2000 in Binary

The process of converting 2000 from decimal to binary involves dividing the number 2000 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 2000 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 2000 by 2 until getting 0 as the quotient is 11111010000. 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 2000.

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 11111010000 in binary is indeed 2000 in the decimal number system.

2000 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 2000 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 ...

2¹⁰ = 1024

2¹¹ = 2048

Since 2048 is greater than 2000, we stop at 2¹⁰ = 1024.

Step 2 - Identify the largest power of 2:

In the previous step, we stopped at 2¹⁰ = 1024.

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, 2000.

Since 2¹⁰ is the number we are looking for, write 1 in the 2¹⁰ place.

Now the value of 2¹⁰, which is 1024, is subtracted from 2000. 2000 - 1024 = 976.

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, 976.

The next largest power of 2 is 2^9, which is 512. Now, we write 1 in the 2⁹ place.

And then subtract 512 from 976. 976 - 512 = 464.

Step 4 - Continue identifying the largest power of 2: The next power is 2⁸ = 256, which fits into 464. 464 - 256 = 208.

Step 5 - Continue the process: The next power is 2⁷ = 128. 208 - 128 = 80.

Step 6 - Continue the process: The next power is 2⁶ = 64. 80 - 64 = 16.

Step 7 - Continue the process: The next power is 2⁴ = 16. 16 - 16 = 0.

Step 8 - Identify the unused place values: In the steps above, we wrote 1 in the 2¹⁰, 2⁹, 2⁸, 2⁷, 2⁶, and 2⁴ places.

Now, we can just write 0s in the remaining places.

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 1 in the 2⁴ place 0 in the 2⁵ place 1 in the 2⁶ place 1 in the 2⁷ place 1 in the 2⁸ place 1 in the 2⁹ place 1 in the 2¹⁰ place

Step 9 - Write the values in reverse order: We now write the numbers upside down to represent 2000 in binary. Therefore, 11111010000 is 2000 in binary.

Grouping Method: In this method, we divide the number 2000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 2000 by 2. 2000 / 2 = 1000. Here, 1000 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (1000) by 2. 1000 / 2 = 500. Here, the quotient is 500 and the remainder is 0.

Step 3 - Repeat the previous step. 500 / 2 = 250. Now, the quotient is 250, and 0 is the remainder.

Step 4 - Repeat the previous step. 250 / 2 = 125. Now, the quotient is 125, and 0 is the remainder.

Step 5 - Repeat the previous step. 125 / 2 = 62. Now, the quotient is 62, and 1 is the remainder.

Step 6 - Repeat the previous step. 62 / 2 = 31. Now, the quotient is 31, and 0 is the remainder.

Step 7 - Repeat the previous step. 31 / 2 = 15. Now, the quotient is 15, and 1 is the remainder.

Step 8 - Repeat the previous step. 15 / 2 = 7. Now, the quotient is 7, and 1 is the remainder.

Step 9 - Repeat the previous step. 7 / 2 = 3. Now, the quotient is 3, and 1 is the remainder.

Step 10 - Repeat the previous step. 3 / 2 = 1. Now, the quotient is 1, and 1 is the remainder.

Step 11 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 12 - Write down the remainders from bottom to top.

Therefore, 2000 (decimal) = 11111010000 (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 2000. Since the answer is 2¹⁰, write 1 next to this power of 2.

Subtract the value (1024) from 2000. So, 2000 - 1024 = 976.

Find the largest power of 2 less than or equal to 976.

The answer is 2⁹. So, write 1 next to this power.

Now, 976 - 512 = 464. Continue this process for each result using decreasing powers of 2, until the remainder is 0.

Final conversion will be 11111010000.

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, 2000 is divided by 2 to get 1000 as the quotient and 0 as the remainder.

Now, 1000 is divided by 2. Here, we will get 500 as the quotient and 0 as the remainder.

Dividing 500 by 2, we get 250 as the quotient and 0 as the remainder.

Continue this division process until the quotient becomes 0.

Now, we write the remainders upside down to get the binary equivalent of 2000, which is 11111010000.

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., 2¹⁰, 2⁹, 2⁸, etc.

Find the largest power that fits into 2000.

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 2000, we use a combination of 1s and 0s based on the powers of 2.

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 2000.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 2000. 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 ... 1024 + 1024 = 2048 → 100000000000

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 2000 is even and its binary form is 11111010000.

Here, the binary of 2000 ends 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 in 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.

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

• Quotient and Remainder: In division, the quotient is the result of division, and the remainder is what is left over after dividing.