8128 in Binary

The process of converting 8128 from decimal to binary involves dividing the number 8128 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 8128 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 8128 by 2 until getting 0 as the quotient is 1111111100000. 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 1111111100000.

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 1111111100000 in binary is indeed 8128 in the decimal number system.

8128 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 8128 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

2¹⁰ = 1024

2¹¹ = 2048

2¹² = 4096

2¹³ = 8192

Since 8192 is greater than 8128, we stop at 2¹² = 4096.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹² = 4096. 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, 8128. Since 2¹² is the number we are looking for, write 1 in the 2¹² place. Now the value of 2¹², which is 4096, is subtracted from 8128. 8128 - 4096 = 4032.

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, 4032. Continue identifying and subtracting the largest possible power of 2 until the remainder is 0.

Step 4 - Identify the unused place values: In the steps above, we wrote 1s for the corresponding powers of 2. Now, we can just write 0s in the remaining unused places.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 8128 in binary. Therefore, 1111111100000 is 8128 in binary.

Grouping Method: In this method, we divide the number 8128 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 8128 by 2. 8128 / 2 = 4064. Here, 4064 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (4064) by 2. 4064 / 2 = 2032. Here, the quotient is 2032 and the remainder is 0.

Step 3 - Repeat the previous step. 2032 / 2 = 1016. Now, the quotient is 1016, and 0 is the remainder.

Step 4 - Repeat the previous step. Continue dividing the quotient by 2, recording the remainders, until the quotient is 0.

Step 5 - Write down the remainders from bottom to top. Therefore, 8128 (decimal) = 1111111100000 (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 8128. Since the answer is 2¹², write 1 next to this power of 2. Subtract the value (4096) from 8128. So, 8128 - 4096 = 4032. Continue finding the largest power of 2 less than or equal to the new number and repeat the process. Final conversion will be 1111111100000.

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, 8128 is divided by 2 to get 4064 as the quotient and 0 as the remainder. Now, 4064 is divided by 2. Here, we will get 2032 as the quotient and 0 as the remainder. Continue dividing the quotient by 2 until the quotient is 0. Now, we write the remainders upside down to get the binary equivalent of 8128, 1111111100000.

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., 2¹², 2¹¹, 2¹⁰, etc. Find the largest power that fits into 8128. 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 8128, we use appropriate 0s and 1s for 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 8128.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 16. 1 → 1, 2 → 10, 3 → 11, 4 → 100, 5 → 101, 6 → 110, 7 → 111, 8 → 1000, 9 → 1001, 10 → 1010, 11 → 1011, 12 → 1100, 13 → 1101, 14 → 1110, 15 → 1111, 16 → 10000.

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 example, 8128 is even and its binary form is 1111111100000. Here, the binary of 8128 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 1111111100000 (binary), each 1 or 0 has a value based on its position.

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

• Power of 2: In binary, each position represents a power of 2, such as 2⁰, 2¹, 2², etc.