8421 in Binary

The process of converting 8421 from decimal to binary involves dividing the number 8421 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 8421 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 8421 by 2 until getting 0 as the quotient is 1000011110101. 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 1000011110101. 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 1000011110101 in binary is indeed 8421 in the decimal number system.

8421 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 8421 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

Since 8192 is less than 8421, we include 2^13, which is 8192.

Step 2 - Identify the largest power of 2: In the previous step, we identified 2¹³ = 8192 as the largest power of 2 less than or equal to 8421. Write 1 in the 2¹³ place and subtract this value from 8421: 8421 - 8192 = 229.

Step 3 - Identify the next largest power of 2: Now, find the largest power of 2 that fits into 229, which is 2⁷ = 128. Write 1 in the 2⁷ place. Subtract 128 from the result of the previous step: 229 - 128 = 101.

Step 4 - Continue the process: Repeat the process for 101. 2⁶ = 64 fits into 101, so write 1 in the 2⁶ place. 101 - 64 = 37. 2⁵ = 32 fits into 37, so write 1 in the 2⁵ place. 37 - 32 = 5. 2² = 4 fits into 5, so write 1 in the 2² place. 5 - 4 = 1. 2⁰ = 1 fits into 1, so write 1 in the 2⁰ place. 1 - 1 = 0.

Step 5 - Identify the unused place values: Fill 0s in the places where no power of 2 was used. Now, by substituting the values, we get: 1 in the 2¹³ place 0 in the 2¹² place 0 in the 2¹¹ place 0 in the 2¹⁰ place 0 in the 2⁹ place 1 in the 2⁸ place 1 in the 2⁷ place 1 in the 2⁶ place 1 in the 2⁵ place 0 in the 2⁴ place 1 in the 2³ place 0 in the 2² place 1 in the 2¹ place 1 in the 2⁰ place Therefore, 1000011110101 is 8421 in binary.

Grouping Method: In this method, we divide the number 8421 by 2. Let us see the step-by-step conversion.

Step 1 - Divide 8421 by 2. 8421 / 2 = 4210 remainder 1.

Step 2 - Divide the previous quotient (4210) by 2. 4210 / 2 = 2105 remainder 0.

Step 3 - Repeat the previous step. 2105 / 2 = 1052 remainder 1.

Step 4 - Repeat the previous step. 1052 / 2 = 526 remainder 0.

Step 5 - Repeat the previous step. 526 / 2 = 263 remainder 0.

Step 6 - Repeat the previous step. 263 / 2 = 131 remainder 1.

Step 7 - Repeat the previous step. 131 / 2 = 65 remainder 1.

Step 8 - Repeat the previous step. 65 / 2 = 32 remainder 1.

Step 9 - Repeat the previous step. 32 / 2 = 16 remainder 0.

Step 10 - Repeat the previous step. 16 / 2 = 8 remainder 0.

Step 11 - Repeat the previous step. 8 / 2 = 4 remainder 0.

Step 12 - Repeat the previous step. 4 / 2 = 2 remainder 0.

Step 13 - Repeat the previous step. 2 / 2 = 1 remainder 0.

Step 14 - Repeat the previous step. 1 / 2 = 0 remainder 1. Write down the remainders from bottom to top.

Therefore, 8421 (decimal) = 1000011110101 (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 8421. Since the answer is 2¹³, write 1 next to this power of 2. Subtract the value (8192) from 8421. So, 8421 - 8192 = 229. Find the largest power of 2 less than or equal to 229. The answer is 2⁷. So, write 1 next to this power. Now, 229 - 128 = 101. Continue the process until you reach 0. Final conversion will be 1000011110101.

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, 8421 is divided by 2 to get 4210 as the quotient and 1 as the remainder. Now, 4210 is divided by 2. Here, we will get 2105 as the quotient and 0 as the remainder. Dividing 2105 by 2, we get 1052 as the quotient and 1 as the remainder. Continue the division process until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 8421, 1000011110101.

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., 2¹³, 2^12, 2¹¹, etc. Find the largest power that fits into 8421. 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.

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 8421.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 16 for quick reference.
   
• 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, 8421 is odd, and its binary form is 1000011110101, ending in 1. If the number is even, then its binary equivalent will end in 0.
   
• 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. 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.

• Quotient: The result obtained by dividing one number by another. In binary conversion, it is important to continue dividing until the quotient is 0.