Beyond the number line: complex numbers
Discover a new type of number that is outside previously known number systems.
In this section, a new type of number is necessitated and then defined collaboratively. By hinging their understanding on their previous knowledge of numbers and then posing a question that doesn’t fit into that system, students hit a point of disequilibrium where they need to define imaginary numbers and then seek to understand how they behave, particularly taking note of patterns that emerge with them.
WHAT: This task serves as a possible first student exploration after an initial introduction to the form and arithmetic of complex multiplication. Students need to understand that every complex number can be expressed in the form $a+bi$, and understand multiplication of complex numbers at least well enough to compute, for example, $i^3= i^2\cdot i = −1\cdot i = −i$ (and understand that this is in the form $a + bi$). The task also (optionally) provides an instance where the formula of a finite geometric series can be used to explain an experimentally observed result N-CN.A.1.
WHY: The task is an excellent example of MP.8, look for and express regularity in repeated reasoning, as students are led to make conjectures about patterns based on experimental calculations. Students can then practice their algebraic manipulation in justifying their observation, a potentially very satisfying academic endeavor.
WHAT: In this activity students are introduced to complex numbers by first placing real numbers on a number line and then reading through a story where imaginary numbers are introduced by asking the question, “What number, times itself, gives -1?” The story continues to define imaginary and complex numbers N-CN.A.1.
WHY: This activity brings about the disequilibrium necessary to define the new number $i$. Though the story is quirky, it asks important questions in an engaging context. By the end of the story (and possible intermittent class discussions) the students will have defined both i as imaginary and $a + bi$ as a complex number and have reasoned through the placement of these on a number line (complex plane) MP.8.