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Last updated on July 4th, 2025

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Binary Number System

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Binary number system is a method used to define or represent numbers using only 0s and 1s. Since binary is the fundamental language understood by the computers, we need this number system to make computers work. We will now see more about the binary number system in this article.

Binary Number System for Canadian Students
Professor Greenline from BrightChamps

What is a Binary Number System?

The binary number system is a base 2 numeral system that uses only two digits: 0 and 1.

 

It is the foundation of digital computing and electronic systems, as computers process and store data in binary form.

 

Each binary digit (bit) represents a power of 2, with the rightmost bit being the least significant.

 

Binary is used in arithmetic operations, data encoding, logic circuits, and memory storage.

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Binary Number Table

Binary numbers are always represented using a 4-bit format. The table given below gives the binary equivalent of base 10 numbers: 

Decimal Number

Binary Number

Decimal Number

Binary Number

1

0001

11

1011

2

0010

12

1100

3

0011

13

1101

4

0100

14

1110

5

0101

15

1111

6

0110

16

10000

7

0111

17

10001

8

1000

18

10010

9

1001

19

10011

10

1010

20

10100

 

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How to Convert from Binary to Decimal and Vice Versa?


Binary to Decimal Conversion:

 

To convert a binary to a decimal number, multiply each binary digit by 1, 2, 4, 8, and so on from right to left, and then add the answers. You can also use a formula to get the conversion right. The formula used is:


D = (an-1 x 2n-1) +...+ (a3 x 23) + (a2 x 22) + (a1 x 21) + (a0 x 20)

For example, convert (10011)2 to a decimal number:

(10011)2 = (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)

= 16 + 0 + 0 + 2 + 1 = (19)10

Hence, the binary number (10011)2 is expressed as (19)10.

 

Decimal to Binary Conversion:

 

To convert a decimal number into binary, keep dividing the number by 2 until the quotient becomes 0. We need to ensure we note down all the remainders that we get during this process. When the quotient becomes 0, just write down the remainders (either 1 or 0) upside down to get the final answer.

For example, convert (28)10 into a binary number:

Using the above division method, we find that (28)10 = (11100)2

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Arithmetic Operations on Binary Numbers

Basic mathematical operations like addition, subtraction, and multiplication are also applicable to the binary number system. 

Let us see the operations we use for the binary number system:



Binary Addition:


In binary addition, we add each digit one by one, and carry over when needed. The table below shows the addition of two given numbers.

Binary Number 1

Binary Number 2

Addition

Carry

0 0 0 0
1 0 1 0
0 1 1 0
1 1 0 1



Binary Subtraction:


The binary numbers are subtracted digit by digit and the answer is obtained. The table below shows the subtraction of two given numbers.

 

Binary Number 1

Binary Number 2

Subtraction

Carry

0 0 0 0
1 0 1 0
0 1 1 1
1 1 0 0



Binary Multiplication:


The multiplication of two binary numbers is shown below:

 

Binary Number 1

Binary Number 2

Multiplication

0 0 0
1 0 0
0 1 0
1 1 1


 

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1’s and 2’s Complement of Binary Numbers

1’s complement and 2’s complement are simple methods to represent negative numbers in binary and help with binary subtraction. These methods are used widely in computers and digital devices. Let us now see the steps involved in finding the 1’s complement and 2’s complement .
 

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1’s complement:

Let’s look at the steps to find 1’s complement:


Step 1: We must write down the binary number for which we need the 1’s complement.


Step 2: Change all 0s to 1 and 1s to 0.

 

This is how we find 1’s complement of a given binary. Let’s understand this better with an example.
Let’s find the 1’s complement of 101010
Now, all we have to do is flip the bits. So, the 1’s complement of 101010 is 010101. As you can see, we have only interchanged the 1’s and 0’s in the binary.  

 

Now, there is some vital information about 1’s complement that we need to know:
 

  • In older computers, 1’s complement played and still plays a significant role representing negative numbers.

     
  • In 1’s complement, zero can be represented in two different ways: 0000 (positive zero) and 1111 (negative zero). This causes confusion and creates a problem. 

     
  • Due to this drawback, many modern systems use 2’s complement instead of 1’s complement.
     
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2’s complement:

These are the steps to find the 2’s complement of a binary: 


Step 1: First, find the 1’s complement. 


Step 2: Just add 1 to the 1’s complement. 


Here, let’s find the 2’s complement of 101010 

To do that, let’s first find its complement.

            So, 1’s complement of 101010 is 010101

Now adding 1 to the 1’s complement, we get, 010101 + 1 = 010110
 

Remember these points about 2’s complement:
 

  • It is used in modern computers.

     
  • It has only one representation of zero 

     
  • The sign is determined by the most significant bit (MSB). If the MSB is 0, it means positive, if it’s 1, then the number is negative.
     
Professor Greenline from BrightChamps

Real-life Applications of Binary Number System

There are numerous real life applications of the binary number system. Some of them are discussed below:

 

  • Computer Systems and Digital Electronics:

    Without the binary number system, we wouldn’t even have computers. From processors to the logic circuits, almost everything in a computer needs the binary number system to function. 

     
  • Data Storage and Transmission:

    Electronic devices like USB flash drives, hard drives, CDs, SSDs, and DVDs use binary to store and transmit data. Every bit of data is stored as a binary. The number system is also used by digital communication services to transmit data via wired or wireless networks.  

     
  • Digital Communication Systems:

    The modern communication system, which includes mobile phones and satellites, uses binary to transmit data. For e.g., video streams and audio signals are converted into binary before transmitted as digital signals. 
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Common Mistakes and How to Avoid Them in Binary Number System

Students can make mistakes and it’s part of the learning process. However, some mistakes can be easily avoided if we take the right precautions. Here are some common mistakes that can be avoided: 
 

Mistake 1

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Not Correctly Converting Binary to Decimal

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Starting from the right, multiply each binary digit’s place value by the correct power of 2. Then, add all the answers to get the converted decimal number. Always double-check the steps.

Mistake 2

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Borrowing Mistakes in Binary Subtraction
 

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Borrowing must always happen from the nearest 1 so that we can convert it to 10 in binary before subtracting. Double-check the steps before writing the final answer. 
 

Mistake 3

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Misinterpreting Two’s Complement Representation
 

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Students should always find 1’s complement of the given binary and add 1 to find 2’s complement. Practicing this will improve our representation of 2’s complement.

Mistake 4

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Incorrect Bitwise Shift Operations
 

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Remember that a left shift (<<) moves all bits to the left and fills the right side with 0s. A right shift (>>) moves bits to the right and drops the least significant bits on the right.
 

Mistake 5

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Incorrect Error Checking in Data Transmission 

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 Students should carefully calculate parity bits to ensure accuracy. They can use techniques like hamming codes or CRC for error detection and correction.
 

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Solved Examples on Binary Number System

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Problem 1

Convert the binary number 1011 to its decimal equivalent.

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11.

Explanation

 Write down the binary digits with their positional powers of 2 (rightmost digit is 2⁰):


1 × 2³ = 1 × 8 = 8

0 × 2² = 0 × 4 = 0

1 × 2¹ = 1 × 2 = 2

1 × 2⁰ = 1 × 1 = 1

Sum the results: 8 + 0 + 2 + 1 = 11
 

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Problem 2

Convert the decimal number 13 into binary.

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1101.
 

Explanation

Divide 13 by 2:

13 ÷ 2 = 6 remainder 1

Divide 6 by 2:

6 ÷ 2 = 3 remainder 0

Divide 3 by 2:

3 ÷ 2 = 1 remainder 1

Divide 1 by 2:

1 ÷ 2 = 0 remainder 1

Write the remainders in reverse order: 1101
 

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Problem 3

Add the binary numbers 1010 and 1101.

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10111.
 

Explanation

Rightmost column: 0 + 1 = 1

Next column: 1 + 0 = 1

Next column: 0 + 1 = 1

Leftmost column: 1 + 1 = 10 (write 0 and carry 1)

Since there’s a carry, place it at the next left position.

Final result: 10111

When adding bit by bit, remember to carry over any value exceeding 1.
 

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Problem 4

Subtract 1100 from 10101.

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1001 
 

Explanation

Start from right:

Column 1: 1 – 0 = 1

Column 2: 0 – 0 = 0

Column 3: 1 – 1 = 0

Column 4: 0 – 1 → borrow from left:

Borrowing: (10 in binary becomes 2 in decimal), so 10 – 1 = 1

Column 5: After borrow, 0 (which became 0) minus 0 = 0; but note the leftmost digit from 10101 was 1, so adjusting gives:

Alternatively, perform full subtraction:

10101

-01100

= 01001
 

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Problem 5

Multiply the binary numbers 101 and 11.

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1111.

Explanation

Write the numbers:

Multiply by the rightmost digit (1):

101 × 1 = 101

Multiply by the next digit (1) and shift one position to the left:

101 × 1 = 101 becomes 1010

Add the partial products:
 

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FAQs on Binary Number System

1.What is the binary number system?

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2.Why is binary used in computers?

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3.How does binary conversion work?

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4.What are bits and bytes?

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5.How is binary used to represent data other than numbers?

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6.How can children in Canada use numbers in everyday life to understand Binary Number System ?

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7.What are some fun ways kids in Canada can practice Binary Number System with numbers?

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8.What role do numbers and Binary Number System play in helping children in Canada develop problem-solving skills?

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9.How can families in Canada create number-rich environments to improve Binary Number System skills?

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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