10 in Binary

The process of converting 10 from decimal to binary involves dividing the number 10 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 10 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 10 by 2 until getting 0 as the quotient is 1010. 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 10. 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 1010 in binary is indeed 10 in the decimal number system.

10 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 10 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

Since 16 is greater than 10, we stop at 2³ = 8. 

Step 2: Identify the largest power of 2: In the previous step, we stopped at 2³ = 8. This is because in this step, we have to identify the largest power of 2, which is lesser or equal to the given number, 10. 

Since 2³ is the number we are looking for, write 1 in the 2³ place. Now the value of 2³, which is 8, is subtracted from 10. 

10 - 8 = 2. 

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, 2. So, the next largest power of 2 is 2¹, which is less or equal to 2 (in this case equal). 

Now, we have to write 1 in the 2¹ places. And then subtract 2 from 2.

2 - 2 = 0. We need to stop the process here since the remainder is 0.

Step 4: Identify the unused place values: In step 2 and step 3, we wrote 1 in the 2³ and 2¹ places. Now, we can just write 0s in the remaining places, which are 20 and 22.

Now, by substituting the values, we get,

0 in the 2⁰ place
1 in the 2¹ place
0 in the 2² place
1 in the 2³ place

Step 5: Write the values in reverse order: We now write the numbers upside down to represent 10 in binary.

Therefore, 1010 is 10 in binary.

Grouping Method:

In this method, we divide the number 10 by 2. Let us see the step-by-step conversion. 

Step 1: Divide the given number 10 by 2.

10 / 2 = 5. Here, 5 is the quotient and 0 is the remainder.

Step 2: Divide the previous quotient (5) by 2.

5 / 2 = 2. Here, the quotient is 2 and the remainder is 1.

Step 3: Repeat the previous step.

2 / 2 = 1. Now, the quotient is 1 and 0 is the remainder.

Step 4: Repeat the previous step.
1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 5: Write down the remainders from bottom side up.

Therefore, 10 (decimal) = 1010 (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 10.

• Since the answer is 2³, write 1 next to this power of 2.

• Subtract the value (8) from 10. So, 10 - 8 = 2.

• Find the largest power of 2 less than or equal to 2.

• The answer is 2¹. So, write 1 next to this power.

• Now, 2 - 2 = 0. Since there is no remainder, we can write 0 next to the remaining powers (2⁰ and 2²).

• Final conversion will be 1010.

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, 10 is divided by 2 to get 5 as the quotient and 0 as the remainder.

• Now, 5 is divided by 2. Here, we will get 2 as the quotient and 1 as the remainder.

• Dividing 2 by 2, we get 0 as the remainder and 1 as the quotient.

• Divide 1 by 2 to get 1 as the remainder and 0 as the quotient. 

• We stop the division once the quotient becomes 0. 

• Now, we write the remainders upside down to get the binary equivalent of 10, 1010.

Rule 3: Representation Method

This rule also involves the breaking of the number into powers of 2.

• Identify the powers of 2 and write it down in the decreasing order i.e., 2³, 2², 2¹, and 2⁰.

• Find the largest power that fits into 10.

• 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 10, we use 0s for 2² and 2⁰ and 1s for 2³ and 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 till 10.

• Memorize to speed up conversions: We can memorize the binary forms for numbers 1 to 10.
  1 → 1, 2 → 10, 3 → 11, 4 → 100, 5 → 101, 6 → 110, 7 → 111, 8 → 1000, 9 → 1001, 10 → 1010.

• 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 e.g., 10 is even and its binary form is 1010. Here, the binary of 10 ends in 0. If the number is odd, then its binary equivalent will end in 1. For e.g., the binary of 17 (odd number) is 10001. As you can see, the last digit here is 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 as base 2 number system.

• Place value: Every digit has a value based on its position in a given number. For e.g., 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.

• Hexadecimal: This number system has 16 digits. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here, A to F represent the values 10 to 15.