157 in Binary

The process of converting 157 from decimal to binary involves dividing the number 157 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 157 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 157 by 2 until getting 0 as the quotient is 10011101. 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 10011101. 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 10011101 in binary is indeed 157 in the decimal number system.

157 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 157 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 less than 157, 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, 157. 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 157. 157 - 128 = 29.

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, 29. So, the next largest power of 2 is 2^4, which is less than or equal to 29. Now, we have to write 1 in the 2^4 places. And then subtract 16 from 29. 29 - 16 = 13.

Step 4 - Repeat the process: Now, find the largest power of 2 that fits into 13. The answer is 2^3. Write 1 in the 2^3 place. 13 - 8 = 5.

Step 5 - Repeat the process: Now, find the largest power of 2 that fits into 5. The answer is 2^2. Write 1 in the 2^2 place. 5 - 4 = 1.

Step 6 - Repeat the process: Now, find the largest power of 2 that fits into 1. The answer is 2^0. Write 1 in the 2^0 place. 1 - 1 = 0. We need to stop the process here since the remainder is 0.

Step 7 - Identify the unused place values: In the steps above, we wrote 1 in the 2^7, 2^4, 2^3, 2^2, and 2^0 places. Now, we can just write 0s in the remaining places, which are 2^6 and 2^1. Now, by substituting the values, we get, 1 in the 2^0 place 0 in the 2^1 place 1 in the 2^2 place 1 in the 2^3 place 1 in the 2^4 place 0 in the 2^5 place 0 in the 2^6 place 1 in the 2^7 place

Step 8 - Write the values in reverse order: We now write the numbers upside down to represent 157 in binary. Therefore, 10011101 is 157 in binary.

Grouping Method: In this method, we divide the number 157 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 157 by 2. 157 / 2 = 78. Here, 78 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (78) by 2. 78 / 2 = 39. Here, the quotient is 39 and the remainder is 0.

Step 3 - Repeat the previous step. 39 / 2 = 19. Now, the quotient is 19, and 1 is the remainder. Step 4 - Repeat the previous step. 19 / 2 = 9. Here, the remainder is 1.

Step 5 - Repeat the previous step. 9 / 2 = 4. Here, the remainder is 1.

Step 6 - Repeat the previous step. 4 / 2 = 2. Here, the remainder is 0. Step 7 - Repeat the previous step. 2 / 2 = 1. Here, the remainder is 0.

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, 157 (decimal) = 10011101 (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 157. Since the answer is 2^7, write 1 next to this power of 2. Subtract the value (128) from 157. So, 157 - 128 = 29. Find the largest power of 2 less than or equal to 29. The answer is 2^4. So, write 1 next to this power. Now, 29 - 16 = 13. Find the largest power of 2 less than or equal to 13. The answer is 2^3. So, write 1 next to this power. 13 - 8 = 5. Find the largest power of 2 less than or equal to 5. The answer is 2^2. So, write 1 next to this power. 5 - 4 = 1. Find the largest power of 2 less than or equal to 1. The answer is 2^0. So, write 1 next to this power. Now, 1 - 1 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2^6 and 2^1). Final conversion will be 10011101.

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, 157 is divided by 2 to get 78 as the quotient and 1 as the remainder. Now, 78 is divided by 2. Here, we will get 39 as the quotient and 0 as the remainder. Dividing 39 by 2, we get 1 as the remainder and 19 as the quotient. Divide 19 by 2 to get 1 as the remainder and 9 as the quotient. Continue dividing until you reach 0 as the quotient. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 157, 10011101.

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., 2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, and 2^0. Find the largest power that fits into 157. 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 157, we use 1s for 2^7, 2^4, 2^3, 2^2, and 2^0, and 0s for 2^6, 2^5, and 2^1.

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 157.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers to help with larger conversions. 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 e.g., 156 is even and its binary form is 10011100. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 157 (an odd number) is 10011101. 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.

• Hexadecimal: This number system has 16 digits. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here, A to F represent the values 10 to 15.