Decimal number system is a universal way of representing integer and fractional values, which has become an international standard in mathematics and everyday life. This positional system ensures simplicity of calculations and intuitive understanding for users.
What is the decimal system?
The decimal system uses base 10 and includes digits from 0 to 9. Each position of a number has a specific weight:
- Units
- Tens
- Hundreds
- Thousands and so on
Let's consider the example of the number 1245:
Mathematical expression of the value: (1 × 1000) + (2 × 100) + (4 × 10) + (5 × 1) = 1245
Practical Mastery of Number System Conversions
Converting to the decimal system from other number systems is one of the most sought-after skills in computer science.
Let's look at a specific example: converting the number 128 from the octal system to the decimal system:
The octal number 128₈ is converted to decimal as follows:
1×8² + 2×8¹ + 8×8⁰ = 64 + 16 + 8 = 88₁₀
Another interesting case is converting the number 304 to the decimal system. If this number is in the quinary system (304₅):
3×5² + 0×5¹ + 4×5⁰ = 75 + 0 + 4 = 79₁₀
For fractional values, such as converting the number 304.5 (assuming 304.5₆ in the senary system) to the decimal system:
3×6² + 0×6¹ + 4×6⁰ + 5×6⁻¹ = 108 + 0 + 4 + 0.833 = 112.833₁₀
Key Features of the System
The base of the system is 10, which determines its name. Ten symbols are used to represent numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Conversion to Other Number Systems
How to Convert from Decimal to Binary
The algorithm for converting integers:
- Divide the number by 2 sequentially
- Record the remainders of division
- Repeat until a quotient of zero is obtained
- Write down the remainders in reverse order
Example: Convert 87 from decimal to binary
- 87 ÷ 2 = 43 (remainder 1)
- 43 ÷ 2 = 21 (remainder 1)
- 21 ÷ 2 = 10 (remainder 1)
- 10 ÷ 2 = 5 (remainder 0)
- 5 ÷ 2 = 2 (remainder 1)
- 2 ÷ 2 = 1 (remainder 0)
- 1 ÷ 2 = 0 (remainder 1)
- Result: 1010111₂
How to Convert from Binary to Decimal
The reverse conversion is done by summing the products of digits by powers of two:
Binary number 1101₂:
1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13₁₀
Working with Hexadecimal Numbers
Converting the number 4C16 to the decimal system requires consideration of alphanumeric notations:
4×16³ + C×16² + 1×16¹ + 6×16⁰ = 4×4096 + 12×256 + 16 + 6 = 16384 + 3072 + 16 + 6 = 19478₁₀
Conversion Table to Decimal Number System
| Original Number | Number System | Decimal Equivalent |
|---|---|---|
| 1010111₂ | Binary | 87 |
| 128₈ | Octal | 88 |
| 304₅ | Quinary | 79 |
| 4C16₁₆ | Hexadecimal | 19478 |
| 304.5₆ | Senary | 112.833 |
Conclusion
Understanding the principles of the decimal system and methods of converting numbers to the decimal number system opens the way to mastering other number systems and delving deeper into mathematics, computer science, and modern technologies. Regular practice of number system conversions develops logical thinking and helps to better understand the architecture of computing systems.