The square root of -1, denoted i, was introduced to extend the real number system to the complex numbers. The complex numbers are necessary to solve certain equations that have no solutions in the real numbers.
For example, the equation x2 = -1 has no real number solutions, because a real number squared is always positive or zero. But we can solve this by introducing a new number i such that i2 = -1. This lets us express the solutions as x = ±i.
The complex numbers are useful in many areas of mathematics, science, and engineering. They are essential for analyzing oscillations and waves, and for solving many equations. Without the complex numbers, we would not have some mathematical models that are crucial for applications like electronics, signal processing, and control systems.
So while i may seem strange, it was necessary to expand the number system and enable further progress in mathematics and its applications. The complex numbers are a consistent and coherent system, as long as we accept i and its properties.
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