Binary Calculator
Add, subtract, multiply, and convert binary numbers (Base 2) to Decimal (Base 10) and Hexadecimal.
Binary Calculator & Converter
Add, subtract, multiply, and divide binary numbers (Base 2), or convert freely between Binary and Decimal with step-by-step mathematical breakdowns and Two's Complement notation.
1Binary Calculation — Add, Subtract, Multiply, or Divide
Perform base-2 arithmetic with automatic decimal, hex, and octal cross-conversions.
2Convert Binary Value to Decimal Value
Convert base-2 binary strings to base-10 decimal numbers with positional power expansions.
3Convert Decimal Value to Binary Value
Convert base-10 decimal numbers to base-2 binary strings with continuous division by 2.
Comprehensive Guide to the Binary Number System (Base 2) & Digital Logic
The binary number system is a base-2 positional numeral system that uses only two symbols: 0 and 1 (binary digits or bits). Modern electronic computers use binary logic internally because electrical transistors can reliably represent two physical voltage states: ON (1) and OFF (0).
Fundamental Binary Arithmetic Rules
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (0 with carry 1)
- 1 + 1 + 1 = 11 (1 with carry 1)
- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 × 1 = 1
Frequently Asked Questions
What is Two's Complement and why is it used?
Two's complement is a mathematical technique used in computer architecture to represent signed negative integers. It allows computer hardware to perform addition and subtraction using the exact same arithmetic logic unit (ALU) circuitry without requiring separate subtraction hardware.
How do binary fractions work?
In binary fractions, bits to the right of the radix point represent negative powers of 2. For example, 0.1₂ = 2⁻¹ = 0.5, 0.01₂ = 2⁻² = 0.25, and 0.001₂ = 2⁻³ = 0.125. Therefore, 101.101₂ = 4 + 1 + 0.5 + 0.125 = 5.625.
Why are hexadecimal numbers used alongside binary?
Hexadecimal (Base 16) is a human-friendly compact representation of binary. Since 16 = 2⁴, exactly four binary bits (one nibble) map to a single hexadecimal digit (e.g., 1111₂ = 0xF).
Can this calculator handle large binary strings (e.g. 128-bit or 256-bit)?
Yes. The engine uses native JavaScript BigInt, enabling exact integer calculations for binary strings with hundreds of digits without overflow or truncation.
Understanding Binary Calculator & Formula
The Binary Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Binary Calculator
- 1 Enter Binary Number 1 and Binary Number 2 (consisting strictly of 0s and 1s).
- 2 Select the mathematical or bitwise operation: Addition (+), Subtraction (-), Multiplication (×), Division (/), AND, OR, XOR, NOT, Left Shift, Right Shift.
- 3 Click "Calculate ▶️" to view the binary result, decimal conversion, hexadecimal equivalent, and step-by-step bitwise truth tables.
Binary Calculator — Frequently Asked Questions
Q1 How do you convert binary to decimal and decimal to binary? ▾
To convert binary to decimal, sum the powers of 2 for every 1 bit: e.g. 1011 in binary = (1×8) + (0×4) + (1×2) + (1×1) = 11. To convert decimal to binary, repeatedly divide by 2 and record the remainders in reverse order.
Q2 How does binary addition work? ▾
Binary addition follows four basic rules: 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (0 with a carry of 1 to the next higher place value).
Q3 What is Two's Complement representation? ▾
Two's Complement is the standard method computers use to represent signed negative integers. To find the two's complement of a binary number, invert all bits (1s become 0s and vice versa) and add 1.
Q4 What are bitwise AND, OR, XOR, and NOT operations? ▾
AND produces 1 only if both bits are 1. OR produces 1 if either bit is 1. XOR produces 1 if exactly one bit is 1. NOT inverts each bit.
More in Numeral systems converters
Roman Numeral Converter
PopularConvert between standard Arabic numbers (1-3,999) and Roman numerals (I, V, X, L, C, D, M) or convert dates.
Unit Converter (Universal Conversions)
Most UsefulUniversal unit converter for length, weight, height, volume, temperature, area, speed, energy, power, pressure, time, cooking, and fuel economy.
Kilograms to Pounds Converter (kg to lbs)
Most UsefulConvert kilograms to pounds and ounces (kg, lbs, oz) instantly with exact mathematical conversion formulas and reference charts.
Acceleration Converter
PopularConvert between meters/sec², feet/sec², standard gravity (g), gals (cm/s²), and km/h/s.
Area Converter
Most UsefulConvert between square feet, square meters, acres, hectares, square inches, square yards, and cents.
Gold Weight Converter
PopularConvert gold weight between grams, troy ounces, tolas, pennyweights, standard 400 oz delivery bars, and 24k/22k/18k purity values.
Explore All 4,700+ Calculators by Category
Genetics, molecular laboratory science, veterinary dosages, animal gestation, agronomy, and forestry.
Stoichiometry, chemical solutions, thermodynamics, kinetics, electrochemistry, and organic reactions.
Concrete, roofing, brick, framing, lumber spans, driveway paving, gravel, and material specifications.
Precision metric and imperial unit converters for length, area, volume, mass, pressure, temperature, and speed.
Carbon footprint, solar energy payback, tree sequestration, recycling offsets, and sustainable living metrics.
Travel durations, clothing and shoe sizes, fuel economy, home budgeting, sleep cycles, and daily routines.
Mortgage amortization, compound interest, loans, retirement planning, taxes, investments, and salary projections.
Culinary volume-to-weight conversions, baking hydration, ingredient scaling, pizza dough, and party food planning.
BMI, daily caloric needs (TDEE/BMR), body fat percentage, pregnancy due dates, clinical dosage, and pediatric metrics.
Scientific calculators, fractions, percentages, geometry, trigonometry, algebra, linear systems, and matrices.
Kinematics, Newtonian dynamics, thermodynamics, fluid mechanics, electromagnetism, optics, and relativity.
Heart rate training zones, running pace, triathlon pacing, cycling power, calories burned, and 1RM lifting benchmarks.
Standard deviation, normal distribution (Z-scores), confidence intervals, sample size, and probability odds.
Education GPA, photography exposures, music BPM, data storage, network bandwidth, and specialty calculations.