99999 in Binary

The process of converting 99999 from decimal to binary involves dividing the number 99999 by 2. Here, it is getting divided by 2 because the binary number system uses only two 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 99999 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 99999 by 2 until getting 0 as the quotient is 11000011010011111. Remember, the remainders here have been written upside down.

In the table shown below, the first column shows the binary digits (0 and 1) as 11000011010011111.

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 11000011010011111 in binary is indeed 99999 in the decimal number system.

99999 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 99999 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

2⁹ = 512

2¹⁰ = 1024

2¹¹ = 2048

2¹² = 4096

2¹³ = 8192

2¹⁴ = 16384 2¹⁵ = 32768

2¹⁶ = 65536

Since 65536 is the largest power of 2 less than or equal to 99999, we stop at 2¹⁶.

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, 99999. Since 65536 is the number we are looking for, write 1 in the 2¹⁶ place. Now the value of 65536 is subtracted from 99999. 99999 - 65536 = 34463.

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, 34463. So, the next largest power of 2 is 2¹⁵ = 32768. Now, we have to write 1 in the 2¹⁵ place. And then subtract 32768 from 34463. 34463 - 32768 = 1695.

Step 4 - Identify the next largest power of 2: Continuing this process, we find which powers of 2 fit into the remaining number, 1695. Repeat this process until the remainder is 0: 1695 - 1024 (2¹⁰) = 671 671 - 512 (2⁹) = 159 159 - 128 (2⁷) = 31 31 - 16 (2⁴) = 15 15 - 8 (2³) = 7 7 - 4 (2²) = 3 3 - 2 (2¹) = 1 1 - 1 (2⁰) = 0

Step 5 - Identify the unused place values: In the previous steps, we wrote 1 in the places corresponding to the powers of 2 that fit into the number. Now, we can just write 0s in the remaining places. Now, by substituting the values, we get, 1 in the 2¹⁶ place 1 in the 2¹⁵ place 0 in the 2¹⁴ place 0 in the 2¹³ place 0 in the 2¹² place 0 in the 2¹¹ place 1 in the 2¹⁰ place 1 in the 2⁹ place 0 in the 2⁸ place 1 in the 2⁷ place 0 in the 2⁶ place 0 in the 2⁵ place 1 in the 2⁴ place 1 in the 2³ place 1 in the 2² place 1 in the 2¹ place 1 in the 2⁰ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 99999 in binary. Therefore, 11000011010011111 is 99999 in binary.

Grouping Method: In this method, we divide the number 99999 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 99999 by 2. 99999 / 2 = 49999. Here, 49999 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (49999) by 2. 49999 / 2 = 24999. Here, the quotient is 24999 and the remainder is 1.

Step 3 - Repeat the previous step. Continue dividing by 2, noting down the remainders, until the quotient becomes 0.

Step 4 - Write down the remainders from bottom to top. Therefore, 99999 (decimal) = 11000011010011111 (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 99999. Since the answer is 2¹⁶, write 1 next to this power of 2. Subtract the value (65536) from 99999. So, 99999 - 65536 = 34463. Find the largest power of 2 less than or equal to 34463. The answer is 2¹⁵. So, write 1 next to this power. Continue this process until the remainder is 0. Write 0 next to the remaining powers. Final conversion will be 11000011010011111.

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, 99999 is divided by 2 to get 49999 as the quotient and 1 as the remainder. Now, 49999 is divided by 2. Here, we will get 24999 as the quotient and 1 as the remainder. Continue this process until the quotient becomes 0. We stop the division once the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 99999, 11000011010011111.

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., 2¹⁶, 2¹⁵, 2¹⁴, and so on. Find the largest power that fits into 99999. 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 99999, 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 99999.

Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 99999.

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 example, 99998 is even and its binary form ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, 99999 is odd and its binary form ends 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 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 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.

• Quotient: The result obtained when one number is divided by another. In binary conversion, it is used as the next dividend in the division process.