765 in Binary

The process of converting 765 from decimal to binary involves dividing the number 765 by 2. Here, 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 765 to binary. In the last step, the remainder is noted down in reverse order, and that becomes the converted value.

For example, the remainders noted down after dividing 765 by 2 until getting 0 as the quotient is 1011111101. 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 765.

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 1011111101 in binary is indeed 765 in the decimal number system.

765 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 765 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 765 and the next power, 1024, is greater, we stop at 2⁹ = 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, 765. 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 765. 765 - 512 = 253.

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, 253. So, the next largest power of 2 is 2⁸, which is 256. Since 2⁸ is slightly more than 253, we move to 2⁷ = 128. We write 1 in the 2⁷ place. Now subtract 128 from 253. 253 - 128 = 125.

Step 4 - Continue identifying and subtracting powers of 2:

2⁶ = 64. Write 1 in the 2⁶ place. 125 - 64 = 61. 2⁵ = 32. Write 1 in the 2⁵ place. 61 - 32 = 29. 2⁴ = 16. Write 1 in the 2⁴ place. 29 - 16 = 13. 2³ = 8. Write 1 in the 2³ place. 13 - 8 = 5. 2² = 4. Write 1 in the 2² place. 5 - 4 = 1. 2¹ = 2. Write 0 in the 2¹ place (as 1 is less than 2). 2⁰ = 1. Write 1 in the 2⁰ place. 1 - 1 = 0.

Step 5 - Write the values in reverse order:

We now write the numbers upside down to represent 765 in binary. Therefore, 1011111101 is 765 in binary.

Grouping Method: In this method, we divide the number 765 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 765 by 2. 765 / 2 = 382 with a remainder of 1.

Step 2 - Divide the previous quotient (382) by 2. 382 / 2 = 191 with a remainder of 0.

Step 3 - Repeat the previous step. 191 / 2 = 95 with a remainder of 1.

Step 4 - Repeat the previous step. 95 / 2 = 47 with a remainder of 1.

Step 5 - Repeat the previous step. 47 / 2 = 23 with a remainder of 1.

Step 6 - Repeat the previous step. 23 / 2 = 11 with a remainder of 1.

Step 7 - Repeat the previous step. 11 / 2 = 5 with a remainder of 1.

Step 8 - Repeat the previous step. 5 / 2 = 2 with a remainder of 1.

Step 9 - Repeat the previous step. 2 / 2 = 1 with a remainder of 0.

Step 10 - Repeat the previous step. 1 / 2 = 0 with a remainder of 1.

Step 11 - Write down the remainders from bottom to top. Therefore, 765 (decimal) = 1011111101 (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 765. Since the answer is 2⁹, write 1 next to this power of 2. Subtract the value (512) from 765. So, 765 - 512 = 253. Find the largest power of 2 less than or equal to 253. The answer is 2⁷. So, write 1 next to this power. Continue this process until the remainder is 0. Final conversion will be 1011111101.

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, 765 is divided by 2 to get 382 as the quotient and 1 as the remainder. Now, 382 is divided by 2. Here, we will get 191 as the quotient and 0 as the remainder. Continue this division process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 765, 1011111101.

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 765. 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 765, we use 0s and 1s according to 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 765.

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

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. For e.g., 20 is even and its binary form is 10100. Here, the binary of 20 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 obtained by dividing one number by another. In binary conversion, it is used repeatedly in the division-by-2 method.