8634 in Binary

The process of converting 8634 from decimal to binary involves dividing the number 8634 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 8634 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 8634 by 2 until getting 0 as the quotient is 10000110101010. 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 10000110101010.

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 10000110101010 in binary is indeed 8634 in the decimal number system.

8634 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 8634 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. 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256 2^9 = 512 2^10 = 1024 2^11 = 2048 2^12 = 4096 2^13 = 8192 Since 8192 is less than 8634, we stop at 2^13.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^13 = 8192. 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, 8634. Since 2^13 is the number we are looking for, write 1 in the 2^13 place. Now the value of 2^13, which is 8192, is subtracted from 8634. 8634 - 8192 = 442.

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, 442. So, the next largest power of 2 is 2^8, which is less than or equal to 442 (in this case equal). Now, we have to write 1 in the 2^8 places. And then subtract 256 from 442. 442 - 256 = 186.

Step 4 - 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, 186. So, the next largest power of 2 is 2^7, which is less than or equal to 186 (in this case equal). Now, we have to write 1 in the 2^7 places. And then subtract 128 from 186. 186 - 128 = 58.

Step 5 - 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, 58. So, the next largest power of 2 is 2^5, which is less than or equal to 58 (in this case equal). Now, we have to write 1 in the 2^5 places. And then subtract 32 from 58. 58 - 32 = 26.

Step 6 - 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, 26. So, the next largest power of 2 is 2^4, which is less than or equal to 26 (in this case equal). Now, we have to write 1 in the 2^4 places. And then subtract 16 from 26. 26 - 16 = 10.

Step 7 - 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, 10. So, the next largest power of 2 is 2^3, which is less than or equal to 10 (in this case equal). Now, we have to write 1 in the 2^3 places. And then subtract 8 from 10. 10 - 8 = 2.

Step 8 - 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, 2. So, the next largest power of 2 is 2^1, which is less than or equal to 2 (in this case equal). Now, we have to write 1 in the 2^1 places. And then subtract 2 from 2. 2 - 2 = 0. We need to stop the process here since the remainder is 0.

Step 9 - Identify the unused place values: In steps 2, 3, 4, 5, 6, 7, and 8, we wrote 1 in the 2^13, 2^8, 2^7, 2^5, 2^4, 2^3, and 2^1 places. Now, we can just write 0s in the remaining places, which are 2^12, 2^11, 2^10, 2^9, 2^6, 2^2, and 2^0. Now, by substituting the values, we get, 0 in the 2^12 place 0 in the 2^11 place 0 in the 2^10 place 0 in the 2^9 place 1 in the 2^8 place 1 in the 2^7 place 0 in the 2^6 place 1 in the 2^5 place 1 in the 2^4 place 0 in the 2^3 place 1 in the 2^2 place 0 in the 2^1 place 0 in the 2^0 place

Step 10 - Write the values in reverse order: We now write the numbers upside down to represent 8634 in binary. Therefore, 10000110101010 is 8634 in binary.

Grouping Method: In this method, we divide the number 8634 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 8634 by 2. 8634 / 2 = 4317. Here, 4317 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (4317) by 2. 4317 / 2 = 2158. Here, the quotient is 2158 and the remainder is 1.

Step 3 - Repeat the previous step. 2158 / 2 = 1079. Now, the quotient is 1079, and 0 is the remainder.

Step 4 - Repeat the previous step. 1079 / 2 = 539. Here, the remainder is 1.

Step 5 - Repeat the previous step. 539 / 2 = 269. Here, the remainder is 1.

Step 6 - Repeat the previous step. 269 / 2 = 134. Here, the remainder is 1.

Step 7 - Repeat the previous step. 134 / 2 = 67. Here, the remainder is 0. Step 8 - Repeat the previous step. 67 / 2 = 33. Here, the remainder is 1.

Step 9 - Repeat the previous step. 33 / 2 = 16. Here, the remainder is 1.

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

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

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

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

Step 14 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 15 - Write down the remainders from bottom to top. Therefore, 8634 (decimal) = 10000110101010 (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 8634. Since the answer is 2^13, write 1 next to this power of 2. Subtract the value (8192) from 8634. So, 8634 - 8192 = 442. Find the largest power of 2 less than or equal to 442. The answer is 2^8. So, write 1 next to this power. Now, 442 - 256 = 186. Find the largest power of 2 less than or equal to 186. The answer is 2^7. So, write 1 next to this power. Now, 186 - 128 = 58. Find the largest power of 2 less than or equal to 58. The answer is 2^5. So, write 1 next to this power. Now, 58 - 32 = 26. Find the largest power of 2 less than or equal to 26. The answer is 2^4. So, write 1 next to this power. Now, 26 - 16 = 10. Find the largest power of 2 less than or equal to 10. The answer is 2^3. So, write 1 next to this power. Now, 10 - 8 = 2. Find the largest power of 2 less than or equal to 2. The answer is 2^1. So, write 1 next to this power. Now, 2 - 2 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2^12, 2^11, 2^10, 2^9, 2^6, 2^2, and 2^0). Final conversion will be 10000110101010.

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, 8634 is divided by 2 to get 4317 as the quotient and 0 as the remainder. Now, 4317 is divided by 2. Here, we will get 2158 as the quotient and 1 as the remainder. Dividing 2158 by 2, we get 0 as the remainder and 1079 as the quotient. Divide 1079 by 2 to get 539 as the quotient and 1 as the remainder. Divide 539 by 2 to get 269 as the quotient and 1 as the remainder. Divide 269 by 2 to get 134 as the quotient and 1 as the remainder. Divide 134 by 2 to get 67 as the quotient and 0 as the remainder. Divide 67 by 2 to get 33 as the quotient and 1 as the remainder. Divide 33 by 2 to get 16 as the quotient and 1 as the remainder. Divide 16 by 2 to get 8 as the quotient and 0 as the remainder. Divide 8 by 2 to get 4 as the quotient and 0 as the remainder. Divide 4 by 2 to get 2 as the quotient and 0 as the remainder. Divide 2 by 2 to get 1 as the quotient and 0 as the remainder. Divide 1 by 2 to get 1 as the remainder and 0 as the quotient. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 8634, 10000110101010.

Rule 3: Representation Method

This rule also involves breaking of the number into powers of 2. Identify the powers of 2 and write it down in decreasing order i.e., 2^13, 2^12, 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 8634. 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 8634, we use 0s for 2^12, 2^11, 2^10, 2^9, 2^6, 2^2, and 2^0, and 1s for 2^13, 2^8, 2^7, 2^5, 2^4, 2^3, and 2^1.

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 8634.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 10 and beyond 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 e.g., 8634 is even and its binary form is 10000110101010. Here, the binary of 8634 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 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 e.g., 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.

• Quotient: The result of division in a mathematical operation.