Number Systems Calculators
Convert between number systems and formats with our free number system calculators. From binary and hexadecimal conversions to scientific notation and Roman numerals, these tools make it easy to work across different numerical representations used in computing, science, and everyday applications.
Binary Calculator
Perform arithmetic operations on binary numbers and convert between binary and decimal.
Hex Calculator
Perform arithmetic on hexadecimal numbers and convert between hex, decimal, and binary.
Scientific Notation Calculator
Convert numbers to and from scientific notation and perform calculations in scientific form.
Base Converter
Convert numbers between any bases from base 2 to base 36 including binary, octal, and hex.
Roman Numeral Converter
Convert between Roman numerals and decimal numbers with validation and rules explanation.
Number to Words Converter
Convert any number into its written word form for checks, documents, and formal writing.
How to Choose the Right Number Systems Calculator
The binary calculator is the starting point for anyone studying computer science or working with digital systems. Computers store and process all data in binary (base 2), making binary arithmetic fundamental to understanding how software and hardware work at the lowest level.
Use the hex calculator when working with hexadecimal values common in programming. Hex is used for color codes in web design (#FF5733), memory addresses in debugging, MAC addresses in networking, and byte values in data analysis. It provides a more compact way to represent binary data since each hex digit corresponds to exactly four binary digits.
The scientific notation calculator is essential for scientists and engineers who work with very large numbers (speed of light: 3 x 10^8 m/s) or very small numbers (electron mass: 9.1 x 10^-31 kg). It simplifies arithmetic with these extreme values and maintains proper significant figures.
For conversions between any number bases, the base converter handles everything from binary (base 2) through hexadecimal (base 16) and beyond to base 36. The Roman numeral converter is useful for dates, outlines, clock faces, and historical references. The number to words converter transforms digits into written English, essential for writing checks, legal documents, and educational materials.
Understanding Number Systems and Base Conversion
Number systems are methods of representing quantities using different sets of symbols and positional rules. The decimal system (base 10) that we use daily employs ten digits (0-9) with each position representing a power of ten. Other bases follow the same positional principle but use different numbers of symbols and different powers.
Binary (base 2) is the foundation of all digital computing. Using only two digits (0 and 1), binary represents every piece of data in a computer, from simple numbers to complex images and programs. Each binary digit (bit) can represent an on/off state in electronic circuits. Eight bits form a byte, which can represent 256 different values (0-255), the basic unit of computer memory and storage.
Hexadecimal (base 16) uses digits 0-9 and letters A-F. It became popular in computing because it provides a human-readable shorthand for binary: each hex digit represents exactly four binary digits. This makes hex invaluable for representing memory addresses, color values (where #FF0000 means red), and byte sequences in a compact, readable format.
Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10. The speed of light is 2.998 x 10^8 m/s rather than 299,800,000 m/s, and a hydrogen atom's diameter is 1.2 x 10^-10 m rather than 0.00000000012 m. This notation makes extremely large or small numbers manageable and clearly communicates the number of significant figures in a measurement.
Roman numerals, developed in ancient Rome, use letters to represent values: I (1), V (5), X (10), L (50), C (100), D (500), and M (1000). Subtractive notation places a smaller value before a larger one to indicate subtraction (IV = 4, IX = 9). Though no longer used for everyday arithmetic, Roman numerals remain important in formal numbering, clock faces, copyright dates, and outlines.
Frequently Asked Questions
How do I convert binary to decimal?
Multiply each binary digit by its positional power of 2 (starting from 2^0 on the right) and add the results. For example, binary 1011 = (1 x 2^3) + (0 x 2^2) + (1 x 2^1) + (1 x 2^0) = 8 + 0 + 2 + 1 = 11 in decimal. Our binary calculator performs this conversion instantly and shows the positional breakdown.
Why do programmers use hexadecimal?
Hexadecimal provides a compact way to represent binary data because each hex digit maps to exactly four binary digits. Instead of writing 11111111 in binary, you write FF in hex. This makes it much easier to read memory dumps, color codes (#FF5733), MAC addresses (00:1A:2B:3C:4D:5E), and byte values. Hex is standard in programming, networking, and digital forensics.
How do I write a number in scientific notation?
Move the decimal point until you have a number between 1 and 10, then multiply by 10 raised to the number of places you moved the decimal. Move right for large numbers (positive exponent) and left for small numbers (negative exponent). For example, 45,000 becomes 4.5 x 10^4 (moved 4 places left), and 0.0032 becomes 3.2 x 10^-3 (moved 3 places right).
What is the largest Roman numeral?
In standard notation, the largest Roman numeral is 3,999 (MMMCMXCIX). Beyond this, extended notation uses a bar (vinculum) over a numeral to multiply it by 1,000. For practical purposes, most modern uses of Roman numerals stay within the 1-3,999 range. Our converter supports extended notation for larger values up to 3,999,999.
How do I convert between any two number bases?
The general method is to first convert from the source base to decimal (base 10), then from decimal to the target base. To convert to decimal, multiply each digit by its positional base power and sum. To convert from decimal, repeatedly divide by the target base and collect remainders in reverse order. Our base converter handles any base from 2 to 36 and shows the conversion steps.