151 in Binary

The process of converting 151 from decimal to binary involves dividing the number 151 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 151 to binary. In the last step, the remainders are noted down from bottom to top, which becomes the converted value.

For example, the remainders noted down after dividing 151 by 2 until getting 0 as the quotient result in 10010111. Remember, the remainders are read from bottom to top.

In the table shown below, the first column shows the binary digits (1 and 0) as they appear in 10010111.

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 10010111 in binary is indeed 151 in the decimal number system.

151 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 151 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. 2¹ = 1 2² = 2 2³ = 4 2⁴ = 8 2⁵ = 16 2⁶ = 32 2⁷ = 64 2⁸ = 128 Since 128 is less than 151, we stop at 2⁸ = 128.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁸ = 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, 151. Since 2⁸ is the number we are looking for, write 1 in the 2⁸ place. Now the value of 2⁸, which is 128, is subtracted from 151. 151 - 128 = 23.

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, 23. So, the next largest power of 2 is 2⁴ = 16. Now, we have to write 1 in the 2⁴ place and then subtract 16 from 23. 23 - 16 = 7.

Step 4 - Repeat the steps for the remainder: The largest power of 2 less than or equal to 7 is 2³ = 4. Write 1 in the 2³ place. Subtract 4 from 7. 7 - 4 = 3. The next largest power of 2 is 2¹ = 2. Write 1 in the 2¹ place. Subtract 2 from 3. 3 - 2 = 1. Finally, write 1 in the 2⁰ place since 2⁰ = 1.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 151 in binary. Therefore, 10010111 is 151 in binary.

Grouping Method: In this method, we divide the number 151 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 151 by 2. 151 / 2 = 75. Here, 75 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (75) by 2. 75 / 2 = 37. Here, the quotient is 37 and the remainder is 1.

Step 3 - Repeat the previous step. 37 / 2 = 18. Now, the quotient is 18, and 1 is the remainder.

Step 4 - Repeat the previous step. 18 / 2 = 9. Here, the quotient is 9 and the remainder is 0.

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

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

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

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

Step 9 - Write down the remainders from bottom to top. Therefore, 151 (decimal) = 10010111 (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 151. Since the answer is 2⁸, write 1 next to this power of 2. Subtract the value (128) from 151. So, 151 - 128 = 23. Find the largest power of 2 less than or equal to 23. The answer is 2⁴. So, write 1 next to this power. Repeat this process for the remainders until 0 is reached. Final conversion will be 10010111.

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, 151 is divided by 2 to get 75 as the quotient and 1 as the remainder. Now, 75 is divided by 2. Here, we will get 37 as the quotient and 1 as the remainder. Continue dividing each quotient by 2 until the quotient is 0. Write the remainders upside down to get the binary equivalent of 151, which is 10010111.

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., 2⁸, 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 151. 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 151, we use 0s for unused powers and 1s for used powers of 2.

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 151.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers, which helps in quickly converting larger numbers.

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 ...

Recognize patterns for larger numbers: As you become familiar with more numbers, start recognizing patterns for numbers like 32, 64, and 128.

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 16 is even and its binary form is 10000. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 17 is 10001.

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.

• Powers of 2: The basis for the binary number system, where each place value is a power of 2.