Complex numbers in high school

In the United States, complex numbers are a standard part of the typical high school Algebra 2 course. Students (normally in grades 10 or 11, corresponding approximately to ages 15-17) learn to add, multiply, and divide complex numbers; to solve quadratic equations with no real roots; and to find all $n$ roots of an $n$th degree polynomial (usually, carefully chosen so that the rational roots theorem gets you most of the way, and the quadratic formula gets you the rest of the way). They also learn the statement of the Fundamental Theorem of Algebra. The next year, in a Precalculus course, they typically also learn the representation of complex numbers as points in a plane.

Although there is a tradition in the U.S. of nominal mention of complex numbers in the high school curriculum, in my observation it is invariably superficial, and complex numbers are not mentioned subsequently. In particular, there is no mention of Euler's identity expressing sine and cosine in terms of complex exponentials, and, therefore, no mention of the palpable fact that all trig identities are deducible "by algebra" from properties of the exponential function (which itself is not treated in any way that would allow understanding of Euler's identity, unfortunately).

Since high schools in the U.S. do not mention solvability of cubics in radicals, there will be no mention of the "irreducible case", where the discriminant is complex, despite there being three real roots. So the quadratic story can easily be dismissed (in the minds of the students, as in the minds of many over the centuries) by saying that in some cases there simply "are no roots" (meaning no real roots).

That is, both in my direct recollection from decades ago, and from observation in more recent times, kids in high school are not led to take complex numbers seriously, but only to view them as yet one more menu-item on the laundry list... Sort of an "option", or something that might be on the final, ... but is not genuine or manifest in the world.

In fact, mathematics grad students at my R1 university tend to default into a similar attitude, that "complex analysis" is just a pre-specialty thing that presents a hurdle to them, for who-knows-what reason, but is not relevant to much of anything. This is dismaying... as are many things to cranky old people, I guess! :)

Complex numbers are also taught in China. I observed a mathematics classroom in Nanjing, Jiangsu Province, China in 2008-09 for students in their penultimate year of secondary school (their high schools have 3 years, whereas U.S. high schools have 4 years) and the end of the book discussed complex numbers. This was a standard public school in a second-tier Chinese city.

The specific book they used, in case you want to track down a physical copy, was the 2005 edition of:

普通高中课程标准实验教科书

选修2-2

(The remaining parts are my own translations.)

Section 3 is entitled, Generalizing Number Systems and an Introduction to Complex Numbers.

Its three subsections are (pp. 103-118):

3.1 Generalizing Number Systems

3.2 Arithmetic Operations with Complex Numbers

3.3 Geometric Interpretation of Complex Numbers

Below are three pictures from the textbook (of which I have a copy). The first is of the introduction; the second is of four sample problems (that I will translate); and the third is a fleeting mention of Euler's Identity (which paul garrett laments is lacking from many US secondary classes).

Intro:

Problems:

(Translation errors my own!)

6. Suppose $z$ is an imaginary number, and let $w = z + \frac{1}{z}$. Prove: $|z| = 1$ is both a necessary and sufficient condition to conclude that $w \in \mathbb{R}$.

7. Given $z_1, z_2, \in \mathbb{C}$, $|z_1| = |z_2| = 1$, $|z_1 + z_2| = \sqrt{3}$, compute $|z_1 - z_2|$.

8. Given an ellipse with major axis of length $2a$, and the coordinates of its foci $F_1$ and $F_2$ at $z_1$ and $z_2$, respectively, pick any point $z$ on the ellipse, and write out the relationship satisfied by $z$, $z_1$, and $z_2$.

9. Let $z$ be an imaginary number, and let $z$, $\bar{z}$, $\frac{1}{z}$ correspond to the vectors $\vec{OA}$, $\vec{OB}$, and $\vec{OC}$. Write out:

(1) the relation between $\vec{OA}$ and $\vec{OB}$;

(2) the relation between $\vec{OB}$ and $\vec{OC}$.

Euler's Identity:

These are essentially historical notes (the chapter begins with a quotation from Leibniz and a mention of Cardano) and do not have corresponding problems. I am not sure whether Euler's Identity would be pursued in the students' next, and last, year of secondary school mathematics.

In Norway, it's part of the curriculum for the optional high school course "Matematikk X", together with some elementary number theory and probability with continuous distributions (especially the normal distribution). Unfortunately, it's not a course many take, since if you do, it will still come on top of the normal mths courses, and most people who take math prioritize physics and chem

In Italy complex numbers are commonly taught in science and technically oriented high schools, like the liceo scientifico or the various types of technical institutes (e.g., on electronics, electrotechnics, etc.).

For the technical institutes, the teaching of complex numbers if fundamental for the understanding of circuit theory in the AC regime (phasors).

Complex numbers are usually taught during the 3rd year (16-year old students), and common topics are:

1. Representations of complex numbers
2. Operations on complex numbers
3. Euler's identity
4. Powers and roots of complex numbers
5. Complex exponential
6. Complex logarithm

In most Israeli hign schools, mathematics can be taken on one of 3 levels. The highest level, known as "5 units" and taken by about 10% of students in the country (generally those planning a career in science/engineering) includes the topic of complex numbers.

These students are then tested on complex numbers in the matriculation exam (bagrut), but the questions are usually quite easy (compared to the other topics on the exam), requiring not much beyond addition and multiplication of such numbers, and being able to convert between the rectangular and polar forms with a bit of algebra and trigonometry.

For example, the 2016 summer bagrut (807 questionnaire) includes the following section (comprising a sixth of the questionnaire's total marks):

Given the complex number $z$, $$z = \frac{(\cos\frac{\pi}{9}+i\sin\frac{\pi}{9})^3}{(\cos\frac{\pi}{12}-i\sin\frac{\pi}{12})^2}$$

(1) Find $|z|$ and the argument (angle) of $z$.

(2) Find all values $n$ ($n$ is a natural number) for which $z^n$ is a purely imaginary number.

Here is a screenshot from the 11th grade textbook Chapter 6 in India. Sometimes complex numbers are also needed for some physics problems so students actually needed to know complex numbers a little earlier.

In my opinion, the syllabus has been simplified and math reduced compared to when I was a student. The following topics are included in the CBSE class XI syllabus quoted from here:

2. Complex Numbers and Quadratic Equations

Need for complex numbers, especially √1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system. Square root of a complex number.

Based on this knowledge, students appear in entrance examinations for engineering colleges roughly after two years of learning above. The following is a sample question that they must answer in less than a minute.

In the United Kingdom complex numbers are taught to pupils who stay in school from age 16 to 18 to do their GCE A or AS levels in maths. This may help: https://www.heacademy.ac.uk/system/files/pre-university-maths-guide.pdf