10000 in Binary

The process of converting 10000 from decimal to binary involves dividing the number 10000 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 10000 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 10000 by 2 until getting 0 as the quotient is 10011100010000. 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 10000.

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 10011100010000 in binary is indeed 10000 in the decimal number system.

10000 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 10000 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

Since 16384 is greater than 10000, we stop at 2¹³ = 8192.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2¹³ = 8192. 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, 10000. Since 2¹³ is the number we are looking for, write 1 in the 2¹³ place. Now the value of 2¹³, which is 8192, is subtracted from 10000. 10000 - 8192 = 1808.

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, 1808. So, the next largest power of 2 is 2¹⁰, which is less than or equal to 1808. Now, we have to write 1 in the 2¹⁰ place. And then subtract 1024 from 1808. 1808 - 1024 = 784.

Step 4 - Continue the process: Repeat the steps by finding the next largest powers of 2 that fit into the remaining number, and write 1s in those places, subtracting each time. Fill in 0s for unused place values. Now, by substituting the values, we get, 1 in the 2¹³ place 0 in the 2¹² place 0 in the 2¹¹ place 1 in the 2¹⁰ place 1 in the 2⁹ place 1 in the 2⁸ place 0 in the 2⁷ place 0 in the 2⁶ place 0 in the 2⁵ place 1 in the 2⁴ place 0 in the 2³ place 0 in the 2² place 0 in the 2¹ place 0 in the 2⁰ place.

Step 5 - Write the values in reverse order: We now write the numbers upside down to represent 10000 in binary. Therefore, 10011100010000 is 10000 in binary.

Grouping Method: In this method, we divide the number 10000 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 10000 by 2. 10000 / 2 = 5000. Here, 5000 is the quotient and 0 is the remainder.

Step 2 - Divide the previous quotient (5000) by 2. 5000 / 2 = 2500. Here, the quotient is 2500 and the remainder is 0.

Step 3 - Repeat the previous step. 2500 / 2 = 1250. Now, the quotient is 1250, and 0 is the remainder.

Step 4 - Continue dividing until the quotient becomes 0.

1250 / 2 = 625. Remainder 0.

625 / 2 = 312. Remainder 1.

312 / 2 = 156. Remainder 0.

156 / 2 = 78. Remainder 0.

78 / 2 = 39. Remainder 0.

39 / 2 = 19. Remainder 1.

19 / 2 = 9. Remainder 1.

9 / 2 = 4. Remainder 1.

4 / 2 = 2. Remainder 0.

2 / 2 = 1. Remainder 0.

1 / 2 = 0. Remainder 1.

Step 5 - Write down the remainders from bottom to top. Therefore, 10000 (decimal) = 10011100010000 (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 10000. Since the answer is 2¹³, write 1 next to this power of 2. Subtract the value (8192) from 10000. So, 10000 - 8192 = 1808. Find the largest power of 2 less than or equal to 1808. The answer is 2¹⁰. So, write 1 next to this power. Repeat the process until you reach a remainder of 0, filling in 0s for unused powers. Final conversion will be 10011100010000.

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, 10000 is divided by 2 to get 5000 as the quotient and 0 as the remainder. Now, 5000 is divided by 2. Here, we will get 2500 as the quotient and 0 as the remainder. Continue dividing each quotient by 2, noting down remainders, until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 10000, 10011100010000.

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., 2¹³, 2¹², ..., 2⁰. Find the largest power that fits into 10000. 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 10000, we use 0s for unused powers and 1s for powers that fit into the number.

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 10000.

Memorize to speed up conversions: We can memorize the binary forms for smaller numbers and patterns for larger ones.

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

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 10000 is even, and its binary form is 10011100010000. Here, the binary of 10000 ends in 0. 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 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.

• Power of 2: A mathematical expression in which 2 is raised to an exponent, used to determine place values in binary conversion.