Binary Systems and Hexadecimal: Converting Denary to Binary
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:
- Subtraction of Powers of 2 (Trial and Error Method)
- 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:
- Identify the largest power of 2 that is less than or equal to 107 → 64 107 − 64 = 43
- The next power of 2 that fits into 43 is 32 43 − 32 = 11
- 16 does not fit → mark 0
- 8 fits → 11 − 8 = 3
- 4 does not fit → mark 0
- 2 fits → 3 − 2 = 1
- 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
- The binary number system is based on powers of ____.
- In the trial-and-error method, we subtract powers of ____ from the denary number.
- Each power of 2 corresponds to a binary ____.
- In the division method, the remainders are read from ____ to ____.
- When converting 107 to binary, the result is ____.
- The largest power of 2 less than 107 is ____.
- In the subtraction method, each used power of 2 is marked with a ____.
- Binary representation is built from combinations of ____ and ____.
- The division method is especially suitable for ____ numbers.
- Both conversion methods rely on the principle of representing values using ____ of 2.
