Complex numbers in high school

Are complex numbers taught in high school in other countries? I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for example.

I guess complex numbers also won't be discussed in high schools in other countries and are simply neglected and never mentioned. However, I think it is important to at least let students know that complex numbers exist. I know that discussing complex numbers in full detail would require a lot of time which is often not available. Yet, without ever mentioning complex numbers, a teacher is leaving out an important piece of information. For example, when students are learning to find roots of polynomials they will usually be searching for them in the "wrong" set ($\mathbb{R}$). Very often examples are picked where a polynomial will have only real roots which guarantees the teacher to avoid the complex numbers. However, I think that it is also important to make students aware that things might not always be as easy as in the examples picked by the teacher and that a polynomial might very well have complex roots. The fundamental theorem of algebra, for example, is a theorem which is very easy to state and to discuss in school. The theorem will require some knowledge about the existence of complex numbers, yet, I think it is well worth discussing in high school: as the theorem's name suggests: it is fundamental to algebra.

What are your thoughts about the topic? Should complex numbers be discussed in high school? Or are there even countries where they are part of the curriculum?

• "I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for example." Are you sure about that? It seems to me they are in some curricula; and if not then they got removed only very recently. – quid Dec 29 '16 at 22:00
• In some curricula it is indeed mentioned as a possible topic. However, I am not aware of any state in Germany where it is actually obligatory for the teacher to discuss the topic. – YukiJ Dec 29 '16 at 22:57
• Complex numbers are taught in Greece as well, in the last year(s) of high school. It used to be a mandatory subject during some years, then it was only for students aiming for math/science/technical universities. Not sure about now, I think it's mandatory again. – ypercubeᵀᴹ Dec 30 '16 at 9:53
• Well, they are taught here in Italy in the last year of high school, but of course not very thoroughly. The Fundamental Theorem of Algebra is mentioned, and students learn to manipulate expressions and solve simple equations; the exponential form is mentioned but not explained or used extensively. – Riccardo Orlando Dec 30 '16 at 10:08
• I come from BaWü, Germany. First question would be: Is "high-school" equal to "Realschule" or to "Gymnasium"? If Gymnasium: When I went to school you had to select two subjects (so-called "Leistungskurse"). If one of the two subjects was maths complex numbers were subject at school for you. If you selected two other subjects (e.g. physics and biology) you heared nothing about complex numbers at school. (In Realschule you didn't hear about complex numbers, either (of course).) – Martin Rosenau Dec 30 '16 at 12:38

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! :)

• I don't disagree with any of this, and I think it is a good complement to my answer. I think it is quite correct to say that although students are taught to work with numbers are taught in US high schools, the treatment is rather superficial, and students come out of it believing that they are nothing more than "make believe" (a situation which is undoubtedly exacerbated by the unfortunate terminology history has saddled us with). – mweiss Dec 30 '16 at 1:43
• Yep. Also, high schools don't seem to teach students the geometrical interpretation of complex multiplication. All they learn to do is multiply out (a+bi)(c+di). – Ben Crowell Dec 30 '16 at 4:47
• @BenCrowell: The geometrical interpretation of complex multiplication is indeed taught in the precalculus textbook I teach from. That is part of mweiss's statement "they typically also learn the representation of complex numbers as points in a plane." I agree with this answer that the significance of complex numbers is not well explained, and I take time to do this when I teach Algebra 2. – Rory Daulton Dec 30 '16 at 12:58
• @RoryDaulton: Good for you, I'm glad you do such a rigorous and in-depth treatment of complex numbers. However, I think it's wildly unrealistic to leave the German OP with the impression that this is the standard mathematical experience of US secondary school students. Most get their high school diploma with at best a shaky understanding of basic arithmetic. My statement about students' lack of understanding of the geometrical interp. of multiplication is not a guess. It is based on experience with community college students from a variety of local high schools, good and bad school districts. – Ben Crowell Dec 31 '16 at 2:55
• @paul "There is no mention of Euler's identity expressing sine and cosine in terms of complex exponentials". WHAT. Dealing with complex numbers and not showing this jewel? It's the obvious choice for getting students interested in complex numbers, but apparently it is "too complicated for students to grasp". Of course, they can't if teachers don't explain the intuition behind it. – Jaime Gallego Dec 31 '16 at 19:49

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, if they continue on to a Precalculus course, they typically also learn the representation of complex numbers as points in a plane.

• Wow, this is great. I wish this was also the case in Germany. It's sensible that the exercises you mentioned are chosen in such a way that the rational roots theorem can be used as it can be exhaustive work to find all $n$ roots of an arbitrary polynomial of degree $n$ by hand. Thanks for the good news. :-) – YukiJ Dec 29 '16 at 21:52
• Also perhaps worth mentioning is that a knowledge of basic arithmetic operations with complex numbers is expected for the (P)SAT and ACT. – Benjamin Dickman Dec 30 '16 at 2:09
• As a reference, the U.S. Common Core standards on complex numbers: corestandards.org/Math/Content/HSN/CN – Daniel R. Collins Dec 30 '16 at 2:46
• When you say "standard," what standard are you referring to? The next year, in a Precalculus course, they typically also learn the representation of complex numbers as points in a plane. Wait, what percentage of US high school students take a precalculus course? Frankly, I think this answer describes some kind of educational dream world, not the real world of US K-12 math education. Most of my community college students are shaky on percentages and fractions. – Ben Crowell Dec 31 '16 at 2:44
• @BenCrowell Algebra 2 is a required course for HS graduation for roughly half of all students in the country. For those who go on to a follow-up course, Precalculus is the usual next course. I think teaching in a community college there is a selection bias effect -- you are much more likely to encounter students who were not successful in high school mathematics, but that does not mean your students are representative of the population at large. – mweiss Jan 1 '17 at 0:22

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:

(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!)

1. 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 $$w \in \mathbb{R}$$.

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

3. 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 $$z_2$$.

4. 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.

• my wife took advanced level math in her Hong Kong highschool education. She solved the cubic and quartic, was taught some of the trig. substitution solution stuff, but, that might just be an isolated event with an excellent instructor. The larger thing, they are taught how to write math neatly with an expectation of good handwriting. It seems we are not allowed to be critical of sad handwriting. Perhaps if elementary school teachers made more a point of fixing it there would be hope... I seem to have digressed with my original anecdote, oh well. – James S. Cook Jan 3 '17 at 6:29

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

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.

• What country is this from? What textbook? – mweiss Dec 30 '16 at 17:30
• Good point. Adding that. – wander95 Dec 30 '16 at 18:11

In most Israeli high 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.

I am a computer engineering student from Egypt.

In Egypt, you choose first what branch you want to study in the high school: maths, science (biology), or arts (history, philosophy, etc). Complex numbers are taught to 3rd year math students (the year just before the university). You choose this branch only if want to study engineering, computer science, mathematics, etc.

As an engineering student, I find complex numbers very important. They're used in engineering maths and physics. They're heavily used in, for example, AC circuit analysis, and Control theory. Fourier series and transform are used a lot in signal processing courses.

Complex numbers are a must for many engineering departments.

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

• Not quite correct. Complex numbers are part of the FP1 module (then developed in FP2 and 3), and so are only taught to those who opt for the Further Mathematics A-level. This is not offered at all schools, and even in those schools that do offer it, is is a small minority who take this course – James K Dec 31 '16 at 17:54

I'm a college student in China. As Benjamin has mentioned, complex numbers are taught in China. But in a rather implicit way.We are taught that they exit, but we don't have enough background knowledge to understand the reason of introducing such an weird "i"...lol.What we're taught is to do the calculation as accurate as possible as well as understanding and using the geometric property of the complex plane which is frequently associated with vectors.

What I'm trying to state is that it's hardly possible to dipict an explicit picture of complex numbers to senior high students.Think about this question: If you just tell them that for some practical reasons, we have to define an i as the sqrt of -1,what if they ask you why we can't define a number which yield 1 when it's multiplied by 0?How will you answer that question?Such thing did happen,and as far as I am concerned,the teacher simply can't answer that question.In fact it's not until that I went to college that I know the answer to that using abstract algebra.

So there're some problems when you introduce such a weird thing to students,especially curious ones.My point is:It's good to introduce it,but we have to figure out a better way to define it well.

• This is probably more of a comment than an answer. If you think that it belongs under my post, then maybe you can put an abbreviated version of it there. – Benjamin Dickman Jan 1 '17 at 20:56
• @BenjaminDickman I mentioned your answer as a reference...I'm new here,such reference is proper in some similar communities back in China.My apology if it doesn't meet the regulations of SE, I'm still getting used to it.Just now,I've tried to add an comment,but I guess my SE reputation is still not enough to do that.(ー_ー)!! – Elementary Sherrinford Jan 2 '17 at 4:51