Why Does the Conversion Between Decimal and Any Base by Repeated Division Work?

Understanding the conversion between decimal and any base using repeated division involves delving into the fundamental principles of positional number systems. This technique is not only efficient but also deeply rooted in the properties of division and modular arithmetic. In this article, we'll break down the process and explore the underlying reasons why this method works.

Understanding Base Systems

In a positional number system, each digit is assigned a value based on its position. The positional value of each digit is a power of the base. For example, in a base ( b ) numeral system, the number 345 can be expressed as:

3 * b2 4 * b1 5 * b0

This concept is crucial for understanding how numbers are represented and converted between different bases.

The Conversion Process

To convert a decimal number ( N ) to another base ( b ) using repeated division, you follow these steps:

Divide the Number: Start with the decimal number ( N ) and divide it by the base ( b ). Record the Remainder: The remainder from this division is the least significant digit (rightmost) of the number in base ( b ). Update the Quotient: Replace ( N ) with the quotient obtained from the division. Repeat: Repeat the process until the quotient becomes zero. Read the Digits: The base ( b ) representation is formed by reading the remainders from the last division to the first.

Why It Works

The repeated division method works because it systematically extracts the digits of the number in the new base. This method is valid due to the properties of division and modular arithmetic.

For a given number ( N ) and a base ( b ), when you divide ( N ) by ( b ), you can express ( N ) as:

N q * b r

where ( q ) is the quotient and ( r ) is the remainder, with ( 0 leq r

In each subsequent division, you are essentially peeling off the digits of ( N ) in base ( b ). Each division reduces the number, providing you with the next digit in the base representation.

Example: Converting Decimal 13 to Base 2

Let's illustrate the process with an example. We'll convert the decimal number 13 to binary (base 2).

( 13 , text{div} , 2 6 ) remainder 1 (least significant bit) ( 6 , text{div} , 2 3 ) remainder 0 ( 3 , text{div} , 2 1 ) remainder 1 ( 1 , text{div} , 2 0 ) remainder 1 (most significant bit)

Reading the remainders from bottom to top, we get 11012. Therefore, 1310 11012.

Conclusion

The repeated division method effectively ensures that each digit of the decimal number is accurately extracted and represented in the new base. This process leverages the properties of division and remainders to provide a robust and reliable conversion technique.