Base N converter
Convert base nconverter between units instantly. Includes conversion tables, common equivalents, and calculation formulas.
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Convert to decimal by multiplying each digit by the source base raised to the power of its position (rightmost is 0) and summing. Then convert from decimal to the target base by repeatedly dividing by the target base and collecting remainders in reverse order.
Last reviewed: December 2025
Worked Examples
Example 1: Decimal to Hexadecimal
Example 2: Binary to Octal
Background & Theory
The Base Nconverter applies the following established principles and formulas. Unit conversion is the process of expressing a quantity in a different unit of measurement while preserving its physical meaning. At the foundation of modern measurement lies the International System of Units (SI), which defines seven base units: the meter for length, kilogram for mass, second for time, ampere for electric current, kelvin for thermodynamic temperature, mole for amount of substance, and candela for luminous intensity. All other units, called derived units, are defined as algebraic combinations of these seven. Dimensional analysis is the principal method for performing unit conversions. By treating units as algebraic quantities that can be multiplied, divided, and cancelled, a conversion factor chain allows a value expressed in one unit to be rewritten in another without altering its physical magnitude. For example, to convert 60 miles per hour to meters per second, one multiplies by a chain of conversion factors each equal to one: (1609.34 m / 1 mile) × (1 hour / 3600 s). Metric prefixes enable compact expression of quantities across extreme ranges of magnitude. Standard prefixes span from nano (10^-9) through micro (10^-6) and milli (10^-3) up through kilo (10^3), mega (10^6), and giga (10^9), and beyond in both directions. These prefixes are strictly multiplicative and apply consistently to any SI base or derived unit. Temperature conversions require affine transformations rather than simple scaling. To convert Celsius to Fahrenheit the formula is °F = (°C × 9/5) + 32, while the conversion to the absolute Kelvin scale is K = °C + 273.15. These formulas reflect the different zero points and degree-size conventions of each scale. Significant figures govern how precision is preserved through calculations. A result should not express more precision than the least precise input value permits. In digital storage, IEEE and IEC standards distinguish between decimal prefixes (kilobyte = 1000 bytes) and binary prefixes (kibibyte = 1024 bytes), a distinction that has practical consequences for how storage capacity is reported by manufacturers versus operating systems. Unit coherence — ensuring that all quantities in an equation share a consistent unit system — is essential for obtaining correct results.
History
The history behind the Base Nconverter traces back through the following developments. Human beings have been measuring and comparing quantities since before recorded history. The earliest known measurement units were body-based: the cubit (the distance from elbow to fingertip), the foot, the hand, and the digit. The furlong originated as the length of a furrow a team of oxen could plow without resting. These anthropomorphic standards were practical for local use but differed between regions and kingdoms, creating persistent difficulties in trade and construction. The ancient Egyptians standardized the royal cubit at approximately 52.4 centimeters and distributed calibrated granite rods to ensure consistency across building projects, including the pyramids. Roman engineers used the mile (mille passuum, one thousand double paces) and spread these standards throughout their empire via road networks. Despite these efforts, measurement diversity persisted across medieval Europe, hampering commerce. The French Revolution created political will for radical standardization. In 1795 France officially adopted the metric system, defining the meter as one ten-millionth of the distance from the equator to the North Pole along the Paris meridian. This gave the world its first fully decimal, rationally constructed measurement system. The Metre Convention of 1875 established the International Bureau of Weights and Measures (BIPM) in Sevres, France, creating a permanent international body to maintain physical artifact standards and coordinate global metrology. For over a century, the kilogram was defined by a platinum-iridium cylinder locked in a vault near Paris. In 1999, a stark demonstration of what unit inconsistency costs occurred when NASA's Mars Climate Orbiter was lost because one engineering team used pound-force seconds while another used newton seconds. The spacecraft entered the Martian atmosphere at the wrong angle and was destroyed, at a cost of 327 million dollars. In 2019 the SI underwent its most significant revision, redefining all seven base units in terms of fixed numerical values of fundamental physical constants such as the speed of light, Planck's constant, and the elementary charge. This eliminated any reliance on physical artifacts and made the measurement system permanently stable and universally reproducible.
Frequently Asked Questions
Sources & References
Formula
Decimal = sum of (digit x base^position); Target = repeated division by target base
Convert to decimal by multiplying each digit by the source base raised to the power of its position (rightmost is 0) and summing. Then convert from decimal to the target base by repeatedly dividing by the target base and collecting remainders in reverse order.
Worked Examples
Example 1: Decimal to Hexadecimal
Problem: Convert decimal 255 to hexadecimal (base 16).
Solution: 255 / 16 = 15 remainder 15 (F)\n15 / 16 = 0 remainder 15 (F)\nReading remainders bottom to top: FF
Result: 255 (base 10) = FF (base 16)
Example 2: Binary to Octal
Problem: Convert binary 11010110 to octal (base 8).
Solution: First convert binary to decimal:\n1x128 + 1x64 + 0x32 + 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = 214\nThen 214 to octal: 214 / 8 = 26 r 6, 26 / 8 = 3 r 2, 3 / 8 = 0 r 3\nResult: 326
Result: 11010110 (base 2) = 326 (base 8)
Frequently Asked Questions
What is a number base or radix?
A number base (also called radix) is the number of unique digits used to represent numbers in a positional numeral system. Base 10 (decimal) uses digits 0-9, base 2 (binary) uses 0-1, base 16 (hexadecimal) uses 0-9 and A-F. Each position in a number represents a power of the base. For example, 42 in base 10 means 4 times 10 to the first power plus 2 times 10 to the zero power, equaling forty-two.
Why is base 16 (hexadecimal) commonly used in computing?
Hexadecimal is popular in computing because it provides a compact representation of binary data. Each hex digit corresponds to exactly 4 binary bits, so one byte (8 bits) is represented by exactly two hex digits. This makes it easy to read and write memory addresses, color codes (like FF0000 for red), and byte values. Compared to binary, hex reduces the length of a representation by a factor of four while maintaining a simple digit-to-bits mapping.
What bases are supported by this converter?
This converter supports any base from 2 to 36. Bases 2 through 10 use numeric digits 0-9 as needed. Bases 11 through 36 extend the digit set with letters A through Z, where A represents 10, B represents 11, and so on up to Z representing 35. Common bases include binary (2), octal (8), decimal (10), duodecimal (12), hexadecimal (16), and base 36 which is the maximum single-character digit base.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
What inputs do I need to use Base N converter accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
Can I use Base N converter on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy