Binary is the fundamental language of computers. Every operation a computer performs—whether it’s arithmetic, graphics, or communication—is executed using binary digits (bits): 0 and 1. To understand how data is represented and processed, it’s essential to learn how to convert a denary (base-10) number into its binary (base-2) equivalent.

There are two major methods used for this conversion:

  1. Subtraction of Powers of 2 (Trial and Error Method)
  2. Successive Division by 2 Method

Method 1: Subtraction of Powers of 2 (Trial and Error Method)

In this method, we represent a denary number as a sum of selected powers of 2. The available powers of 2 are: 1, 2, 4, 8, 16, 32, 64, 128, 256, …

To convert 107 into binary:

  1. Identify the largest power of 2 that is less than or equal to 107 → 64 107 − 64 = 43
  2. The next power of 2 that fits into 43 is 32 43 − 32 = 11
  3. 16 does not fit → mark 0
  4. 8 fits → 11 − 8 = 3
  5. 4 does not fit → mark 0
  6. 2 fits → 3 − 2 = 1
  7. 1 fits → 1 − 1 = 0

Now assign 1 for used powers and 0 for unused powers:

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

Hence, 107 = 1101011 in binary.

This method helps visualize how binary numbers are built directly from powers of 2.

Method 2: Successive Division by 2

This approach involves dividing the number repeatedly by 2 and recording the remainder each time. The remainders form the binary digits when read from bottom to top.

Example: Convert 107 to binary

107 ÷ 2 = 53 remainder 1  
53 ÷ 2 = 26 remainder 1  
26 ÷ 2 = 13 remainder 0  
13 ÷ 2 = 6  remainder 1  
6 ÷ 2  = 3  remainder 0  
3 ÷ 2  = 1  remainder 1  
1 ÷ 2  = 0  remainder 1

Now, read the remainders upward: 1 1 0 1 0 1 1

Therefore, 107 in binary = 1101011.

Comparison of Both Methods

Both methods lead to the same result because they rely on the same principle — representing numbers using powers of 2. The Subtraction of Powers of 2 method is more intuitive for smaller numbers, while the Successive Division by 2 method is more systematic and easier for larger values.

Summary

  • Binary uses 0s and 1s to represent data.
  • Denary numbers can be converted into binary using two main techniques.
  • The Trial and Error method uses powers of 2 subtraction.
  • The Successive Division method uses repeated division and remainder tracking.
  • Both methods produce the same binary result.
  • Example: 107 = 1101011 in binary.

Review Questions — Fill in the Gaps

  1. The binary number system is based on powers of ____.
  2. In the trial-and-error method, we subtract powers of ____ from the denary number.
  3. Each power of 2 corresponds to a binary ____.
  4. In the division method, the remainders are read from ____ to ____.
  5. When converting 107 to binary, the result is ____.
  6. The largest power of 2 less than 107 is ____.
  7. In the subtraction method, each used power of 2 is marked with a ____.
  8. Binary representation is built from combinations of ____ and ____.
  9. The division method is especially suitable for ____ numbers.
  10. Both conversion methods rely on the principle of representing values using ____ of 2.
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