Complex Dot to Dot Puzzle

Algebra Level pending

Let $z=a+bi, z\neq0$

If you multiply $z$ by $i$ 4 times and plot the result on the complex plane after each multiplication, what shape do you get?

A dot to dot letter

i

A spiral It is random! It converges to

0

A square A Pascals triangle A circle

1 solution

Ved Pradhan
Jun 17, 2020

Multiplication by $i$ implies a 90 degree counter clockwise rotation. I recommend watching 3Blue1Brown's high school lecture on complex trigonometry (I think it's the second or third one) to get a clear explanation, but here is a quick one:

$i \times (a+bi) =$ $ai + bi^{2} =$ $-b + ai$

You should've have learned in your math class that switching the coordinates and multiplying the new first one by negative -1 should result in a 90 degrees counter clockwise rotation. If you haven't, it is super easy to prove. Just get out a piece of graph paper, draw a ray in the first quadrant, rotate it 90 degrees counter clockwise, and find the new coordinates.

Anyway, since we apply this transformation four times, we get four 90 degree rotations. And since the magnitude of the complex number is not changed (rotation is a rigid transformation), all lengths stay the same. Thus, the four complex numbers form a $\boxed{\text{Square}}$ .

Here is 3Blue1Brown's video on complex trigonometry: https://www.3blue1brown.com/videos-blog/complex-number-fundamentals-lockdown-math-ep-3

You can also watch it on YouTube or ItemPool. ItemPool doesn't seem to be working, but if you watch it there, you can also do interactive quizzes in the middle.

Ved Pradhan - 12 months ago

Complex Dot to Dot Puzzle

1 solution

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