1984 in Binary

The process of converting 1984 from decimal to binary involves dividing the number 1984 by 2. Here, it is getting 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 1984 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 1984 by 2 until getting 0 as the quotient is 11111000000. 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 11111000000. 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 11111000000 in binary is indeed 1984 in the decimal number system.

1984 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 1984 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

Since 2048 is greater than 1984, we stop at 2^10 = 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, 1984. Since 2^10 is the number we are looking for, write 1 in the 2^10 place. Now the value of 2^10, which is 1024, is subtracted from 1984. 1984 - 1024 = 960.

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, 960. So, the next largest power of 2 is 2^9 = 512. Now, we have to write 1 in the 2^9 place. And then subtract 512 from 960. 960 - 512 = 448.

Step 4 - Continue the process: Repeat the process for 448, using the next largest power of 2, which is 2^8 = 256. 448 - 256 = 192. Next, use 2⁷ = 128. 192 - 128 = 64. Finally, use 2⁶ = 64. 64 - 64 = 0. We need to stop the process here since the remainder is 0.

Step 5 - Identify the unused place values: We wrote 1 in the 2^10, 2^9, 2^8, 2^7, and 2^6 places. Now, we can just write 0s in the remaining places, which are 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Now, by substituting the values, we get, 0 in the 2^0 place 0 in the 2^1 place 0 in the 2^2 place 0 in the 2^3 place 0 in the 2^4 place 0 in the 2^5 place 1 in the 2^6 place 1 in the 2^7 place 1 in the 2^8 place 1 in the 2^9 place 1 in the 2^10 place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 1984 in binary. Therefore, 11111000000 is 1984 in binary.

Grouping Method: In this method, we divide the number 1984 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 1984 by 2. 1984 / 2 = 992. Here, 992 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (992) by 2. 992 / 2 = 496. Here, the quotient is 496 and the remainder is 0.

Step 3 - Repeat the previous step. 496 / 2 = 248. Now, the quotient is 248 and 0 is the remainder.

Step 4 - Repeat the previous step. 248 / 2 = 124. Here, the quotient is 124 and the remainder is 0.

Step 5 - Repeat the previous step. 124 / 2 = 62. Here, the quotient is 62 and the remainder is 0.

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

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

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

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

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

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, 1984 (decimal) = 11111000000 (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 1984. Since the answer is 2^10, write 1 next to this power of 2. Subtract the value (1024) from 1984. So, 1984 - 1024 = 960. Find the largest power of 2 less than or equal to 960. The answer is 2^9. So, write 1 next to this power. Now, 960 - 512 = 448. Continue the process for 448, 192, and 64, using 2^8, 2^7, and 2^6 respectively. Since there is no remainder, we can write 0 next to the remaining powers (2^5, 2^4, 2^3, 2^2, 2^1, and 2^0). Final conversion will be 11111000000.

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, 1984 is divided by 2 to get 992 as the quotient and 0 as the remainder. Now, 992 is divided by 2. Here, we will get 496 as the quotient and 0 as the remainder. Dividing 496 by 2, we get 0 as the remainder and 248 as the quotient. Continue dividing until the quotient becomes 0. Write the remainders upside down to get the binary equivalent of 1984, 11111000000.

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., 2^11, 2^10, 2^9, 2^8, 2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Find the largest power that fits into 1984. 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 1984, we use 0s for 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0, and 1s for 2^10, 2^9, 2^8, 2^7, and 2^6.

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 1984.

• Memorize to speed up conversions: We can memorize the binary forms for numbers like 1, 2, 4, 8, 16, 32, 64, etc., up to 1984.
   
• Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary. For example: 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 …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, 1984 is even, and its binary form is 11111000000. Here, the binary of 1984 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 nswers 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 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.

• Base: The base of a number system indicates the number of unique digits, including zero, that a positional numeral system uses to represent numbers.