Each number is represented by its base in the number system concept. If the base is 2, it is a binary number; if the base is 8, it is an octal number; if the base is 10, then it is called the decimal number system; and if the base is 16, it is part of the hexadecimal number system.
The conversion of decimal numbers to any other number system is an easy method.
Converting other base number systems into decimal numbers requires practice. In this article, let us learn more about the decimal number system and the conversion from the decimal number system to other systems in detail. Base-10 is called the decimal system because a digit’s value in a number is determined by where it lies about the decimal point.
What is the Decimal Number System or Base-10 Number System?
The Decimal Number System is the standard system used for counting and calculations in everyday life. Also known as the base-10 system, it uses 10 digits, 0 through 9, to represent all numbers. Each digit in a number has a place value based on powers of 10, increasing from right to left. For example, in the number 472, the digit 4 represents 400 (4 × 100), the 7 represents 70 (7 × 10), and the 2 represents 2 (2 × 1).
This system is widely used in mathematics, commerce, science, and digital applications due to its simplicity and historical roots in human counting (10 fingers). Its structure makes it easy to perform arithmetic operations like addition, subtraction, multiplication, and division.
Examples of the Decimal Number System
The decimal number system is based on place values, where each digit’s position determines its value, increasing in powers of 10 from right to left. Here are some examples that show how it works:
1. Basic Example
Let’s take the number 547:
5 is in the hundreds place → 5 × 100 = 500
4 is in the tens place → 4 × 10 = 40
7 is in the ones place → 7 × 1 = 7
So, 547 = 500 + 40 + 7
2. Larger Number Example
Number: 3,208
3 × 1000 = 3000
2 × 100 = 200
0 × 10 = 0
8 × 1 = 8
3,208 = 3000 + 200 + 0 + 8
3. Decimal (Fractional) Number Example
Decimal numbers can also represent fractions in base-10. Take 45.67:
4 × 10 = 40
5 × 1 = 5
6 × 0.1 = 0.6
7 × 0.01 = 0.07
45.67 = 40 + 5 + 0.6 + 0.07
The Powers of 10
The decimal number system is built on the powers of 10. Each position (or digit) in a number has a value that is 10 times the value of the position to its right. These are called place values.
| Position | Power of 10 | Place Value |
| Thousands | 10³ | 1,000 |
| Hundreds | 10² | 100 |
| Tens | 10¹ | 10 |
| Units (Ones) | 10⁰ | 1 |
| Tenths | 10⁻¹ | 0.1 |
| Hundredths | 10⁻² | 0.01 |
Example: In the number 2,345.67:
2 × 1000 (10³) = 2000
3 × 100 (10²) = 300
4 × 10 (10¹) = 40
5 × 1 (10⁰) = 5
6 × 0.1 (10⁻¹) = 0.6
7 × 0.01 (10⁻²) = 0.07
→ Total = 2,345.67
Using Decimals in Daily Life
The decimal system is the default way humans count and calculate. It’s so common that we rarely notice we’re using it. Here’s where it shows up:
Money: Dollars and cents follow base-10 (e.g., $10 = 10 × $1).
Time (in decimal-based systems): Decimal hours, like 2.5 hours = 2 hours and 30 minutes.
Measurements: Meters, grams, and liters in the metric system use base-10 conversions (1 km = 1000 m).
Education: Math in schools is taught using base-10 blocks and charts.
Its simplicity makes it perfect for mental math, written calculations, and real-life problem solving.
Origin of decimal
The base-10 system originated from ancient civilizations, particularly the Hindu-Arabic numeral system developed in India around the 6th century. The use of zero (0) as both a digit and a proxy was revolutionary. The system was later transmitted to the Islamic world and then to Europe, becoming the global standard. This system introduced two key ideas:
Digits 0–9: A set of reusable symbols
Place Value: The value of a digit depends on its position
Rules of the Decimal Number System
To use the decimal number system effectively, several fundamental rules apply:
Ten Digits Only: Only the digits 0 through 9 are used.
Place Value Rule: Each digit’s value depends on its position, based on powers of 10.
Zero as a Placeholder: 0 shows the absence of a value in a place (e.g., 204 vs. 24).
Decimal Point Rule: Used to separate the whole number part from the fractional part (e.g., 12.5).
Value from Left to Right:
1. Digits to the left of the decimal point increase in powers of 10 (units, tens, hundreds…).
2. Digits to the right of the decimal point decrease in powers of 10 (tenths, hundredths…).
1. Conversion from Other Bases to Decimal Number System
To convert a number from any base (e.g., binary, octal, or hexadecimal) to decimal (base-10), apply the positional value method:
Formula:
Decimal=dn×bn+dn−1×bn−1+⋯+d0×b0\text{Decimal} = d_n \times b^n + d_{n-1} \times b^{n-1} + \dots + d_0 \times b^0Decimal=dn×bn+dn−1×bn−1+⋯+d0×b0
Where:
- ddd = digit at position
- bbb = base of the number
- nnn = position index (starting from 0 on the right)
2. Binary to Decimal Conversion (Base-2 → Base-10)
Binary numbers only use digits 0 and 1. Convert by multiplying each digit by 2 raised to its position, starting from the right.
Example:
Convert 10110210110_2101102 to decimal:
1×24+0×23+1×22+1×21+0×20=16+0+4+2+0=22101×2^4 + 0×2^3 + 1×2^2 + 1×2^1 + 0×2^0 = 16 + 0 + 4 + 2 + 0 = \mathbf{22}_{10}1×24+0×23+1×22+1×21+0×20=16+0+4+2+0=2210
3. Octal to Decimal Conversion (Base-8 → Base-10)
Octal numbers use digits from 0 to 7. Multiply each digit by 8 raised to the power of its position.
Example:
Convert 7258725_87258 to decimal:
7×82+2×81+5×80=448+16+5=469107×8^2 + 2×8^1 + 5×8^0 = 448 + 16 + 5 = \mathbf{469}_{10}7×82+2×81+5×80=448+16+5=46910
4. Hexadecimal to Decimal Conversion (Base-16 → Base-10)
Hexadecimal uses digits 0–9 and letters A–F, where A=10 through F=15. Multiply each by 16 raised to its position.
Example:
Convert 3A163A_163A16 to decimal:
A = 10
3×161+10×160=48+10=58103×16^1 + 10×16^0 = 48 + 10 = \mathbf{58}_{10}3×161+10×160=48+10=5810
5. Conversion from Decimal Number System to Other Bases
To convert from decimal to binary, octal, or hexadecimal:
- Divide the decimal number by the base.
- Record the remainder.
- Repeat until the quotient is 0.
- Read the remainders in reverse — that’s your converted number.
6. Decimal to Binary Conversion (Base-10 → Base-2)
Use division by 2 and collect remainders in reverse order.
Example:
Convert 251025_{10}2510 to binary:
25 ÷ 2 = 12 R1
12 ÷ 2 = 6 R0
6 ÷ 2 = 3 R0
3 ÷ 2 = 1 R1
1 ÷ 2 = 0 R1
→ Binary: 110012\mathbf{11001}_2110012
7. Decimal to Octal Conversion (Base-10 → Base-8)
Use division by 8, remainders in reverse give the octal number.
Example:
Convert 781078_{10}7810 to octal:
78 ÷ 8 = 9 R6
9 ÷ 8 = 1 R1
1 ÷ 8 = 0 R1
→ Octal: 1168\mathbf{116}_81168
8. Decimal to Hexadecimal Conversion (Base-10 → Base-16)
Use division by 16, and convert remainders ≥10 into A–F.
Example:
Convert 25410254_{10}25410 to hexadecimal:
254 ÷ 16 = 15 R14 → E
15 ÷ 16 = 0 R15 → F
→ Hexadecimal: FE16\mathbf{FE}_{16}FE16
Summary Table
| Conversion Type | Method | Example | Result |
| Binary to Decimal | Positional (Base 2) | 1011 | 11 |
| Octal to Decimal | Positional (Base 8) | 157 | 111 |
| Hex to Decimal | Positional (Base 16) | 2F | 47 |
| Decimal to Binary | Divide by 2 | 25 | 11001 |
| Decimal to Octal | Divide by 8 | 78 | 116 |
| Decimal to Hex | Divide by 16 | 254 | FE |
Decimal vs. Binary vs. Octal vs. Hexadecimal Number Systems
| Decimal | Binary (Base-2) | Octal (Base-8) | Hexadecimal (Base-16) |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
Decimal is base-10 (uses digits 0–9).
Binary is base-2 (uses 0 and 1).
Octal is base-8 (uses digits 0–7).
Hexadecimal is base-16 (uses 0–9 and A–F).
If you’re curious about how binary numbers work and why they are the foundation of all digital systems, be sure to check out our detailed guide on the Binary Number System. It explains the concept, structure, and real-world applications of binary in an easy-to-understand way.
Conclusion
The Decimal Number System is the most widely used numbering system in everyday life, primarily due to its intuitive base-10 structure that aligns with our ten fingers. However, understanding how it relates to other systems like Binary (base-2), Octal (base-8), and Hexadecimal (base-16) is crucial, especially in fields like computer science, electronics, and digital systems.
Each numbering system serves a specific purpose. Binary is the language of computers. Octal and Hexadecimal offer script representations of binary for easier human interpretation. Decimal remains dominant in arithmetic, finance, and daily calculations. By mastering conversions between these systems, you not only improve your number literacy but also strengthen your foundation for more advanced topics like programming, data encoding, and network addressing.
Frequently Asked Questions (FAQs)
What is Decimal Number System?
The decimal number system is a base-10 number system that uses ten digits (0 to 9) to represent all numeric values.
What is the Place Value System in Decimal?
Each digit in a decimal number has a place value based on powers of 10 (e.g., units, tens, hundreds, etc.).
How Do You Convert Binary to Decimal?
Multiply each binary digit by 2 raised to the power of its position, then sum all the results.
How Do You Convert Decimal to Binary?
Divide the decimal number by 2 repeatedly, recording remainders in reverse order.
What is the Hexadecimal Equivalent of Decimal 255?
The hexadecimal equivalent of 255 (decimal) is FF.
Is Decimal a Positional Number System?
Yes, it is a positional number system where the position of each digit determines its value.
What is the Smallest Decimal Digit?
The smallest digit in the decimal system is 0.
Can Decimal Numbers Be Negative?
Yes, decimal numbers can be positive or negative, depending on the context.
Who Invented the Decimal Number System?
The system originated in ancient India, where mathematicians like Aryabhata and Brahmagupta contributed to its development.