### Explanation of Decimal Conversion

Decimal conversion is the process of converting a number from one base to another. This is particularly useful in various fields such as computer science, mathematics, and engineering. The most common conversions involve:

- Decimal to binary
- Decimal to hexadecimal
- Decimal to octal

#### Converting Decimal to Binary

Binary is base-2, which means it uses only 0 and 1. To convert a decimal number to binary, you repeatedly divide the number by 2 and write down the remainder. The binary number is the series of remainders read from the bottom up.

**Example: Convert 13 to binary**

- $13 \div 2 = 6$ remainder $1$
- $6 \div 2 = 3$ remainder $0$
- $3 \div 2 = 1$ remainder $1$
- $1 \div 2 = 0$ remainder $1$

Reading from bottom to top, $13$ in binary is $1101$.

#### Converting Decimal to Hexadecimal

Hexadecimal is base-16, using digits 0-9 and letters A-F. The conversion process is similar to binary but divides by 16 instead of 2.

**Example: Convert 254 to hexadecimal**

- $254 \div 16 = 15$ remainder $14$

Since 14 in hexadecimal is 'E', $254$ in hexadecimal is $FE$.

#### Converting Decimal to Octal

Octal is base-8, so it uses digits 0-7. The conversion is done by repeatedly dividing by 8.

**Example: Convert 83 to octal**

- $83 \div 8 = 10$ remainder $3$
- $10 \div 8 = 1$ remainder $2$

So, $83$ in octal is $123$.

### Mathematical Representation

For a more formal approach, the conversion from decimal $D$ to any base $b$ can be represented as:

$D = d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \cdots + d_1 \cdot b + d_0$
where $d$ are the digits in the new base.

For example, converting a decimal number $46$ to base $3$:

$46 = 1 \cdot 3^3 + 2 \cdot 3^2 + 1 \cdot 3^1 + 1 \cdot 3^0$
Thus, $46$ in base 3 is $1211_3$.

### Important Points

**Understanding positional value**: Each digit represents a power of the base.
**Repeated division/remainder method**: A practical way to handle conversions manually.
**Calculator tools and algorithms**: Useful for handling large numbers or automated conversions.

This fundamental concept is crucial for various applications, especially in computing, where binary and hexadecimal systems are prevalent.