Base Converter

Formula

Decimal = sum of (digit x base^position) | Target base = repeated division by that base, reading remainders bottom to top

Changing number base is a re-notation, not a unit conversion, so no multiplying factor is involved. To read a number out of any base, multiply each digit by the base raised to its position index counting from zero on the right and add the products: hexadecimal 1A is 1 x 16 + 10 = 26 in decimal. To write a decimal number into a target base, divide repeatedly by that base and read the remainders from last to first: 26 divided by 2 gives remainders 0, 1, 0, 1, 1, so 26 is 11010 in binary. Because 16 and 8 are both powers of 2, binary regroups straight into hexadecimal four bits at a time and into octal three bits at a time with no arithmetic at all.

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).

Why do computers use binary?

Computers use binary (base 2) because it maps directly to the physical state of transistors: on (1) or off (0). It is the most reliable way to store and process data electronically.

What is Hexadecimal used for?

Hexadecimal (base 16) is used to represent binary data in a more human-readable format. One hex digit represents exactly 4 binary bits (a nibble). It's commonly used for color codes (#FF5733) and memory addresses.

How to convert binary to decimal?

Multiply each bit by 2 raised to the power of its position (from right, starting at 0). For 101: (1×2^2) + (0×2^1) + (1×2^0) = 4 + 0 + 1 = 5.

What is Octal used for?

Octal (base 8) was popular in early computing (like the PDP-8) and is still used in Unix/Linux file permissions (e.g., chmod 755). Each octal digit represents exactly 3 bits.

Why are letters used in Hexadecimal?

Base 16 needs 16 digits. We use 0-9 for the first ten, then A=10, B=11, C=12, D=13, E=14, F=15. This allows representing values up to 15 with a single character.

What is the maximum value of 8 bits (1 byte)?

The maximum value is 255 (binary 11111111, hex FF). Including 0, a byte can represent 256 unique values.

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.