Universal Number Base Converter

The Universal Number Base Converter is a powerful tool that allows you to convert numbers between different number bases. This calculator supports conversions between bases 2 to 36, including common bases like binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16).

What is a Number Base?

A number base, or radix, is the number of unique digits used to represent numbers in a positional numeral system. For example, the decimal system uses base 10 (digits 0-9), while binary uses base 2 (digits 0 and 1).

Conversion Formula

The general formula for converting a number from one base to another involves two steps:

1. Convert the original number to decimal (base 10)
2. Convert the decimal number to the target base

Step 1: Converting to Decimal

For a number \(a_n a_{n-1} ... a_1 a_0\) in base \(b\), the decimal value is:

\[a_n \cdot b^n + a_{n-1} \cdot b^{n-1} + ... + a_1 \cdot b^1 + a_0 \cdot b^0\]

Where:

• \(a_i\) represents each digit in the number
• \(b\) is the original base
• \(n\) is the position of the digit (starting from 0 for the rightmost digit)

Step 2: Converting from Decimal to Target Base

To convert from decimal to the target base, we repeatedly divide by the target base and keep track of the remainders. The remainders in reverse order form the digits of the number in the new base.

Calculation Steps

Let's convert the number 1234 from base 10 to base 16 (hexadecimal) as an example:

1. 1234 ÷ 16 = 77 remainder 2
2. 77 ÷ 16 = 4 remainder 13 (D in hexadecimal)
3. 4 ÷ 16 = 0 remainder 4

Reading the remainders from bottom to top, we get: 4D2

Therefore, \((1234)_{10} = (4D2)_{16}\)

Example with Visual Representation

Let's visualize the conversion of (101101)₂ to decimal:

This visual representation shows how each bit in the binary number corresponds to a power of 2, and the sum of the active bits (1's) gives us the decimal equivalent.

The Universal Number Base Converter simplifies these calculations, allowing you to quickly and accurately convert between any supported bases.