120 in Binary

The process of converting 120 from decimal to binary involves dividing the number 120 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 120 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 120 by 2 until getting 0 as the quotient is 1111000. 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 1111000.

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 1111000 in binary is indeed 120 in the decimal number system.

120 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 120 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 120, we stop at 2^6 = 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, 120.

Since 2⁶ is the number we are looking for, write 1 in the 2^6 place. Now the value of 2⁶, which is 64, is subtracted from 120. 120 - 64 = 56.

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, 56. So, the next largest power of 2 is 2⁵ = 32. Now, we have to write 1 in the 2⁵ place. And then subtract 32 from 56. 56 - 32 = 24.

Step 4 - Identify the next largest power of 2: Repeat the process for 24. The next largest power of 2 is 2⁴ = 16. Write 1 in the 2⁴ place and subtract 16 from 24. 24 - 16 = 8.

Step 5 - Identify the next largest power of 2: For 8, the largest power is 2³ = 8. Write 1 in the 2³ place and subtract 8 from 8. 8 - 8 = 0. We need to stop the process here since the remainder is 0.

Step 6 - Identify the unused place values: In steps 2, 3, 4, and 5, we wrote 1 in the 2⁶, 2⁵, 2⁴, and 2³ places. Now, we can just write 0s in the remaining places, which are 2², 2¹, and 2⁰. Now, by substituting the values, we get: 0 in the 2² place 0 in the 2¹ place 0 in the 2⁰ place

Step 7 - Write the values in reverse order: We now write the numbers upside down to represent 120 in binary.

Therefore, 1111000 is 120 in binary. Grouping Method: In this method, we divide the number 120 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 120 by 2. 120 / 2 = 60. Here, 60 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (60) by 2. 60 / 2 = 30. Here, the quotient is 30 and the remainder is 0.

Step 3 - Repeat the previous step. 30 / 2 = 15. Now, the quotient is 15, and 0 is the remainder.

Step 4 - Repeat the previous step. 15 / 2 = 7. Here, the remainder is 1.

Step 5 - Continue the process. 7 / 2 = 3. Here, the remainder is 1.

Step 6 - Continue the process. 3 / 2 = 1. Here, the remainder is 1.

Step 7 - Divide the remaining quotient. 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, 120 (decimal) = 1111000 (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 120. Since the answer is 2⁶, write 1 next to this power of 2. Subtract the value (64) from 120. So, 120 - 64 = 56. Find the largest power of 2 less than or equal to 56. The answer is 2⁵. So, write 1 next to this power. Continue this process until you reach 0. Final conversion will be 1111000.

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, 120 is divided by 2 to get 60 as the quotient and 0 as the remainder.

Now, 60 is divided by 2. Here, we will get 30 as the quotient and 0 as the remainder. Continue dividing until the quotient becomes 0. Write the remainders upside down to get the binary equivalent of 120, 1111000.

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, etc. Find the largest power that fits into 120. 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 120, we use 0s for 2², 2¹, and 2⁰ and 1s for 2⁶, 2⁵, 2⁴, and 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 120. Memorize to speed up conversions: We can memorize the binary forms for key numbers.

Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary. 1 → 1 2 → 10 4 → 100 8 → 1000 16 → 10000 32 → 100000 64 → 1000000 128 → 10000000…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, 120 is even and its binary form is 1111000. Here, the binary of 120 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 121 (an odd number) is 1111001. 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.

• Power of 2: Each position in a binary number represents a power of 2, starting from 2⁰ from right to left.

• 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 which is used in determining the binary form of a number.