Introduction

Binary strings and decimal strings are two different ways to represent numbers in computing. While binary is the fundamental language of computers, decimal is more intuitive for humans. The ability to convert between these two numeral systems is crucial in various computing applications, especially in software development, data analysis, and systems engineering.

Understanding Binary and Decimal Systems

The Binary System

The binary numeral system is based on two digits: `0` and `1`. Each digit is a bit, and multiple bits can define larger, more complex values. The value of a binary number is determined by its positions, which are powers of two, starting from the rightmost digit. For example, the binary string `1011` can be interpreted as:

• `1` × $2^3$ (8)
• `0` × $2^2$ (0)
• `1` × $2^1$ (2)
• `1` × $2^0$ (1)

Adding these together, `1011` in binary equals `11` in decimal.

The Decimal System

The decimal numeral system (base 10) is the standard system for denoting integer and non-integer numbers. It employs ten symbols: `0` through `9`. This system is based on powers of 10. A number like `234` in decimal can be expressed as:

• `2` × $10^2$ (200)
• `3` × $10^1$ (30)
• `4` × $10^0$ (4)

Summing these, the decimal value `234` is straightforward in decimal.

Conversion from Binary String to Decimal String

Converting binary to decimal involves evaluating the sum of powers of two, based on the binary digits' position. Here's how you can perform this conversion:

Step-by-Step Conversion

1. Write Down the Binary Number:
   • Consider the binary string `1101`.
2. Identify Bit Position and Power of Two:
   • From right to left, label each bit with its respective power of two:
     • Rightmost bit (position 0): $2^0$
     • Next to the right (position 1): $2^1$
     • Continue labeling: $2^2$, $2^3$, and so on.
3. Multiply Each Bit by its Power of Two:
   • `1` × $2^3$ = 8
   • `1` × $2^2$ = 4
   • `0` × $2^1$ = 0
   • `1` × $2^0$ = 1
4. Add the Results:
   • 8 + 4 + 0 + 1 = 13

Thus, the binary string `1101` is equivalent to the decimal `13`.

Conversion Algorithm

For programmatic conversion, the algorithm follows an iterative or recursive process evaluating each bit in the string, multiplying by its respective power of two, and accumulating the sum: