Convert 121 into base 10.
To convert a number from a given base \(n\) to base \(10\), we interpret the number using the powers of the base. A number expressed as \(d_k d_{k-1} \dots d_1 d_0\) in base \(n\) represents the value:
\[d_k \cdot n^k + d_{k-1} \cdot n^{k-1} + \dots + d_1 \cdot n^1 + d_0 \cdot n^0\]
For the given number \(121_n\), the digits are \(d_2=1\), \(d_1=2\), and \(d_0=1\). Since the digits must be smaller than the base, we require \(n > 2\).
Using the positional notation expansion, we substitute the digits into the formula:
\[121_n = 1 \cdot n^2 + 2 \cdot n^1 + 1 \cdot n^0\]
This simplifies to:
\[n^2 + 2n + 1\]
We recognize the expression \(n^2 + 2n + 1\) as a standard algebraic identity, which is the square of a binomial:
\[n^2 + 2n + 1 = (n+1)^2\]
This result shows that any number written as \(121\) in base \(n\) is simply the square of the integer one greater than the base.
Base conversion is a fundamental concept in computing, where data is often represented in binary (base 2), octal (base 8), or hexadecimal (base 16). Understanding how to convert these representations to decimal (base 10) allows computers to process information and humans to interpret it. For instance, if you have a memory address stored as a base-16 number, knowing how to convert it to base 10 helps in debugging software or analyzing hardware performance. Similarly, the square pattern \(121_n = (n+1)^2\) is a helpful shortcut for quick calculations in different numeral systems, reinforcing how algebraic patterns persist across various mathematical structures.