105 in Binary

The process of converting 105 from decimal to binary involves dividing the number 105 by 2. Here, it is divided by 2 because the binary number 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 is a commonly used method to convert 105 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 105 by 2 until getting 0 as the quotient is 1101001. 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 1101001. 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 1101001 in binary is indeed 105 in the decimal number system.

105 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 105 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 Since 64 is less than 105, we stop at 2⁶ = 64.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁶ = 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, 105. Since 2⁶ is the number we are looking for, write 1 in the 2⁶ place. Now the value of 2⁶, which is 64, is subtracted from 105. 105 - 64 = 41.

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, 41. So, the next largest power of 2 is 2⁵ = 32. Write 1 in the 2⁵ place. Subtract 32 from 41. 41 - 32 = 9.

Step 4 - Continue with smaller powers: Next, find the largest power of 2 that fits into 9, which is 2³ = 8. Write 1 in the 2³ place. Subtract 8 from 9. 9 - 8 = 1.

Step 5 - Continue to the smallest power: The last remaining value is 1, which is 2⁰. Write 1 in the 2⁰ place. Subtract 1 from 1. 1 - 1 = 0. We need to stop the process here since the remainder is 0.

Step 6 - Identify the unused place values: In the previous steps, we wrote 1s in the 2⁶, 2⁵, 2³, and 2⁰ places. Now, we can just write 0s in the remaining places, which are 2⁴ and 2². Now, by substituting the values, we get: 0 in the 2⁴ place 0 in the 2² place

Step 7 - Write the values in reverse order: We now write the numbers upside down to represent 105 in binary. Therefore, 1101001 is 105 in binary.

Grouping Method: In this method, we divide the number 105 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 105 by 2. 105 / 2 = 52. Here, 52 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (52) by 2. 52 / 2 = 26. Here, the quotient is 26 and the remainder is 0.

Step 3 - Repeat the previous step. 26 / 2 = 13. Now, the quotient is 13, and 0 is the remainder.

Step 4 - Repeat the previous step. 13 / 2 = 6. Here, the quotient is 6, and 1 is the remainder.

Step 5 - Repeat the previous step. 6 / 2 = 3. Here, the quotient is 3, and 0 is the remainder.

Step 6 - Repeat the previous step. 3 / 2 = 1. Here, the quotient is 1, and 1 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, 105 (decimal) = 1101001 (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 105. Since the answer is 2⁶, write 1 next to this power of 2. Subtract the value (64) from 105. So, 105 - 64 = 41. Find the largest power of 2 less than or equal to 41. The answer is 2⁵. So, write 1 next to this power. Now, 41 - 32 = 9. Continue with 9, the largest power of 2 is 2³. Write 1 next to it. 9 - 8 = 1. The largest power of 2 for 1 is 2⁰. Write 1 next to it. Since there is no remainder, we write 0 next to the remaining powers (2⁴ and 2²). Final conversion will be 1101001.

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, 105 is divided by 2 to get 52 as the quotient and 1 as the remainder. Now, 52 is divided by 2. Here, we will get 26 as the quotient and 0 as the remainder. Dividing 26 by 2, we get 13 as the quotient and 0 as the remainder. Divide 13 by 2 to get 6 as the quotient and 1 as the remainder. Divide 6 by 2 to get 3 as the quotient and 0 as the remainder. Divide 3 by 2 to get 1 as the quotient and 1 as the remainder. Divide 1 by 2 to get 1 as the remainder and 0 as the quotient. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 105, 1101001.

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., 2⁶, 2⁵ ,2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 105. 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 105, we use 0s for 2⁴ and 2², and 1s for 2⁶, 2⁵, 2³, and 2⁰

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 105.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 105.
   
• 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 ... Continue this pattern to recognize binary equivalents.
   
• Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 16 is even and its binary form is 10000. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 105 is 1101001. 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 105 (base 10), 1 has occupied the hundreds place, 0 is in the tens place, and 5 is in the ones place.

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

• Powers of 2: In binary, each place value represents a power of 2, which is critical for conversion from decimal to binary.