Updated: July 19, 2025

Numbers are fundamental to mathematics and computing, but they can be represented in various numeration systems. Each system uses a different base or radix and a set of symbols to express numbers. The most common numeration systems include decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Understanding how to convert numbers between these systems is essential for fields like computer science, electronics, and mathematics.

In this article, we will explore the principles behind different numeration systems and provide detailed methods for converting numbers between them. Whether you are a student, programmer, or enthusiast, mastering these conversions will enhance your numerical literacy and problem-solving skills.


Understanding Numeration Systems

A numeration system is a way of expressing numbers using a consistent set of digits or symbols. The key characteristic of any numeration system is its base, which determines how many unique digits it has before “rolling over” to the next positional place.

Common Numeration Systems

  • Decimal (Base 10): The most widely used system in daily life. It uses digits from 0 to 9.
  • Binary (Base 2): Used extensively in computing and digital electronics. Uses only two symbols: 0 and 1.
  • Octal (Base 8): Occasionally used in computing as a shorthand for binary. Uses digits from 0 to 7.
  • Hexadecimal (Base 16): Commonly used in programming and digital systems, especially for representing memory addresses. Uses digits from 0 to 9 and letters A to F (representing values 10 to 15).

Each positional digit in these systems represents the digit multiplied by the base raised to the power of its position index, counting from right to left starting at zero.


General Principles of Number Conversion

Converting numbers between bases involves changing the representation without altering the value. There are two broad approaches:

  1. Converting from any base to decimal: This involves expanding the number according to its place values and summing the results.
  2. Converting from decimal to any other base: This involves repeatedly dividing the number by the new base and recording remainders.

For conversions between non-decimal bases (e.g., binary to hexadecimal), it can be easier to convert first into decimal, then into the target base, or use shortcut methods where applicable.


Converting From Any Base to Decimal

The general formula for converting a number ( N ) from base ( b ) with digits ( d_n d_{n-1} \ldots d_1 d_0 ) is:

[
N = \sum_{i=0}^{n} d_i \times b^i
]

Where:
– ( d_i ) is the digit at position ( i ) (starting from 0 on the right),
– ( b ) is the base,
– ( n ) is the highest digit position.

Example: Convert Binary 1101 to Decimal

Digits: (1,1,0,1)
Positions: (3,2,1,0)

Calculation:

[
(1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 8 + 4 + 0 + 1 = 13
]

Thus, binary 1101 equals decimal 13.

Step-by-Step Procedure

  1. Write down the digits of the original number.
  2. Assign positional powers from right-to-left starting at zero.
  3. Multiply each digit by its base raised to its positional power.
  4. Sum all these values.
  5. The sum is the equivalent decimal value.

This procedure works for fractional numbers as well; positions after the decimal point use negative powers.


Converting From Decimal to Any Base

Converting a decimal number ( N ) into another base ( b ) is typically done via repeated division:

Algorithm:

  1. Divide ( N ) by base ( b ).
  2. Record the remainder (this becomes one digit).
  3. Update ( N ) as the quotient from step 1.
  4. Repeat steps until ( N = 0 ).
  5. The remainders collected represent the digits in reverse order.

Example: Convert Decimal 45 to Binary

Step-by-step division:

Division Quotient Remainder
45 / 2 22 1
22 / 2 11 0
11 / 2 5 1
5 / 2 2 1
2 / 2 1 0
1 / 2 0 1

Collecting remainders bottom-up: 101101

Therefore, decimal 45 equals binary 101101.


Binary to Octal and Hexadecimal Conversions

Binary numbers can be converted directly into octal or hexadecimal systems without going through decimal by grouping bits.

Binary to Octal

Since octal is base-8, which is (2^3), group binary digits in sets of three starting from right:

Example: Convert binary 101101011 to octal

Group bits:
000 101 101 011

Groups correspond to octal digits:

  • 000 = 0
  • 101 = 5
  • 101 = 5
  • 011 = 3

So octal number is 0553.

Drop leading zeros if desired: 553.

Binary to Hexadecimal

Hexadecimal is base-16 or (2^4), so group binary digits in sets of four:

Example: Convert binary 110111101010 to hexadecimal

Group bits:
1101 1110 1010

Convert each group:

  • 1101 = D (13)
  • 1110 = E (14)
  • 1010 = A (10)

Hexadecimal representation: DEA


Octal and Hexadecimal Back To Binary

Conversion back from octal or hex uses reverse grouping.

Octal To Binary

Convert each octal digit into its three-bit binary equivalent.

Example: Octal 345

Digits:
3011
4100
5101

Binary equivalent: 011100101

Hexadecimal To Binary

Convert each hex digit into its four-bit binary equivalent.

Example: Hexadecimal 9F

Digits:
91001
F1111

Binary equivalent: 10011111


Converting Between Octal and Hexadecimal via Binary

Because both octal and hexadecimal map neatly onto groups of bits, conversion between them is easiest via binary as an intermediate step.

Example: Convert Octal 725 To Hexadecimal

Step-by-step:

  1. Convert octal digits into binary:
  2. 7: 111
  3. 2: 010
  4. 5: 101

Combined binary: 111010101

  1. Group into fours from right for hex:
  2. Add leading zeros if needed – 000111010101
  3. Groups:

    • 0001: 1
    • 1101: D
    • 0101: 5
  4. Resulting hex: 1D5


Working With Fractional Numbers

Fractional parts can also be converted between bases using similar principles but in reverse order for non-decimal bases.

Converting Fractional Part From Base-N To Decimal

Use negative powers for positions after decimal point:

[
N_{\text{fraction}} = \sum_{i=1}^{m} d_{-i} \times b^{-i}
]

Example: Convert binary fraction .101 to decimal:

[
(1 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3}) = \frac{1}{2} + 0 + \frac{1}{8} = 0.625
]

Converting Decimal Fraction To Base-N

Multiply fractional part by new base repeatedly:

Algorithm:
1. Multiply fraction by base ( b ).
2. Extract integer part – this becomes next digit.
3. Keep fractional part for next iteration.
4. Repeat until fraction becomes zero or desired precision reached.

Example: Convert decimal fraction .625 to binary:

Step-by-step:

Step Multiply Integer Part Fraction Remaining
1 .625 * 2 = 1.25 1 .25
2 .25 * 2 = 0.5 0 .5
3 .5 * 2 = 1.0 1 .0

Fraction terminates; digits are .101.


Practical Tips for Conversion

  • Use grouping by bits when switching between bases that are powers of two.
  • For large numbers, rely on calculators or scripting tools with built-in base conversion capabilities.
  • When working with fractions that do not terminate naturally in target base, specify precision or rounding rules.
  • Learning shortcuts such as memorizing hex-digit-to-binary mappings improves speed and accuracy.

Applications of Number Base Conversion

Conversions between numeration systems are widely applied in areas like:

  • Computer Programming: Representing data and memory addresses in hex.
  • Digital Electronics: Designing logic circuits with binary signals.
  • Networking: IP addressing often requires conversions between formats.
  • Cryptography: Manipulating keys encoded in various bases.
  • Mathematics Education: Understanding place value concepts and numeral properties.

Conclusion

Converting numbers between different numeration systems may seem challenging initially but becomes manageable once you understand the underlying principles of positional notation and arithmetic operations involved.

By mastering conversion techniques such as expanding numbers into powers of their bases or performing repeated division/remainder extraction, you can effortlessly translate numbers among decimal, binary, octal, and hexadecimal systems.

With practice and application of grouping strategies for bases that are powers of two, these conversions become straightforward tools essential for computing professionals, engineers, educators, and students alike.

Take time to practice converting diverse numbers , integers and fractions , across various bases until you gain confidence and fluency in these fundamental numerical transformations!