Binary Converter
Instantly convert binary with our free converter. See conversion tables, formulas, and step-by-step explanations. Free to use with no signup required.
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Formula
Each binary digit (bit) represents a power of 2 based on its position from right to left, starting at position 0. To convert binary to decimal, multiply each bit by 2 raised to its position power and sum the results. For example, binary 1010 = 1x8 + 0x4 + 1x2 + 0x1 = 10.
Last reviewed: December 2025
Worked Examples
Example 1: Decimal 200 to Binary
Example 2: Binary 10110 to Other Bases
Background & Theory
The Binary Converter 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 Binary Converter 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 (bit x 2^position)
Each binary digit (bit) represents a power of 2 based on its position from right to left, starting at position 0. To convert binary to decimal, multiply each bit by 2 raised to its position power and sum the results. For example, binary 1010 = 1x8 + 0x4 + 1x2 + 0x1 = 10.
Worked Examples
Example 1: Decimal 200 to Binary
Problem: Convert decimal number 200 to binary.
Solution: 200 / 2 = 100 r 0\n100 / 2 = 50 r 0\n50 / 2 = 25 r 0\n25 / 2 = 12 r 1\n12 / 2 = 6 r 0\n6 / 2 = 3 r 0\n3 / 2 = 1 r 1\n1 / 2 = 0 r 1\nBinary: 11001000
Result: 200 = 11001000 (binary) = 310 (octal) = C8 (hex)
Example 2: Binary 10110 to Other Bases
Problem: Convert binary 10110 to decimal, octal, and hexadecimal.
Solution: Decimal: 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = 22\nOctal: 22 / 8 = 2 r 6 -> 26\nHex: 22 / 16 = 1 r 6 -> 16
Result: 10110 (binary) = 22 (decimal) = 26 (octal) = 16 (hex)
Frequently Asked Questions
What is binary and why do computers use it?
Binary is a base-2 number system that uses only two digits: 0 and 1. Computers use binary because their fundamental building blocks (transistors) have two states: on and off. These two states map perfectly to the binary digits 1 and 0. Every piece of data a computer processes, from text to images to programs, is ultimately represented as sequences of binary digits (bits). This simplicity makes digital circuits reliable and efficient to manufacture.
How do I convert decimal to binary?
To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainder. Continue until the quotient is 0. Then read the remainders from bottom to top. For example, to convert 13: 13/2 = 6 remainder 1, 6/2 = 3 remainder 0, 3/2 = 1 remainder 1, 1/2 = 0 remainder 1. Reading bottom to top gives 1101. So decimal 13 equals binary 1101.
What is the difference between binary, octal, and hexadecimal?
These are all positional number systems with different bases. Binary is base 2 (digits 0-1), octal is base 8 (digits 0-7), and hexadecimal is base 16 (digits 0-9 plus A-F). They are related by powers of 2: each octal digit corresponds to exactly 3 binary bits, and each hexadecimal digit corresponds to exactly 4 binary bits. This relationship makes conversions between them straightforward and is why all three are commonly used in computing.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Does Binary Converter work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy