143 in Binary

The process of converting 143 from decimal to binary involves dividing the number 143 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 143 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 143 by 2 until getting 0 as the quotient is 10001111. 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 10001111.

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 10001111 in binary is indeed 143 in the decimal number system.

143 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 143 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 2^8 = 256 Since 256 is greater than 143, we stop at 2^7 = 128.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^7 = 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, 143. Since 2^7 is the number we are looking for, write 1 in the 2^7 place. Now the value of 2^7, which is 128, is subtracted from 143. 143 - 128 = 15.

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, 15. So, the next largest power of 2 is 2^3, which is 8. Now, we have to write 1 in the 2^3 places. And then subtract 8 from 15. 15 - 8 = 7.

Step 4 - Identify the next largest power of 2 for the remainder: Now, for 7, the largest power of 2 is 2^2, which is 4. Write 1 in the 2^2 place and subtract 4 from 7. 7 - 4 = 3.

Step 5 - Continue until the remainder is zero: For 3, the largest power of 2 is 2^1, which is 2. Write 1 in the 2^1 place and subtract 2 from 3. 3 - 2 = 1. Finally, for 1, the largest power of 2 is 2^0, which is 1. Write 1 in the 2^0 place and subtract 1 from 1. 1 - 1 = 0. We stop the process here since the remainder is 0.

Step 6 - Write the values: We place 0s in the unused place values, which are 2^6, 2^5, and 2^4. Now, by substituting the values, we get, 1 in the 2^7 place 0 in the 2^6 place 0 in the 2^5 place 0 in the 2^4 place 1 in the 2^3 place 1 in the 2^2 place 1 in the 2^1 place 1 in the 2^0 place Therefore, 10001111 is 143 in binary. Grouping Method: In this method, we divide the number 143 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 143 by 2. 143 / 2 = 71. Here, 71 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (71) by 2. 71 / 2 = 35. Here, the quotient is 35 and the remainder is 1.

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 this process. 8 / 2 = 4. Here, the quotient is 4, and 0 is the remainder.

Step 6 - Divide again. 4 / 2 = 2. The quotient is 2, and 0 is the remainder.

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

Step 8 - Write down the remainders from bottom to top. Therefore, 143 (decimal) = 10001111 (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 143.

Since the answer is 2⁷, write 1 next to this power of 2. Subtract the value (128) from 143. So, 143 - 128 = 15. Find the largest power of 2 less than or equal to 15. The next largest is 2³, so write 1 next to this power. Continue the process for 2², 2¹, and 2⁰. Final conversion will be 10001111.

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, 143 is divided by 2 to get 71 as the quotient and 1 as the remainder.

Now, 71 is divided by 2. Here, we will get 35 as the quotient and 1 as the remainder. Continue dividing and writing the remainders until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 143, 10001111.

Rule 3: Representation Method This rule also involves breaking 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 143. 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 143, we use 0s for 2⁶, 2⁵, and 2⁴ and 1s for 2⁷, 2³, 2², 2¹, 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 143.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers and build upon them. Recognize the patterns:

There is a 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, 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 143 (an odd number) is 10001111. 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 10001111 (base 2), each digit has a place value that is a power of 2.

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

• Quotient and Remainder: These are results obtained from dividing one number by another. In binary conversion, they help determine the binary digits.