682 in Binary

The process of converting 682 from decimal to binary involves dividing the number 682 by 2. 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 682 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 682 by 2 until getting 0 as the quotient is 1010101010. 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 1010101010.

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 1010101010 in binary is indeed 682 in the decimal number system.

682 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 682 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 Since 512 is less than 682 and 1024 is greater than 682, we stop at 2^9 = 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, 682. Since 2⁹ is the number we are looking for, write 1 in the 2⁹ place. Now the value of 2⁹, which is 512, is subtracted from 682. 682 - 512 = 170.

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, 170. So, the next largest power of 2 is 2⁷, which is 128. Now, we have to write 1 in the 2⁷ place. And then subtract 128 from 170. 170 - 128 = 42.

Step 4 - Continue with the next largest power of 2: The next largest power of 2 that fits into 42 is 2⁵, which is 32. Write 1 in the 2⁵ place. Then subtract 32 from 42. 42 - 32 = 10.

Step 5 - Continue with the next largest power of 2: The next largest power of 2 that fits into 10 is 2³, which is 8. Write 1 in the 2³ place. Then subtract 8 from 10. 10 - 8 = 2.

Step 6 - Final step: The next largest power of 2 that fits into 2 is 2¹, which is 2. Write 1 in the 2¹ place. Then subtract 2 from 2. 2 - 2 = 0. We need to stop the process here since the remainder is 0.

Step 7 - Identify the unused place values: In the previous steps, we wrote 1s in the 2⁹, 2⁷, 2⁵, 2³, and 2¹ places. Now, we can just write 0s in the remaining places, which are 2⁸, 2⁶, 2⁴, 2², and 2⁰. Now, by substituting the values, we get: 0 in the 2⁰ place 1 in the 2¹ place 0 in the 2² place 1 in the 2³ place 0 in the 2⁴ place 1 in the 2⁵ place 0 in the 2⁶ place 1 in the 2⁷ place 0 in the 2⁸ place 1 in the 2⁹ place Therefore, 1010101010 is 682 in binary.

Grouping Method: In this method, we divide the number 682 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 682 by 2. 682 / 2 = 341. Here, 341 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (341) by 2. 341 / 2 = 170. Here, the quotient is 170 and the remainder is 1.

Step 3 - Repeat the previous step. 170 / 2 = 85. Now, the quotient is 85 and 0 is the remainder.

Step 4 - Repeat the previous step. 85 / 2 = 42. Here, the quotient is 42 and 1 is the remainder.

Step 5 - Repeat the previous step. 42 / 2 = 21. The quotient is 21 and the remainder is 0.

Step 6 - Repeat the previous step. 21 / 2 = 10. The quotient is 10 and the remainder is 1.

Step 7 - Repeat the previous step. 10 / 2 = 5. The quotient is 5 and the remainder is 0.

Step 8 - Repeat the previous step. 5 / 2 = 2. The quotient is 2 and the remainder is 1.

Step 9 - Repeat the previous step. 2 / 2 = 1. The quotient is 1 and the remainder is 0.

Step 10 - Repeat the previous step. 1 / 2 = 0. The quotient is 0 and the remainder is 1. We stop the division here because the quotient is 0.

Step 11 - Write down the remainders from bottom to top. Therefore, 682 (decimal) = 1010101010 (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 682. Since the answer is 2⁹, write 1 next to this power of 2. Subtract the value (512) from 682. So, 682 - 512 = 170. Find the largest power of 2 less than or equal to 170. The answer is 2⁷. So, write 1 next to this power. Continue the process until you reach a remainder of 0. Since there is no remainder, we can write 0 next to the remaining powers. The final conversion will be 1010101010.

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, 682 is divided by 2 to get 341 as the quotient and 0 as the remainder. Now, 341 is divided by 2. Here, we will get 170 as the quotient and 1 as the remainder. Continue this process until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 682, which is 1010101010.

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., 2⁹, 2⁸, 2⁷, and so on. Find the largest power that fits into 682. 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 682, we use 0s and 1s appropriately 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 682.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers.

Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary.

Even and odd rule: Whenever a number is even, its binary form will end 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.

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

• Power of 2: In the binary system, each place value is a power of 2, which is essential in binary conversion.