How to convert decimal to base 2 (binary)?

To convert decimal to base 2: Continuously divide the integer by 2, recording each remainder (0 or 1). Read the remainders in reverse order (bottom to top). For example, 13 / 2 = 6 R1, 6 / 2 = 3 R0, 3 / 2 = 1 R1, 1 / 2 = 0 R1 => binary 1101.

How to convert base to base (arbitrary base N)?

To convert from base A to base B: First convert the number from base A to decimal (base 10) by multiplying each digit by A raised to its positional power. Then convert that decimal number to base B by repeated division by B.

How does the decimal to floating point converter work?

Our decimal to floating point converter follows the IEEE 754 32-bit single-precision standard. It encodes the number into 1 sign bit (0 for positive, 1 for negative), 8 exponent bits with a bias of 127, and 23 normalized mantissa (fractional) bits.

Decimal to Base-N Number Converter

Complete Guide: How to Convert Base Numbers & Converting to Base 10

Welcome to the definitive number base converter and base to base calculator. Throughout history and computer science, civilizations and computing architectures have used varying radix systems: ancient Babylonians used sexagesimal (base 60), Mayans used vigesimal (base 20), digital electronics use binary (base 2), and modern humans standardly use the denary decimal system (base 10). When working with computer architectures, bitmasks, memory addressing, or cryptography, using an accurate number base conversion calculator and decimal to base n converter is vital.

Mathematical Principles: Converting Bases Calculator

Any integer N in base b can be expressed through positional polynomials:

Value = d_n×bⁿ + d_n-1×b^n-1 + ... + d₁×b¹ + d₀×b⁰

To convert from base to base:

Convert Source Base to Base 10: Multiply each digit by the source base raised to its zero-indexed position from right to left.
Convert Decimal to Target Base: Repeatedly divide the decimal number by the target base, writing down the remainders until the quotient reaches 0. Reading remainders from last to first gives the final result.

Standard Number Bases Comparison Table

Base (Radix)	System Name	Allowed Digits	Typical Technology Application
Base 2	Binary	`0, 1`	Logic gates, machine code, bitwise arithmetic
Base 8	Octal	`0 – 7`	Unix file permissions (e.g. chmod 755), legacy mainframes
Base 10	Decimal (Denary)	`0 – 9`	Human arithmetic, currency, SI measurements
Base 16	Hexadecimal	`0 – 9, A – F`	RAM memory addresses, CSS hex colors, byte inspection
Base 36	Hexatrigesimal	`0 – 9, A – Z`	Compact URL shorteners, alphanumeric serial keys

Decimal to Floating Point Converter (IEEE 754 Standard)

In modern computing (C, C++, Java, JavaScript, Python), floating-point numbers are represented using the IEEE 754 standard. A 32-bit single-precision float contains:

1 Sign Bit: 0 for positive, 1 for negative.
8 Exponent Bits: Stores the exponent with an offset bias of +127.
23 Mantissa Bits: Stores the significant fractional binary bits with an implicit leading 1.

Frequently Asked Questions (FAQ)

What is a decimal to decimal converter?

A decimal to decimal converter standardizes, formats, or validates decimal notation across differing regional formats (e.g. converting European comma decimals 12,50 to standard dot notation 12.50, or scientific notation 1.25e1 to standard base 10 integer forms).

Why does hexadecimal use letters A through F?

Because base 16 requires 16 unique single-digit symbols. The digits 0 through 9 provide ten symbols, so the first six Latin letters are borrowed: A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.

Can this calculator convert negative numbers between bases?

Yes. For base conversions, negative numbers preserve their negative sign across bases (e.g. -255 in decimal is -FF in hex and -11111111 in binary). In our IEEE 754 floating point inspector, negative numbers set the leading Sign bit to 1.