It’s a mathematical abstraction, and the equations work out. Deal with it.
It’s used in advanced physics, trust us. Just wait until college.
Focusing on relationships, not mechanical formulas.
Seeing complex numbers as an upgrade to our number system, just like zero, decimals and negatives were.
Using visual diagrams, not just text, to understand the idea.
Really Understanding Negative Numbers
Enter Imaginary Numbers
Visual Understanding of Negative and Complex Numbers
We can’t multiply by a positive twice, because the result stays positive
We can’t multiply by a negative twice, because the result will flip back to positive on the second multiplication
i is a “new imaginary dimension” to measure a number
i (or -i) is what numbers “become” when rotated
Multiplying i is a rotation by 90 degrees counter-clockwise
Multiplying by -i is a rotation of 90 degrees clockwise
Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers.
1, -1, 1, -1, 1, -1, 1, -1
x, -x, x, -x, x, -x…
(No questions here)
(Can’t do much)
(That’s what i is all about)
(Ah, 3 rotations counter-clockwise = 1 rotation clockwise. Neat.)
(4 rotations bring us “full circle”)
(Here we go again…)
X, Y, -X, -Y, X, Y, -X, -Y…
Understanding Complex Numbers
a is the real part
b is the imaginary part
A Real Example: Rotations
Multiplying by a complex number rotates by its angle
Original heading: 3 units East, 4 units North = 3 + 4i
Rotate counter-clockwise by 45 degrees = multiply by 1 + i
Complex Numbers Aren’t
Convince you that complex numbers were considered “crazy” but can be useful (just like negative numbers were)
Show how complex numbers can make certain problems easier, like rotations
Epilogue: But they’re still strange!
Other Posts In This Series
A Visual, Intuitive Guide to Imaginary Numbers