100000 in Binary

The process of converting 100000 from decimal to binary involves dividing the number 100000 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 100000 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 100000 by 2 until getting 0 as the quotient is 11000011010100000. 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 11000011010100000. 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 11000011010100000 in binary is indeed 100000 in the decimal number system.

100000 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 100000 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 ... 2¹⁶ = 65536 2¹⁷ = 131072 Since 131072 is greater than 100000, we stop at 2¹⁶ = 65536.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹⁶ = 65536. 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, 100000. Since 2¹⁶ is the number we are looking for, write 1 in the 2¹⁶ place. Now the value of 2¹⁶, which is 65536, is subtracted from 100000. 100000 - 65536 = 34464.

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, 34464. So, the next largest power of 2 is 2¹⁵ = 32768. Now, we have to write 1 in the 2¹⁵ places. And then subtract 32768 from 34464. 34464 - 32768 = 1696. Repeat this process, identifying the largest powers of 2 and writing 1s and 0s in their respective places, until the remainder is 0.

Step 4 - Identify the unused place values: Write 0s in the places where the power of 2 is not used. After all these steps, you will have the binary representation of 100000.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 100000 in binary. Therefore, 11000011010100000 is 100000 in binary.

Grouping Method: In this method, we divide the number 100000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 100000 by 2. 100000 / 2 = 50000. Here, 50000 is the quotient and 0 is the remainder. Continue dividing the quotient by 2 and recording the remainder until the quotient becomes 0.

Step 2 - Write down the remainders from bottom to top. Therefore, 100000 (decimal) = 11000011010100000 (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 100000. Since the answer is 2¹⁶, write 1 next to this power of 2. Subtract the value (65536) from 100000. So, 100000 - 65536 = 34464. Find the largest power of 2 less than or equal to 34464. The answer is 2¹⁵. So, write 1 next to this power. Continue this process until the remainder is 0. Final conversion will be 11000011010100000.

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, 100000 is divided by 2 to get 50000 as the quotient and 0 as the remainder. Now, 50000 is divided by 2. Here, we will get 25000 as the quotient and 0 as the remainder. Continue dividing the quotient by 2, recording remainders, until the quotient is 0. Now, we write the remainders upside down to get the binary equivalent of 100000, 11000011010100000.

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., 2¹⁶, 2¹⁵, 2¹⁴, and so on. Find the largest power that fits into 100000. 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 100000, we use 0s for unused powers and 1s for used powers.

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 100000.

• Memorize to speed up conversions: We can memorize the binary forms for smaller numbers to quickly convert larger numbers.
   
• Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary. For example: 1 → 1 1 + 1 = 2 → 10 2 + 2 = 4 → 100 4 + 4 = 8 → 1000 This pattern can be extended to larger numbers.
   
• Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 100000 is even and its binary form is 11000011010100000. If the number is odd, then its binary equivalent will end in 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 occupies 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.

• Quotient: The result of division that represents how many times the divisor fits into the dividend.