498 in Binary

Converting 498 from decimal to binary involves dividing the number by 2 because the binary system uses only two digits (0 and 1). The quotient becomes the dividend in the next step, and the process continues until the quotient becomes 0.

This method is commonly used to convert any decimal number to binary. In the final step, the remainders are noted down from bottom to top, forming the binary representation.

For 498, the remainders noted after dividing by 2 until getting 0 as the quotient yield 111110010.

The table below shows the binary digits of 498. The first column shows the binary digits (1 and 0) as 111110010.

The second column represents the place values of each digit, and the third column shows the value calculation, where the binary digits are multiplied by their corresponding place values.

The results of the third column can be added to verify that 111110010 in binary is indeed 498 in the decimal number system.

498 can be converted easily from decimal to binary using various methods. Let's see how it is done.

Expansion Method: Let us go through the step-by-step process of converting 498 using the expansion method.

Step 1 - Determine the place values: In the binary system, each place value is a power of 2. Therefore, we will identify the powers of 2.

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

Since 256 is less than 498, we stop at 2⁸ = 256.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁸ = 256 because it is the largest power of 2 less than or equal to 498. Write 1 in the 2⁸ place. Now, subtract the value of 2⁸, which is 256, from 498. 498 - 256 = 242.

Step 3 - Identify the next largest power of 2: Find the largest power of 2 that fits into 242. The next largest power of 2 is 2⁷ = 128. Write 1 in the 2⁷ place. Subtract 128 from 242. 242 - 128 = 114.

Step 4 - Continue identifying powers of 2: Next, identify the largest power of 2 for 114. The largest power of 2 is 2⁶ = 64. Write 1 in the 2⁶ place. Subtract 64 from 114. 114 - 64 = 50.

Step 5 - Continue with the remaining number: For 50, the largest power of 2 is 2⁵ = 32. Write 1 in the 2⁵ place. Subtract 32 from 50. 50 - 32 = 18.

Step 6 - Continue finding powers of 2: For 18, the largest power of 2 is 2^4 = 16. Write 1 in the 2^4 place. Subtract 16 from 18. 18 - 16 = 2.

Step 7 - Final power of 2: For 2, the power is 2¹ = 2. Write 1 in the 2¹ place. Subtract 2 from 2. 2 - 2 = 0. Stop here since the remainder is 0.

Step 8 - Fill unused place values: Write 0s in the remaining places (2³, 2², 2⁰). Write the values in reverse order: 111110010 is the binary equivalent of 498.

Grouping Method: Using this method, divide 498 by 2. Follow the step-by-step conversion:

Step 1 - Divide 498 by 2. 498 / 2 = 249. Quotient is 249, remainder is 0.

Step 2 - Divide the previous quotient (249) by 2. 249 / 2 = 124. Quotient is 124, remainder is 1.

Step 3 - Repeat the division. 124 / 2 = 62. Quotient is 62, remainder is 0.

Step 4 - Repeat the division. 62 / 2 = 31. Quotient is 31, remainder is 0.

Step 5 - Continue dividing. 31 / 2 = 15. Quotient is 15, remainder is 1.

Step 6 - Continue dividing. 15 / 2 = 7. Quotient is 7, remainder is 1.

Step 7 - Continue dividing. 7 / 2 = 3. Quotient is 3, remainder is 1.

Step 8 - Continue dividing. 3 / 2 = 1. Quotient is 1, remainder is 1.

Step 9 - Final step. 1 / 2 = 0. Quotient is 0, remainder is 1. Stop here.

Step 10 - Write remainders from bottom to top. Therefore, 498 (decimal) = 111110010 (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 methods to convert any number to binary. The place value method is similar to the expansion method, where we need to find the largest power of 2. Let's see a brief step-by-step explanation to understand this rule. Find the largest power of 2 less than or equal to 498. Since the answer is 2⁸, write 1 next to this power of 2. Subtract the value (256) from 498. So, 498 - 256 = 242. Find the largest power of 2 less than or equal to 242. The answer is 2⁷. So, write 1 next to this power. Continue the process until the remainder is 0. Final conversion will be 111110010.

Rule 2: Division by 2 Method

The division by 2 method is similar to the grouping method. A brief step-by-step explanation is given below for better understanding. Divide 498 by 2 to get 249 as the quotient and 0 as the remainder. Divide 249 by 2 to get 124 as the quotient and 1 as the remainder. Continue dividing the quotient until it becomes 0. Write the remainders upside down to get the binary equivalent of 498, 111110010.

Rule 3: Representation Method

This rule involves breaking the number into powers of 2. Identify the powers of 2 and write them down in decreasing order, i.e., 2⁸, 2⁷, 2⁶, etc. Find the largest power that fits into 498. 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 498, we use 0s and 1s for the correct 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 498.

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

Recognize the patterns: There is a pattern when converting numbers from decimal to binary, especially for powers of 2. 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 Continue recognizing patterns with larger numbers.

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 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 498 (base 10), 4 is in the hundreds place, 9 is in the tens place, and 8 is in the ones place.

• 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 position represents a power of 2. For example, the digit in the far right (least significant bit) is 2⁰, the next is 2¹, and so on.