In the Argand plane, also known as the complex plane, the complex number $i = \sqrt{-1}$ can be interpreted geometrically as a rotation. Let's break this down step by step.

1. Complex Numbers and the Argand Plane

• The Argand plane is a way to visualize complex numbers. The horizontal axis (real axis) represents the real part of a complex number, and the vertical axis (imaginary axis) represents the imaginary part.
• A complex number $z = a + bi$ is represented as a point $(a, b)$ or as a vector from the origin to the point $(a, b)$.

2. The Number $i$

• The number $i$ is purely imaginary, with no real part. It corresponds to the point $(0, 1)$ on the Argand plane.
• Geometrically, $i$ represents a unit vector pointing upwards along the imaginary axis.

3. Multiplication by $i$

• Multiplying a complex number by $i$ corresponds to a counterclockwise rotation of 90 degrees (or $\frac{\pi}{2}$ radians) around the origin on the Argand plane.

  • If you start with $1$ (which is the point $(1, 0)$ on the Argand plane), multiplying by $i$ gives $i$, moving the point to $(0, 1)$, which is a 90-degree rotation.
  • Multiplying $i$ by $i$ (i.e., $i^2$) results in $-1$, which corresponds to a point at $(-1, 0)$, another 90-degree rotation.
  • Multiplying $i^2$ by $i$ (i.e., $i^3$) results in $-i$, which corresponds to a point at $(0, -1)$, yet another 90-degree rotation.
  • Finally, $i^4 = 1$ returns to the original point $(1, 0)$, completing a full 360-degree rotation.

4. Powers of $i$

• The integral powers of $i$ follow a cyclical pattern every four steps:
  • $i^1 = i$ (90-degree rotation from $1$)
  • $i^2 = -1$ (180-degree rotation from $1$)
  • $i^3 = -i$ (270-degree rotation from $1$)
  • $i^4 = 1$ (360-degree rotation, back to the start)
• This cycle repeats for higher powers of $i$:
  • $i^5 = i$
  • $i^6 = -1$
  • And so on...

5. Geometric Interpretation Summary

• $i = \sqrt{-1}$: A 90-degree counterclockwise rotation in the Argand plane.
• $i^2$: A 180-degree rotation.
• $i^3$: A 270-degree rotation.
• $i^4$: A 360-degree rotation, bringing the point back to its original position.

This geometric interpretation illustrates how multiplication by $i$ operates as a rotation in the complex plane.

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Would you like more details or have any questions about this explanation?

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Related Questions

1. How does the geometric interpretation of complex numbers help in understanding Euler's formula?
2. What is the significance of the Argand plane in visualizing complex functions?
3. How do the rotations by $i$ relate to the concept of complex conjugates?
4. Can you explain the geometric interpretation of complex roots in the Argand plane?
5. How is the magnitude of a complex number represented in the Argand plane?
6. What happens when you multiply a complex number by a real number in the Argand plane?
7. How does the concept of rotation by $i$ apply to solving quadratic equations with complex roots?
8. What is the importance of the Argand plane in electrical engineering?

Tip

Remember, the Argand plane is a powerful tool for visualizing complex number operations, turning abstract algebraic processes into concrete geometric actions like rotations and scalings.