345 in Binary

The process of converting 345 from decimal to binary involves dividing the number 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 345 to binary. In the last step, the remainder is noted down from bottom to top, and that becomes the converted value. For example, the remainders noted down after dividing 345 by 2 until getting 0 as the quotient is 101011001. Remember, the remainders are written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 101011001. 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 101011001 in binary is indeed 345 in the decimal number system.

345 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 345 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

2⁸ = 256

Since 256 is less than 345, we stop at 2⁸ = 256.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2⁸ = 256. This is because we have to identify the largest power of 2, which is less than or equal to the given number, 345. Since 2⁸ is the number we are looking for, write 1 in the 2⁸ place. Now the value of 2⁸, which is 256, is subtracted from 345. 345 - 256 = 89

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, 89. So, the next largest power of 2 is 2⁶, which is 64. Write 1 in the 2⁶ place. And then subtract 64 from 89. 89 - 64 = 25

Step 4 - Continue with remaining powers of 2: Repeat the process with the remaining number 25. The largest power of 2 less than or equal to 25 is 2⁴, which is 16. Write 1 in the 2⁴ place and subtract 16 from 25. 25 - 16 = 9

Step 5 - Repeat the process for 9: The largest power of 2 less than or equal to 9 is 2³, which is 8. Write 1 in the 2³ place and subtract 8 from 9. 9 - 8 = 1

Step 6 - Final step for 1: The largest power of 2 that fits into 1 is 2⁰, which is 1. Write 1 in the 2⁰ place and subtract 1 from 1. 1 - 1 = 0

Step 7 - Fill remaining places with 0: In the remaining places, write 0s. Now, by substituting the values, we get: 0 in the 2¹ place 0 in the 2² place 0 in the 2⁵ place 0 in the 2⁷ place

Step 8 - Write the values in reverse order: We now write the numbers upside down to represent 345 in binary. Therefore, 101011001 is 345 in binary.

Grouping Method: In this method, we divide the number 345 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 345 by 2. 345 / 2 = 172. Here, 172 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (172) by 2. 172 / 2 = 86. Here, the quotient is 86 and the remainder is 0.

Step 3 - Repeat the previous step. 86 / 2 = 43. Now, the quotient is 43, and 0 is the remainder.

Step 4 - Continue dividing. 43 / 2 = 21. The quotient is 21 and the remainder is 1.

Step 5 - Continue dividing. 21 / 2 = 10. The quotient is 10 and the remainder is 1.

Step 6 - Continue dividing. 10 / 2 = 5. The quotient is 5 and the remainder is 0.

Step 7 - Continue dividing. 5 / 2 = 2. The quotient is 2 and the remainder is 1.

Step 8 - Continue dividing. 2 / 2 = 1. The quotient is 1 and the remainder is 0.

Step 9 - Final division. 1 / 2 = 0. The remainder is 1. We stop the division here because the quotient is 0.

Step 10 - Write down the remainders from bottom to top. Therefore, 345 (decimal) = 101011001 (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 345. Since the answer is 2⁸, write 1 next to this power of 2. Subtract the value (256) from 345. So, 345 - 256 = 89. Find the largest power of 2 less than or equal to 89. The answer is 2⁶. So, write 1 next to this power. Now, 89 - 64 = 25. Continue until the remainder is 0. Final conversion will be 101011001.

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, 345 is divided by 2 to get 172 as the quotient and 1 as the remainder. Now, 172 is divided by 2. Here, we will get 86 as the quotient and 0 as the remainder. Dividing 86 by 2, we get 0 as the remainder and 43 as the quotient. Continue until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 345, 101011001.

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 345. 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 345, we use 1s for 2⁸, 2⁶, 2⁴, 2³, 2⁰ and 0s for 2⁷, 2⁵, 2², 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 345.

• Memorize to speed up conversions: We can memorize the binary forms for smaller numbers, as it helps in faster conversion.
   
• Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary.
   
• Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 345 is odd, and its binary form, 101011001, ends in 1. If the number is even, then its binary equivalent will end in 0.
   
• 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 in 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 101011001 (base 2), the leftmost 1 has the highest place value.
   
• Octal: It is the number system with a base of 8. It uses digits from 0 to 7.
   
• Quotient: It is the result obtained by dividing one number by another. In binary conversion, the quotient becomes the new dividend in subsequent division steps.