The i squared rule defines the fundamental property of the imaginary unit i, stating that when i is squared, the result is negative one (i² = -1). This rule is crucial for understanding and working with imaginary numbers and, by extension, complex numbers.
Understanding the Imaginary Unit (i)
The imaginary unit, denoted as i, was introduced in mathematics to provide a solution for the square root of negative numbers, which cannot be found within the realm of real numbers. By definition, i is the square root of -1. This means that i allows us to express the square root of any negative number. For instance, the square root of -9 can be written as √(-1 9) = √(-1) √(9) = 3i.
The Fundamental Rule: i² = -1
The very essence of the imaginary unit i lies in its defining property:
- i² = -1
This rule serves as the cornerstone for all operations involving imaginary numbers. It is not derived from other properties but is the foundational definition that allows the system of complex numbers to exist consistently with other mathematical principles. Without this rule, solving equations like x² + 1 = 0 would be impossible using real numbers alone.
Squaring Imaginary Numbers
An imaginary number is typically expressed in the form bi, where b is a real number and i is the imaginary unit. When an imaginary number is squared, the i squared rule is directly applied.
To square an imaginary number bi:
- Square the real part (b).
- Square the imaginary unit (i).
- Multiply the results.
This leads to the general formula:
*(bi)² = b² i²**
Since we know that i² = -1, the formula simplifies to:
*(bi)² = b² (-1)**
(bi)² = -b²
This means that the square of any imaginary number is always a negative real number.
Practical Examples
Let's illustrate the i squared rule with some common examples:
| Imaginary Number | Calculation | Result |
|---|---|---|
| i | i² | -1 |
| 3i | (3i)² = 3² i² = 9 (-1) | -9 |
| -2i | (-2i)² = (-2)² i² = 4 (-1) | -4 |
| 5i | (5i)² = 5² i² = 25 (-1) | -25 |
| -7i | (-7i)² = (-7)² i² = 49 (-1) | -49 |
These examples demonstrate how applying the i² = -1 rule consistently yields a negative real number when squaring any imaginary number. This rule is fundamental for operations in complex analysis, electrical engineering, quantum mechanics, and various other scientific and engineering fields.
For more details on imaginary numbers, you can refer to resources on Imaginary Numbers.
