1000 in Binary

The process of converting 1000 from decimal to binary involves dividing the number 1000 by 2.

This division is used because the binary number system only employs two 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 1000 to binary.

In the final step, the remainder is noted down bottom side up, forming the converted value.

After dividing 1000 by 2 until reaching 0 as the quotient, the remainders noted down are 1111101000.

Remember, the remainders are recorded from bottom to top.

In the table shown below, the first column shows the binary digits (0 and 1) as 1111101000.

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 1111101000 in binary is indeed 1000 in the decimal number system.

1000 can be easily converted 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 1000 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

Since 1024 is greater than 1000, we stop at 2⁹ = 512.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁹ = 512. 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, 1000.

Since 2⁹ is the number we are looking for, write 1 in the 2⁹ place.

Now subtract the value of 2⁹, which is 512, from 1000. 1000 - 512 = 488.

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, 488. So, the next largest power of 2 is 2⁸, which is less than or equal to 488. Now, we have to write 1 in the 2⁸ place. And then subtract 256 from 488. 488 - 256 = 232.

Step 4 - Continue the process: Repeat the process for the remaining value. The next largest power of 2 is 2⁷ = 128. 232 - 128 = 104. The next largest power of 2 is 2⁶ = 64. 104 - 64 = 40. The next largest power of 2 is 2⁵ = 32. 40 - 32 = 8. The next largest power of 2 is 2³ = 8. 8 - 8 = 0. We need to stop the process here since the remainder is 0.

Step 5 - Identify the unused place values: In the previous steps, we wrote 1 in the 2⁹, 2⁸, 2⁷, 2⁶, 2⁵, and 2³ places. Now, we can just write 0s in the remaining places, which are 2⁴,2², 2¹, and 2⁰. Now, by substituting the values, we get, 0 in the 2⁰ place 0 in the 2¹ place 0 in the 2²place 1 in the 2³ place 0 in the 2⁴ place 1 in the 2⁵ place 1 in the 2⁶ place 1 in the 2⁷place 1 in the 2⁸ place 1 in the 2⁹ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 1000 in binary. Therefore, 1111101000 is 1000 in binary.

Grouping Method: In this method, we divide the number 1000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 1000 by 2. 1000 / 2 = 500. Here, 500 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (500) by 2. 500 / 2 = 250. Here, the quotient is 250 and the remainder is 0.

Step 3 - Repeat the previous step. 250 / 2 = 125. Now, the quotient is 125, and 0 is the remainder.

Step 4 - Continue dividing the quotient by 2 until the quotient becomes 0. 125 / 2 = 62, remainder 1.

62 / 2 = 31, remainder 0.

31 / 2 = 15, remainder 1.

15 / 2 = 7, remainder 1.

7 / 2 = 3, remainder 1.

3 / 2 = 1, remainder 1.

1 / 2 = 0, remainder 1.

Step 5 - Write down the remainders from bottom to top. Therefore, 1000 (decimal) = 1111101000 (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 1000. Since the answer is 2⁹, write 1 next to this power of 2. Subtract the value (512) from 1000. So, 1000 - 512 = 488. Find the largest power of 2 less than or equal to 488. The answer is 2⁸. So, write 1 next to this power. Repeat the process until the remainder is 0. Final conversion will be 1111101000.

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, 1000 is divided by 2 to get 500 as the quotient and 0 as the remainder. Now, 500 is divided by 2. Here, we will get 250 as the quotient and 0 as the remainder. Continue dividing the quotient by 2 until the quotient becomes 0. Record the remainders. Write the remainders upside down to get the binary equivalent of 1000, 1111101000.

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., 2^9, 2^8, 2^7, etc. Find the largest power that fits into 1000. 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 beyond 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 1000, we use 1s for 2⁹, 2⁸, 2⁷, 2⁶, 2⁵ and 2³, and 0s for 2⁴, 2², 2¹, and 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 numbers up to 1000.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 10 and beyond.

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, 1000 is even, and its binary form is 1111101000. Here, the binary of 1000 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 17 (an odd number) is 10001. 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 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 has occupied 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: Each position in a binary number represents a power of 2, starting from 0.