Base Calculator

How to Calculate with Numbers in Different Bases

Performing arithmetic operations on numbers in different bases involves several steps. Here's a comprehensive guide on how to do it:

Base Conversion Formula

To convert a number from base \(b\) to decimal (base 10), use the following formula:

\[ (d_n d_{n-1} ... d_1 d_0)_b = d_n \times b^n + d_{n-1} \times b^{n-1} + ... + d_1 \times b^1 + d_0 \times b^0 \]

Where:

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

Calculation Steps

1. Convert both numbers to decimal (base 10) using the formula above.
2. Perform the desired arithmetic operation (addition, subtraction, multiplication, or division) on the decimal numbers.
3. Convert the result back to the desired base using repeated division by the new base.

Example Calculation

Let's subtract the binary number 1010 (base 2) from the hexadecimal number A5 (base 16), and express the result in decimal (base 10).

Step 1: Convert to Decimal

Binary 1010 to decimal:

\[ 1010_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 0 + 2 + 0 = 10_{10} \]

Hexadecimal A5 to decimal:

\[ A5_{16} = 10 \times 16^1 + 5 \times 16^0 = 160 + 5 = 165_{10} \]

Step 2: Perform the Operation

Subtract the decimal numbers:

\[ 165 - 10 = 155_{10} \]

Step 3: Convert to Desired Base

The result is already in decimal (base 10), so no further conversion is needed.

Final Result

A5 (base 16) - 1010 (base 2) = 155 (base 10)

Visual Representation

This diagram illustrates the process of subtracting a binary number (1010) from a hexadecimal number (A5) and obtaining the result in decimal (155).