119 in Binary

The process of converting 119 from decimal to binary involves dividing the number 119 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 method is commonly used to convert 119 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 119 by 2 until getting 0 as the quotient is 1110111. 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 1110111. 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 1110111 in binary is indeed 119 in the decimal number system.

119 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 119 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

Since 128 is greater than 119, 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, 119. 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 119. 119 - 64 = 55.

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, 55. So, the next largest power of 2 is 2⁵, which is less than or equal to 55. Now, we have to write 1 in the 2⁵ place. And then subtract 32 from 55. 55 - 32 = 23.

Step 4 - Continue the process: Now, find the largest power of 2 less than or equal to 23, which is 2⁴ = 16. Write 1 in the 2⁴ place and subtract 16 from 23. 23 - 16 = 7.

Step 5 - Continue with remaining value: For the remainder 7, the largest power of 2 is 2² = 4. Write 1 in the 22 place and subtract 4 from 7. 7 - 4 = 3. Now, for 3, the largest power of 2 is 2¹ = 2. Write 1 in the 2¹ place and subtract 2 from 3. 3 - 2 = 1. Finally, for 1, 2⁰ = 1. Write 1 in the 2⁰ place. We have reached 0, so the conversion process stops here.

Step 6 - Write the values in order: We now write the numbers to represent 119 in binary. Therefore, 1110111 is 119 in binary.

Grouping Method: In this method, we divide the number 119 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 119 by 2. 119 / 2 = 59. Here, 59 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (59) by 2. 59 / 2 = 29. Here, the quotient is 29 and the remainder is 1.

Step 3 - Repeat the previous step. 29 / 2 = 14. Now, the quotient is 14, and 1 is the remainder.

Step 4 - Repeat the previous step. 14 / 2 = 7. Here, the quotient is 7, and 0 is the remainder.

Step 5 - Continue the process. 7 / 2 = 3. The quotient is 3, and the remainder is 1. 3 / 2 = 1. The quotient is 1, and the remainder is 1. 1 / 2 = 0. The quotient is 0, and the remainder is 1. We stop the division here because the quotient is 0.

Step 6 - Write down the remainders from bottom to top. Therefore, 119 (decimal) = 1110111 (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 119. Since the answer is 2⁶, write 1 next to this power of 2. Subtract the value (64) from 119. So, 119 - 64 = 55. Find the largest power of 2 less than or equal to 55. The answer is 2⁵. So, write 1 next to this power. Now, 55 - 32 = 23. Continue the process for the remaining numbers. Final conversion will be 1110111.

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, 119 is divided by 2 to get 59 as the quotient and 1 as the remainder. Now, 59 is divided by 2. Here, we will get 29 as the quotient and 1 as the remainder. Continue dividing until the quotient is 0. Now, we write the remainders upside down to get the binary equivalent of 119, 1110111.

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 119. 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 119, we use 1s for 2⁶, 2⁵, 2⁴, 2², 2¹, and 2⁰, and 0s for 2³.

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 119.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 119.
   
• 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 example, 8 is even and its binary form is 1000. Here, the binary of 8 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 119 (an odd number) is 1110111. 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 119 (base 10), 1 occupies the hundreds place, 1 is in the tens place, and 9 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 number left after division. For example, in 119 divided by 8, the remainder is 7.