complex numbers

Help, if you don’t mind!

I’d like to flesh out our collection of complex numbers mistakes. If you’re teaching this topic in the next few months, could you send me some pictures of mistakes? I’m looking for all sorts of mistakes — either common or uncommon errors.

In the meantime, would you take a second and comment about a complex numbers mistake that you’re used to seeing from your students?

When you’re done with that, check out some of the new mistakes that I just posted…

Thanks for all the great mistakes and comments lately, guys. Keep up the great work!

  • There’s the paradox:

    1 = sqrt((-1)*(-1)) = sqrt(-1)*sqrt(-1) = i*i = -1

    Not really a mistake, but rather just something to ponder.

    • Jay

      I am pretty sure this is a question about a question about the primary square root, which is defined as such to prevent this confusion. Since the square root is a function, it can only have one result. The question you’re pondering is solving the equation x^2=1 vs. the function of y= sqrt(1). To get even more complex (HA!), consider 1 as the complex number (1+0i) = 1e^(0i) in Euler form. Thus the square root of 1 can be written as

      (1+0i)^(1/2) = [1e^(0i)]^(1/2)
      = [1]^(1/2) e^[(0i)(1/2)]
      = 1^(1/2)e^0
      = 1^(1/2)
      = 1

  • Shelley

    Last year about 10 of my students treated all complex number problems on our quiz as if they were a binomial times a binomial, so
    (5+i)(5-i), (5+i)+(5-i), (5+i)-(5-i) all had an answer of 26.
    Not sure if this is a common mistake, or just me not checking for understanding enough prior to the quiz. Either way, I won’t ever forget that misconception when I teach this concept in the future!

  • One mistake that I see is when students divide two complex numbers, they cancel out the denominator completely, instead of just rationalizing the denominator.

    For example: (3+2i)/(5-4i) = (3+2i)(5+4i)/(5-4i)(5+4i) = 15 +12i +10i – 8 = 7+22i rather than (7+22i)/34

  • Julie

    One mistake that I see this student making is seeing 12 and 3i as the same terms, but they are not because they are two separate terms. Only the same terms can be combining together. The answer should have been 14 because 12 + 2 = 14 and 3i -3i = 0.