The binary system is the foundation of all computer operations. Every image, sound, video, and command in a digital device is represented using combinations of 0s and 1s. This system, known as binary (base-2), uses only two symbols instead of ten, as in our everyday denary (base-10) system.

What is the Binary Number System?

Humans typically count using ten digits — 0 through 9 — because we have ten fingers. This is known as the denary or base-10 system. Computers, however, rely on millions of tiny electronic switches called transistors, each of which can only exist in one of two states:

  • ON, represented by 1
  • OFF, represented by 0

Because these two states correspond perfectly to electrical signals (current flowing or not), computers use the binary system (base-2) to represent all data and instructions.

How Binary Counting Works

In binary, each digit (called a bit) represents a power of 2, starting from the rightmost bit. The value of each position doubles as you move left.

Example of counting:

Denary: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9...
Binary: 0, 1, 10, 11, 100, 101, 110, 111, 1000...

Each position in binary corresponds to a specific power of 2:

Binary Digit 1 1 1 0 1 1 1 0
Power of 2 128 64 32 16 8 4 2 1

Now, add up the values under the digits that contain 1s:

128 + 64 + 32 + 8 + 4 + 2 = 238

Therefore, the binary number 11101110 equals 238 in denary.

Converting from Denary to Binary

To communicate with computers, we often need to convert denary numbers into binary. There are two main methods for doing this:

Method 1: Subtraction of Powers of 2

  1. Identify the largest power of 2 less than or equal to your number.
  2. Subtract it from the number.
  3. Continue subtracting the next largest powers of 2 until the result is zero.
  4. Mark each used power with 1 and unused with 0.

Example: Convert 107 to binary

Power of 2 128 64 32 16 8 4 2 1
Used? 0 1 1 0 1 0 1 1

Result → 01101011 Therefore, 107 in binary is 01101011.

Method 2: Successive Division by 2

  1. Divide the number by 2 repeatedly.
  2. Record the remainder (0 or 1) after each division.
  3. Continue until the quotient becomes 0.
  4. Read the remainders from bottom to top to get the binary number.

Example: Convert 107 to binary

107 ÷ 2 = 53 R1  
53 ÷ 2 = 26 R1  
26 ÷ 2 = 13 R0  
13 ÷ 2 = 6  R1  
6 ÷ 2  = 3  R0  
3 ÷ 2  = 1  R1  
1 ÷ 2  = 0  R1

Reading from bottom to top: 1101011 Therefore, 107 = 1101011 in binary.

Why Computers Use Binary

Binary is the most reliable and efficient system for computers because it aligns with the physical nature of electronic circuits. Transistors are easier to design and control when they only have two possible states (on or off). This simplicity ensures accuracy, stability, and speed in digital processing.

Review Questions — Fill in the Gaps

  1. The binary system is a ____ number system.
  2. Computers represent data using only two digits: ____ and ____.
  3. Each binary digit is called a ____.
  4. A transistor can exist in two states: ____ and ____.
  5. The binary system aligns with the ____ behavior of electronic switches.
  6. In binary, each position represents a ____ of 2.
  7. The binary number 11101110 equals ____ in denary.
  8. In the subtraction method, each used power of 2 is marked with a ____.
  9. In the division-by-2 method, the remainders are read from ____ to ____.
  10. Binary is used in computers because it is ____ and ____ to process electronically.
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