An approach to introducing complex numbers to high school students
In his 2010 paper Giorgio Bagni of the University of Udine wrote:
The introduction of imaginary numbers is an important step of the mathematical curriculum. It is interesting to note that, in the Middle School, pupils are frequently reminded of the impossibility of calculating the square root of negative numbers. Then pupils themselves are asked to accept the presence of a new mathematical object, “√ −1”, named i, and of course this can cause confusion in students’ minds. This situation can be a source of discomfort for some students, who use mathematical objects previously considered illicit and “wrong”. The habit (forced by previous educational experiences) of using only real numbers and the (new) possibility of using complex numbers are conflicting elements.
He proposes using the actual problem from the 1530s, that was then written up clearly with definitions and the method of solving it by Bombelli in 1572. His paper (Bagni, 2010) includes images of the text, and the historical technique is laid out here (Orlando Merino University of Rhode Island, 2006). See the 'History" link below for more details and resources.
The answer is real while the method goes via complex numbers, this presumably helped the original inventors to accept the procedure, though they remained reluctant. Bagni (2000) tested students for their acceptance of procedures using complex numbers and found that they were much more willing to accept the cubic proposed than the quadratic with a complex answer.
A lesson plan along the lines of:
- Start by describing a discovery, with names and dates
- offer this link, a lively story from a rich site, www.storyofmathematics.com
- The difficulty of believing it at the time may help encourage the students to consider their own doubts
- Define operations with this new number i, following Bombelli closely
- Take the historical rules and work through to the real number solution ...
- Use these rules to find (2 + i)3 = 2 + 11i and (2 - i)3 = 2 - 11i
- Use the procedure offered to get x = ³√(2 + 11i) + ³√(2 - 11i)
- Thus get x = (2 + i) + (2 - i) = 4
Clearly the definition involves the square root of negative numbers, but following immediately into a practical problem with a confirm-able real answer is the key here
This should lead to a general discussion …
About the proposition that i is something we "just made up" and after playing around with it, then it turns out to be both beautifully elegant and powerful in an abstract way, and rather useful in solving various problems as well as thinking about space in the (much!) longer term.