If i doesn't equal to r, various implications arise depending on the context, particularly in mathematics and physics. Here’s a detailed exploration of the consequences:

Mathematical Implications
- **Complex Numbers**: In complex number theory, if \( i \) (the imaginary unit) does not equal \( r \) (a real number), it emphasizes that they belong to different sets. \( i \) represents values on the imaginary axis, while \( r \) lies on the real axis.
- **Equations**: Consider the equation \( z = r + i \). If \( i \neq r \), the solutions represent distinct points in the complex plane, leading to unique interpretations and results in problem-solving.

Geometric Interpretation
- **Coordinate System**: In a Cartesian coordinate system, \( r \) represents a point along the x-axis, while \( i \) represents a point along the y-axis. Their inequality signifies they do not intersect, thus portraying different dimensions in graphical representations.
- **Vector Analysis**: If vectors are involved, \( i \) and \( r \) being unequal indicates they have different directions or magnitudes, influencing vector addition or scalar multiplication outcomes.

Physical Contexts
- **Electrical Engineering**: In AC circuits, if \( i \) (current) doesn’t equal \( r \) (resistance), it affects circuit behavior, leading to varying voltage and power calculations.
- **Thermodynamics**: In thermodynamic equations, unequal parameters may indicate different states of a system, impacting energy transfer and efficiency.

Conclusion
In summary, when \( i \) does not equal \( r \), it signals significant distinctions across mathematical, geometric, and physical domains, each leading to unique interpretations and consequences in problem-solving.