480 in Binary

The process of converting 480 from decimal to binary involves dividing the number 480 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 480 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 480 by 2 until getting 0 as the quotient is 111100000. 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 111100000.

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 111100000 in binary is indeed 480 in the decimal number system.

480 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 480 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 Since 256 is the largest power of 2 less than 480, we begin with 2^8.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁸ = 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, 480. Since 2⁸ is the number we are looking for, write 1 in the 2⁸ place. Now the value of 2⁸, which is 256, is subtracted from 480. 480 - 256 = 224.

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, 224. 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 224. 224 - 128 = 96.

Step 4 - Continue the process: Identify the largest power of 2 that fits into 96. It is 2⁶ = 64. Write 1 in the 2⁶ place and subtract 64 from 96. 96 - 64 = 32. Now, 32 fits perfectly into 2⁵, so write 1 in the 2⁵ place and subtract. 32 - 32 = 0.

Step 5 - Identify the unused place values: In previous steps, we wrote 1 in the 2⁸, 2⁷, 2⁶, and 2⁵ places. Now, we can just write 0s in the remaining places, which are 2⁴, 2³, 2², 2¹, and 2⁰. Now, by substituting the values, we get: 0 in the 2⁰ place 0 in the 2¹ place 0 in the 2² place 0 in the 2³ place 0 in the 2⁴ place 1 in the 2⁵ place 1 in the 2⁶ place 1 in the 2⁷ place 1 in the 2⁸ place Therefore, the binary representation of 480 is 111100000.

Grouping Method: In this method, we divide the number 480 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 480 by 2. 480 / 2 = 240. Here, 240 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (240) by 2. 240 / 2 = 120. Here, the quotient is 120 and the remainder is 0.

Step 3 - Repeat the previous step. 120 / 2 = 60. Now, the quotient is 60, and 0 is the remainder.

Step 4 - Continue the division process. 60 / 2 = 30, remainder 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. Write down the remainders from bottom to top. Therefore, 480 (decimal) = 111100000 (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 480. Since the answer is 2⁸, write 1 next to this power of 2. Subtract the value (256) from 480. So, 480 - 256 = 224. Find the largest power of 2 less than or equal to 224. The answer is 2⁷. So, write 1 next to this power. Now, 224 - 128 = 96. Find the largest power of 2 less than or equal to 96. The answer is 2⁶. Write 1 next to this power. 96 - 64 = 32. Find the largest power of 2 less than or equal to 32. The answer is 2⁵. Write 1 next to this power. 32 - 32 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2⁴, 2³, 2², 2¹, and 2⁰). Final conversion will be 111100000.

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, 480 is divided by 2 to get 240 as the quotient and 0 as the remainder. Now, 240 is divided by 2. Here, we will get 120 as the quotient and 0 as the remainder. Dividing 120 by 2, we get 60 as the quotient and 0 as the remainder. Continue dividing until the quotient becomes 0, writing down each remainder. Finally, write the remainders upside down to get the binary equivalent of 480, which is 111100000.

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., 2⁸, 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 480. 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 480, we use 0s for 2⁴, 2³, 2², 2¹, and 2⁰, and 1s for 2⁸, 2⁷, 2⁶, and 2⁵.

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 480.

Memorize to speed up conversions: We can memorize the binary forms for numbers that are powers of 2.

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 ... 256 + 256 = 512 → 1000000000

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 480 is even and its binary form is 111100000. Here, the binary of 480 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, excluding the remainder.