117 in Binary

The process of converting 117 from decimal to binary involves dividing the number 117 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 method is commonly used to convert 117 to binary. In the last step, the remainder is noted down from bottom to top, and that becomes the converted value. For example, the remainders noted down after dividing 117 by 2 until getting 0 as the quotient is 1110101. Remember, the remainders here have been written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) for 117. 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 1110101 in binary is indeed 117 in the decimal number system.

117 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 117 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 2^7 = 128 Since 128 is greater than 117, 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, 117. 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 117. 117 - 64 = 53.

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, 53. So, the next largest power of 2 is 2^5, which is less than or equal to 53. Now, we have to write 1 in the 2^5 place. And then subtract 32 from 53. 53 - 32 = 21.

Step 4 - Continue the process: Find the next largest power of 2 for 21, which is 2^4 = 16. Write 1 in the 2^4 place. 21 - 16 = 5.

Step 5 - Identify the next largest power of 2 for 5, which is 2^2 = 4. Write 1 in the 2^2 place. 5 - 4 = 1.

Step 6 - Finally, write 1 in the 2^0 place for the remainder 1. Now, by substituting the values, we get: 1 in the 2^6 place 1 in the 2^5 place 1 in the 2^4 place 0 in the 2^3 place 1 in the 2^2 place 0 in the 2^1 place 1 in the 2^0 place Therefore, 1110101 is 117 in binary.

Grouping Method: In this method, we divide the number 117 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 117 by 2. 117 / 2 = 58 with a remainder of 1.

Step 2 - Divide the previous quotient (58) by 2. 58 / 2 = 29 with a remainder of 0.

Step 3 - Repeat the previous step. 29 / 2 = 14 with a remainder of 1.

Step 4 - Repeat the previous step. 14 / 2 = 7 with a remainder of 0.

Step 5 - Repeat the previous step. 7 / 2 = 3 with a remainder of 1.

Step 6 - Repeat the previous step. 3 / 2 = 1 with a remainder of 1.

Step 7 - Repeat the previous step. 1 / 2 = 0 with a remainder of 1.

Step 8 - Write down the remainders from bottom to top. Therefore, 117 (decimal) = 1110101 (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 117. Since the answer is 2^6, write 1 next to this power of 2. Subtract the value (64) from 117. So, 117 - 64 = 53. Find the largest power of 2 less than or equal to 53. The answer is 2^5. So, write 1 next to this power. Now, 53 - 32 = 21. Continue the process as outlined in the expansion method. Final conversion will be 1110101.

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, 117 is divided by 2 to get 58 as the quotient and 1 as the remainder. Now, 58 is divided by 2. Here, we will get 29 as the quotient and 0 as the remainder. Dividing 29 by 2, we get 14 as the quotient and 1 as the remainder. Divide 14 by 2 to get 7 as the quotient and 0 as the remainder. Continue this process until the quotient becomes 0. Now, write the remainders upside down to get the binary equivalent of 117, 1110101.

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 117. 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 117, we use 0s and 1s for the respective 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 117.

• Memorize to speed up conversions: We can memorize the binary forms for numbers like 1 to 10 or specific numbers relevant to our needs.
   
• 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, 118 is even and its binary form is 1110110. Here, the binary of 118 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 117 (an odd number) is 1110101. 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 117 (base 10), 1 has occupied the hundreds place, 1 is in the tens place, and 7 is in the ones place.

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

• Remainder: The amount left over after division when one divisor does not divide the dividend exactly.