141 in Binary

The process of converting 141 from decimal to binary involves dividing the number 141 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 is a commonly used method to convert 141 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 141 by 2 until getting 0 as the quotient is 10001101. 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 10001101.

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 10001101 in binary is indeed 141 in the decimal number system.

141 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 141 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 less than 141, we start with 2⁷.

Step 2 - Identify the largest power of 2: In the previous step, we started at 2⁷ = 128. 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, 141. Since 2⁷ is the number we are looking for, write 1 in the 2⁷ place. Now the value of 2⁷, which is 128, is subtracted from 141. 141 - 128 = 13.

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, 13. So, the next largest power of 2 is 2³ = 8. Now, we have to write 1 in the 2³ places. And then subtract 8 from 13. 13 - 8 = 5.

Step 4 - Continue the process: Now, the largest power of 2 that fits into 5 is 2² = 4. Write 1 in the 2^2 place. Subtract 4 from 5. 5 - 4 = 1.

Step 5 - Final adjustment: The largest power of 2 that fits into 1 is 2⁰ = 1. Write 1 in the 2⁰ place. Subtract 1 from 1. 1 - 1 = 0. Now, by substituting the values, we get, 1 in the 2⁷ place 0 in the 2⁶ place 0 in the 2⁵ place 0 in the 2⁴ place 1 in the 2³ place 1 in the 2² place 0 in the 2¹ place 1 in the 2⁰ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 141 in binary. Therefore, 10001101 is 141 in binary.

Grouping Method: In this method, we divide the number 141 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 141 by 2. 141 / 2 = 70. Here, 70 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (70) by 2. 70 / 2 = 35. Here, the quotient is 35 and the remainder is 0.

Step 3 - Repeat the previous step. 35 / 2 = 17. Now, the quotient is 17, and 1 is the remainder.

Step 4 - Repeat the previous step. 17 / 2 = 8. Here, the quotient is 8, and 1 is the remainder.

Step 5 - Repeat the previous step. 8 / 2 = 4. Here, the quotient is 4, and 0 is the remainder.

Step 6 - Repeat the previous step. 4 / 2 = 2. Here, the quotient is 2, and 0 is the remainder.

Step 7 - Repeat the previous step. 2 / 2 = 1. Here, the quotient is 1, and 0 is the remainder.

Step 8 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 9 - Write down the remainders from bottom to top. Therefore, 141 (decimal) = 10001101 (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 141. Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 141. So, 141 - 128 = 13. Find the largest power of 2 less than or equal to 13. The answer is 2³. So, write 1 next to this power. Continue for 2² and 2⁰. Now, 13 - 8 - 4 - 1 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2⁶, 2⁵, and 2⁴). Final conversion will be 10001101.

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, 141 is divided by 2 to get 70 as the quotient and 1 as the remainder. Now, 70 is divided by 2. Here, we will get 35 as the quotient and 0 as the remainder. Dividing 35 by 2, we get 17 as the quotient and 1 as the remainder. Continue this process until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 141, 10001101.

Rule 3: Representation Method

This rule also involves breaking down the number into powers of 2. Identify the powers of 2 and write it down in decreasing order, i.e., 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 141. 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 141, we use 1s for 2⁷, 2³, 2², and 2⁰, and 0s for 2⁶, 2⁵, 2⁴, and 2¹.

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 141.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 15, and then practice extending this to larger numbers like 141.

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, 14 is even, and its binary form is 1110. Here, the binary of 14 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 141 (an odd number) is 10001101. 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 102 (base 10), 1 has occupied the hundreds place, 0 is in the tens place, and 2 is in the ones place.

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

• Power of 2: In the binary system, each place value is a power of 2, such as 2⁰, 2¹, 2², etc.