Base Converter

Formula

Converted Value = Input × Conversion Factor

This Base Converter tool applies a unit-specific conversion factor to transform the input into the target unit.

Worked Examples

Example 1: Convert Decimal to Binary

Problem:Convert the decimal number 10 to binary.

Solution:10 ÷ 2 = 5 remainder 0 5 ÷ 2 = 2 remainder 1 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Read remainders bottom-up: 1010

Result:1010

Example 2: Convert Hex to Decimal

Problem:Convert Hex 'FF' to Decimal.

Solution:F = 15 (15 × 16^1) + (15 × 16^0) = (15 × 16) + (15 × 1) = 240 + 15 = 255

Result:255

Example 3: Convert Binary to Octal

Problem:Convert Binary 110101 to Octal.

Solution:Group bits by 3 from right: 110 101 110 (binary) = 6 (decimal) 101 (binary) = 5 (decimal) Combine: 65

Result:65

Frequently Asked Questions

What are number bases?

A number base (radix) is the number of unique digits used to represent numbers. Decimal is base 10 (0-9), Binary is base 2 (0-1), Octal is base 8 (0-7), and Hexadecimal is base 16 (0-9, A-F).

Can bases have fractional parts?

Yes. In binary, digits to the right of the 'binary point' represent halves, quarters, eighths (2^-1, 2^-2, 2^-3), just as decimal decimals represent tenths and hundredths.

Is base 64 a number system?

Yes, Base64 uses 64 unique characters (A-Z, a-z, 0-9, +, /) to encode binary data into text. It is widely used for email attachments and embedding images in web pages.

Background & Theory

Understanding number bases is fundamental to computer science. A 'base' or 'radix' determines how many unique digits are available to represent values. **Common Bases:** | Base | Name | Digits | Use Case | |------|------|--------|----------| | 2 | Binary | 0, 1 | Machine code, logic gates | | 8 | Octal | 0-7 | Unix permissions, legacy computing | | 10 | Decimal | 0-9 | Human arithmetic | | 16 | Hexadecimal | 0-9, A-F | Color codes, memory addresses | **How Positional Notation Works:** The value of a digit depends on its position. In Decimal 123: (1 × 100) + (2 × 10) + (3 × 1) In Binary 101: (1 × 4) + (0 × 2) + (1 × 1) = 5 (Decimal) **Hexadecimal Mapping:** | Hex | Decimal | Binary | |-----|---------|--------| | 0-9 | 0-9 | 0000-1001 | | A | 10 | 1010 | | B | 11 | 1011 | | C | 12 | 1100 | | D | 13 | 1101 | | E | 14 | 1110 | | F | 15 | 1111 |

History

Number systems have evolved over millennia based on counting needs and available technology. Base 10 (Decimal) became the global standard likely because humans have ten fingers. It originated in India and was spread by Arab mathematicians to Europe. Base 60 (Sexagesimal) was used by ancient Babylonians (c. 2000 BC) and persists today in time (60 seconds/minutes) and geometry (360 degrees). Base 2 (Binary) was described by Leibniz in 1703, inspired by the I Ching. It became the foundation of modern computing in the 20th century because electronic switches have two states: on and off. Base 16 (Hexadecimal) became standard in computing to represent bytes concisely. Since a byte is 8 bits, and a hex digit is 4 bits, two hex digits perfectly represent one byte (00 to FF). Base 8 (Octal) was prominent in early computing (like the 12-bit PDP-8 and 36-bit PDP-10) where word sizes were divisible by 3.