Complex numbers
Lab: first steps with complex numbers
Connect a+bi to its point in the plane and discover when x²+c=0 requires non-real numbers. Every example has a statement and explanation.
Problem statement
Complex plane
In a+bi mode, tap the plane or drag the point. You can also use inputs or sliders. On the plane, arrow keys move by 0.25; Shift+arrow by 1; Home returns to the origin.
Read the number
The coefficient without i gives the horizontal displacement.
It is b, not bi: a real number giving the vertical displacement.
Coordinates (a; b)
The point and number represent the same object. A semicolon separates coordinates to avoid confusion with a decimal comma.
Classification
Build the point in three steps
The hollow circle marks the destination; the filled point marks the current construction step.
Solutions and explanation
Check
We seek every equation solution, not just the principal root. Irrational coordinates in the plot are approximate; the radical expression stays exact.