The binary number system is the foundation of every digital device, from smartphones and laptops to satellites and supercomputers. Most people should be familiar with the decimal (base-10) number system in which the digits are numbered from 0 to 9. The binary system on the other hand is a base-2 system. Computers rely on this system for storage, processing, and transmitting of data.
So why do computers only use the binary system and not any other?
Well, binary uses only two digits (0 and 1) , which actually aligns perfectly with the on/off nature of electronic circuits. Getting to understanding how binary works is not only essential for computer scientists and engineers, but also for anyone looking to build a strong reliable foundation in anything computing.
In this article, we will learn what the binary number system is, how to count using it, how to covert between binary and decimal numbers, and also learn how to perform all the basic arithmetic operations using the system.
Let’s begin!
What is the binary number system?
The binary number system is a base-2 number system that uses only two digits: 0 and 1. Unlike the decimal system, which is based on powers of 10, the binary system is based on powers of 2. Each digit in a binary number represents an increasing power of 2, starting from the rightmost digit.
Binary vs decimal
| Decimal (Base 10) | Binary (Base 2) |
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
How to count in the binary number system
Counting in the binary number system might seem daunting at first since we all grew up counting using the decimal system, but once you understand the pattern, it becomes really easy and useful especially if you are going on to learning other numbering systems. You’ll learn that the pattern is actually used by all numbering systems.
The counting basics
Just like base-10 counting, binary starts at 0 and increments by 1. The point at which the two differ is the limit of the highest number. Binary only allows two digits (0 and 1) whilst the decimal system uses (0 to 9).
Here is a table showing binary counting from 0 to 15:
| Decimal | Binary |
| 0 | 0000 |
| 1 | 0001 |
| 2 | 0010 |
| 3 | 0011 |
| 4 | 0100 |
| 5 | 0101 |
| 6 | 0110 |
| 7 | 0111 |
| 8 | 1000 |
| 9 | 1001 |
| 10 | 1010 |
| 11 | 1011 |
| 12 | 1100 |
| 13 | 1101 |
| 14 | 1110 |
| 15 | 1111 |
Binary conversion
Binary to decimal conversion
Converting from binary to decimal is at the foundation of understanding how numbers are represented inside computers. Once you know the method, it will become a very simple process.
The positional method
Each digit (bit) in a binary number represents a power of 2, this starts from the rightmost digit which is also known as the least significant bit (LSB).
For us to convert binary to decimal, we multiply each bit by \(2n2^n2n\), where n is the position index starting from 0 on the right.
Decimal to binary conversion
Just like how we converted binary numbers to decimal, we can also do it the other way around. This becomes especially useful when it comes to understanding how computers internally store numeric data.
The two most common methods of converting decimal number system to binary number system is via the division by 2 method or the subtraction method using powers of 2.
Division by 2 method
In this method, you repeatedly divide the decimal number by 2 and recording the remainders. The binary result is formed by reading the remainders from the bottom to the top.
Let’s solve an example
Convert 23 to binary:
- 23 ÷ 2 = 11, remainder → 1
- 11 ÷ 2 = 5, remainder → 1
- 5 ÷ 2 = 2, remainder → 1
- 2 ÷ 2 = 1, remainder → 0
- 1 ÷ 2 = 0, remainder → 1
The remainders would be read from bottom to top so, 10111
Subtracting powers of 2
This method starts with listing the powers of 2 from largest to smallest, stopping before you exceed the number. We then subtract the largest power of 2, then mark 1 in that position. Mark a 0 wherever the power of 2 cannot fit.
Let’s solve an example
Convert 19 to binary:
- Closest powers of 2:
\(16 (2^4), 8 (2^3), 4 (2^2), 2 (2^1), 1 (2^0)\) - 19 − 16 = 3 → write 1
- 8 is too big → write 0
- 4 is too big → write 0
- 2 fits → 3 − 2 = 1 → write 1
- 1 fits → 1 − 1 = 0 → write 1
Reading from top to bottom would give you, 10011.
Basic binary number system arithmetic operations
Now that we’ve understood how to convert binary into decimal, Let’s learn how to do some basic arithmetic operations. The rules are the exact same except much simpler because you are working with only two digits (0 and 1).
Let’s get into it starting off with addition.
Binary addition
Binary addition follows four simple rules:
| A | B | A + B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 10 (0 with carry 1) |
Example: Add 1010 + 1101
1010
+ 1101
——-
10111
Explanation:
- 0 + 1 = 1
- 1 + 0 = 1
- 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1
- Final carry = 1 → add it to the front
Result: 10111 (which is 23 in decimal)
Binary subtraction
Binary subtraction also has simple rules:
| A | B | A − B |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| 0 | 1 | 1 (borrow 1) |
Example: Subtract 1101 – 1010
1101
− 1010
——–
0011
Explanation:
- 1 − 0 = 1
- 0 − 1 → borrow → becomes 10 − 1 = 1
- 1 − 0 = 1
- 1 − 1 = 0
Result: 0011 (which is 3 in decimal)
Binary multiplication
Binary multiplication is similar to decimal multiplication but much simpler:
| A | B | A × B |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Example: Multiply 101 × 11
101 (5)
× 11 (3)
——-
101 (101 × 1)
+ 1010 (101 × 1, shifted left)
———
1111 (15 in decimal)
Result: 1111
Binary division
Binary division is like long division in decimal. You find how many times the divisor fits into the dividend and subtract repeatedly.
Divide 1101 ÷ 11
- 1101 (13 in decimal) ÷ 11 (3 in decimal)
- 3 fits into 13 four times with a remainder of 1
Result:
- Quotient: 100 (4 in decimal)
- Remainder: 1
Now that you’ve learned how to count, convert, and compute in binary, let’s learn about how binary is used practically.
The applications of binary
Understanding binary doesn’t just stop at the theoretical aspects, it bridges to real world applications. From storing data to executing instructions, binary is the language of machines
Data storage
Whether it’s text, image, audio, or videos, all data in a computer is stored as binary digits (bits). A bit is a single binary digit either 0 or 1. A byte is 8 bits which is often used to store a single character.
As a matter of fact, files, memory, and disk capacities are all measured in bytes. Some of the most common data representations are as follows:
- 1 KB = 1,024 bytes
- 1 MB = 1,024 KB
- 1 GB = 1,024 MB
Computer processing
The Central Processing Unit (CPU) executes machine instructions in binary. For example, every command like “add,” “load,” or “jump” has a binary equivalent or representation.
These instructions will then be processed by the logic gates, which use binary input and output to perform operations.
Digital images, colours, and pixels
All images are made up of tiny units you might have heard of called pixels, each pixel’s colour is defined using a binary value.
A Red Greed Blue (RGB) colour might be stored as the following:
- Red: 11111111 (255)
- Green: 00000000 (0)
- Blue: 00000000 (0)
This will result in pure red.
Networking and IP addresses
Every device that has the capability to connect to the internet has an IP address, which is just another binary number under the layers.
Let’s look at an example:
IPv4: 192.168.0.1 = 11000000.10101000.00000000.00000001 in binary
As you can see all IP addresses are basically a set of binary numbers made more readable in decimal. Routers and switches use binary to route data efficiently.
In conclusion, binary is not just something to be studied theoretically, it powers everything from the smartphones we use to the internet we access. Once you understand how it works, you’re one step closer to mastering computer science, programming, or electronics.
