 1073741824. 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, 2147483647. Since 2³⁰ is the number we are looking for, write 1 in the 2³⁰ place. Now the value of 2³⁰, which is 1073741824, is subtracted from 2147483647. 2147483647 - 1073741824 = 1073741823.

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, 1073741823. So, the next largest power of 2 is 2²⁹. Repeat this process until all powers of 2 are exhausted, filling in 1s for each used power.

Step 4 - Identify the unused place values: Since all place values are used in this case, the binary number is fully composed of 1s. Now, by substituting the values, we get, 1 in the 2³⁰ place 1 in the 2²⁹ place 1 in the 2²⁸ place ... 1 in the 2⁰ place

Step 5 - Write the values in reverse order: We now write the numbers to represent 2147483647 in binary. Therefore, 1111111111111111111111111111111 is 2147483647 in binary.

Grouping Method: In this method, we divide the number 2147483647 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 2147483647 by 2. 2147483647 / 2 = 1073741823. Here, 1073741823 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (1073741823) by 2. 1073741823 / 2 = 536870911. Here, the quotient is 536870911 and the remainder is 1.

Step 3 - Repeat the previous step. Continue this process until the quotient becomes 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 2147483647 (decimal) = 1111111111111111111111111111111 (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 2147483647. Since the answer is 2³⁰, write 1 next to this power of 2. Subtract the value (1073741824) from 2147483647. So, 2147483647 - 1073741824 = 1073741823. Find the largest power of 2 less than or equal to 1073741823. Repeat this process until the remainder is 0. Final conversion will be 1111111111111111111111111111111.

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, 2147483647 is divided by 2 to get 1073741823 as the quotient and 1 as the remainder. Now, 1073741823 is divided by 2. Here, we will get 536870911 as the quotient and 1 as the remainder. Continue this process of division until the quotient is 0. Now, we write the remainders upside down to get the binary equivalent of 2147483647, 1111111111111111111111111111111.

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., 2³⁰, 2²⁹, 2²⁸, ..., 2⁰. Find the largest power that fits into 2147483647. Repeat the process and allocate 1s 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 2147483647, we use 1s for each power from 2³⁰ to 2⁰.

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 conversions.

• Recognize powers of 2: Memorizing the powers of 2 up to a certain point can greatly speed up conversions.
   
• 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., 2147483646 (an even number) would end in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 2147483647 ends 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, 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 is in 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.

• Quotient: The result of a division problem, representing the number of times one number is contained within another.