95 in Binary

The process of converting 95 from decimal to binary involves dividing the number 95 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 95 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 95 by 2 until getting 0 as the quotient is 1011111. 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 1011111. 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 1011111 in binary is indeed 95 in the decimal number system.

95 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 95 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 Since 128 is greater than 95, we stop at 2^6 = 64.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^6 = 64. 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, 95. Since 2^6 is the number we are looking for, write 1 in the 2^6 place. Now the value of 2^6, which is 64, is subtracted from 95. 95 - 64 = 31.

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, 31. So, the next largest power of 2 is 2^4 = 16, which is less than or equal to 31. Now, we have to write 1 in the 2^4 place. And then subtract 16 from 31. 31 - 16 = 15.

Step 4 - Identify the next largest power of 2: The next largest power of 2 is 2^3 = 8, which is less than or equal to 15. Now, we have to write 1 in the 2^3 place. And then subtract 8 from 15. 15 - 8 = 7.

Step 5 - Continue identifying powers of 2: The next largest power of 2 is 2^2 = 4, which is less than or equal to 7. Now, we have to write 1 in the 2^2 place. And then subtract 4 from 7. 7 - 4 = 3.

Step 6 - Continue identifying powers of 2: The next largest power of 2 is 2^1 = 2, which is less than or equal to 3. Now, we have to write 1 in the 2^1 place. And then subtract 2 from 3. 3 - 2 = 1.

Step 7 - Final power of 2: The next largest power of 2 is 2^0 = 1, which equals 1. Now, we have to write 1 in the 2^0 place. And then subtract 1 from 1. 1 - 1 = 0. We need to stop the process here since the remainder is 0.

Step 8 - Write the binary representation: The binary representation of 95 is constructed by placing 1s in the positions identified in the previous steps. Therefore, 1011111 is 95 in binary.

Grouping Method: In this method, we divide the number 95 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 95 by 2. 95 / 2 = 47. Here, 47 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (47) by 2. 47 / 2 = 23. Here, the quotient is 23 and the remainder is 1.

Step 3 - Repeat the previous step. 23 / 2 = 11. Now, the quotient is 11, and 1 is the remainder.

Step 4 - Repeat the previous step. 11 / 2 = 5. Here, the quotient is 5, and 1 is the remainder.

Step 5 - Repeat the previous step. 5 / 2 = 2. Here, the quotient is 2, and 1 is the remainder.

Step 6 - Repeat the previous step. 2 / 2 = 1. Here, the quotient is 1, and 0 is the remainder.

Step 7 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 8 - Write down the remainders from bottom to top.

Therefore, 95 (decimal) = 1011111 (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 95. Since the answer is 2^6, write 1 next to this power of 2. Subtract the value (64) from 95. So, 95 - 64 = 31. Find the largest power of 2 less than or equal to 31. The answer is 2^4. So, write 1 next to this power. Now, 31 - 16 = 15. Continue the process until the remainder is 0. Final conversion will be 1011111.

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, 95 is divided by 2 to get 47 as the quotient and 1 as the remainder. Now, 47 is divided by 2. Here, we will get 23 as the quotient and 1 as the remainder. Dividing 23 by 2, we get 11 as the quotient and 1 as the remainder. Continue the division process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 95, 1011111.

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., 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Find the largest power that fits into 95. 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 95, we use 1s for 2^6, 2^4, 2^3, 2^2, 2^1, and 2^0.

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 95.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 95. 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, 94 is even and its binary form is 1011110. Here, the binary of 94 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 95 (an odd number) is 1011111. As you can see, the last digit here is 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 1011111 (base 2), 1 occupies the 2^6 place, and so on.

• 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, crucial for understanding binary conversions.