Computers process data using the binary number system, which relies only on two digits: 0 and 1. Humans, however, are used to the denary (decimal) system, which uses digits from 0 to 9. To communicate effectively between these two systems, we must understand how to convert binary numbers into their denary equivalents.

Understanding the Concept

The denary system (base 10) is built around powers of 10. For example:

  • The first place represents 10⁰ = 1
  • The second place represents 10¹ = 10
  • The third place represents 10² = 100, and so on.

The binary system (base 2), on the other hand, uses powers of 2 instead of 10. Each position in a binary number represents a power of 2:

  • First position → 2⁰ = 1
  • Second position → 2¹ = 2
  • Third position → 2² = 4
  • Then 8, 16, 32, 64, 128, and so forth.

Each 1 in a binary number means that the power of 2 in that position is included in the total, while each 0 means it is excluded. This is how binary values are translated into denary numbers.

Worked Example: Converting Binary to Denary

Let’s convert the binary number 11101110 into denary form.

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

To find the denary equivalent, add the values of all the positions marked with 1s:

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

Therefore, the binary number 11101110 equals 238 in denary.

The Binary Switch Analogy

Each binary digit (bit) acts like a small switch that can be either ON (1) or OFF (0). When the switch is ON, its power of 2 contributes to the total. When OFF, it does not. Thus, converting binary to denary simply means adding all the “ON” values together.

Why Binary Works So Well

Binary is efficient for computers because it directly corresponds to the electrical states of their circuits — ON or OFF. These two states are easy to implement, reliable to detect, and perfect for digital processing.

Practice Exercises

Try converting the following binary numbers to denary values:

  1. 10101010
  2. 01110011
  3. What is the 5th power of 2 in binary notation?
  4. What is the denary value of 11111111?
  5. True or False: The 5th column from the right represents 2⁴ in binary.

Review Questions — Fill in the Gaps

  1. The binary system is based on powers of ____.
  2. The denary system is based on powers of ____.
  3. In binary, each position value doubles as you move ____.
  4. Each binary digit is referred to as a ____.
  5. A binary digit of 1 means the corresponding power of 2 is ____.
  6. A binary digit of 0 means the corresponding power of 2 is ____.
  7. The binary number 11101110 equals ____ in denary.
  8. Each bit in binary acts like an electrical ____ that can be on or off.
  9. Binary is efficient because it aligns with the ____ nature of computer circuits.
  10. To convert binary to denary, add all the values corresponding to ____ digits.
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