486 in Binary

The process of converting 486 from decimal to binary involves dividing the number 486 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 486 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 486 by 2 until getting 0 as the quotient is 111100110. 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 111100110.

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 111100110 in binary is indeed 486 in the decimal number system.

486 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 486 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 Since 512 is greater than 486, we stop at 2^8 = 256.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^8 = 256. 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, 486. Since 2^8 is the number we are looking for, write 1 in the 2^8 place. Now the value of 2^8, which is 256, is subtracted from 486. 486 - 256 = 230.

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, 230. So, the next largest power of 2 is 2^7, which is 128. Now, we have to write 1 in the 2^7 place. And then subtract 128 from 230. 230 - 128 = 102.

Step 4 - Continue with the next largest power of 2: The next largest power of 2 that fits into 102 is 2^6, which is 64. Write 1 in the 2^6 place and subtract 64 from 102. 102 - 64 = 38. Continue this process: 2^5 = 32, so write 1 and subtract 32 from 38. 38 - 32 = 6. 2^2 = 4, so write 1 and subtract 4 from 6. 6 - 4 = 2. 2^1 = 2, so write 1 and subtract 2 from 2. 2 - 2 = 0.

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

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 486 in binary. Therefore, 111100110 is 486 in binary.

Grouping Method: In this method, we divide the number 486 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 486 by 2. 486 / 2 = 243. Here, 243 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (243) by 2. 243 / 2 = 121. Here, the quotient is 121 and the remainder is 1.

Step 3 - Repeat the previous step. 121 / 2 = 60. Now, the quotient is 60, and 1 is the remainder.

Step 4 - Repeat the previous step. 60 / 2 = 30. Here, the quotient is 30, and 0 is the remainder.

Step 5 - Continue this process until the quotient becomes 0. 30 / 2 = 15, remainder 0. 15 / 2 = 7, remainder 1. 7 / 2 = 3, remainder 1. 3 / 2 = 1, remainder 1. 1 / 2 = 0, remainder 1.

Step 6 - Write down the remainders from bottom to top. Therefore, 486 (decimal) = 111100110 (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 486. Since the answer is 2^8, write 1 next to this power of 2. Subtract the value (256) from 486. So, 486 - 256 = 230. Find the largest power of 2 less than or equal to 230. The answer is 2^7. So, write 1 next to this power. Continue this process until the remainder is 0. Final conversion will be 111100110.

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, 486 is divided by 2 to get 243 as the quotient and 0 as the remainder. Now, 243 is divided by 2. Here, we will get 121 as the quotient and 1 as the remainder. Continue dividing and recording remainders until the quotient is 0. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 486, 111100110.

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., 2^8, 2^7, 2^6, and so on. Find the largest power that fits into 486. 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 486.

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

Recognize the patterns: There is a pattern when converting numbers from decimal to binary. 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 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, 486 is even and its binary form is 111100110. Here, the binary of 486 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 example, in 486 (base 10), 4 has occupied the hundreds place, 8 is in the tens place, and 6 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 number.