# Are real numbers also complex numbers?

To me, all real numbers $r$ are complex numbers of the form $r + 0i$.

Are there any countries / school systems in which the term "complex numbers" refer to numbers of the form $a+bi$ where $a$ and $b$ are real numbers and $b \neq 0$?

I've been receiving several emails in which students seem to think that complex numbers expressively exclude the real numbers, instead of including them. Note by Calvin Lin
6 years, 2 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

I can't speak for other countries or school systems but we are taught that all real numbers are complex numbers. The set of real numbers is a proper subset of the set of complex numbers. Complex numbers are numbers in the form $a+bi$ where $a,b\in \mathbb{R}$. And real numbers are numbers where the imaginary part, $b=0$.

To avoid such e-mails from students, it is a good idea to define what you want to mean by a complex number under the details and assumption section. Whenever we get a problem about three digit numbers, we always get the example that $012$ is not a three digit number. So you can do something like that.

- 6 years, 2 months ago

Indeed. I have a standard list of definitions for less-known terms like floor function, factorials, digit sum, palindromes. I'm wondering about the extent to which I would expand this list, and if I would need to add a line stating

The real number $r$ is also a complex number of the form $r + 0i$.

While this looks good as a start, it might lead to a lot of extraneous definitions of basic terms. I also get questions like "Is 0 an integer? Is 1 a rational number?". If I also always have to add lines like

0 is an integer.
0 is a rational number.
1 is a rational number.

, then the details and assumptions will be overcrowded, and lose their actual purpose.

Staff - 6 years, 2 months ago

I have a suggestion for that. How about writing a mathematics definition list for Brilliant? Then you can write something like this under the details and assumptions section:

"If you have any problem with a mathematical term, click here (a link to the definition list)."

I know you are busy. But I think there are Brilliant users (including myself) who would be happy to help and contribute.

You can still include the definitions for the less known terms under the details section.

- 6 years, 2 months ago

I agree with you Mursalin, a list of mathematics definitions and assumptions will be very apreciated on Brilliant, mainly by begginers at Math at olympic level.

- 6 years, 2 months ago

I think yes....as a real no. Can be written as r+i0.... Where r is the real part of the no.

- 6 years, 2 months ago

Hmm. I've always been taught that the complex numbers include the reals as well. But then again, some people like to keep number systems separate to make things clearer (especially for younger students, where the concept of a complex number is rather counterintuitive), so those school systems may do this.

- 6 years, 2 months ago

If your students keep misunderstanding this concept, you can create a kind of nomenclature for complex numbers of the form a + bi ; where b is different from zero. In addition, a similar thing that intrigues me like your question is the fact of, for example, zero be included or not in natural numbers set.

- 6 years, 2 months ago

There isn't a standardized set of terms which mathematicians around the world uses. Often, it is heavily influenced by historical / cultural developments.

There is disagreement about whether 0 is considered natural. For that reason, I (almost entirely) avoid the phrase "natural numbers" and use the term "positive numbers" instead. However, it has recently come to my attention, that the Belgians consider 0 a positive number, but not a strictly positive number. This might mean I'd have to use "strictly positive numbers", which would begin to get cumbersome.

Staff - 6 years, 2 months ago

I have not thought about that, I think you right.

That is an interesting fact. However, in my opinion, "positive numbers" is a good term, but can give an idea of inclusion of the zero. Futhermore, the most right term would be "positive and non-null numbers".

- 6 years, 1 month ago

I've never heard about people considering $0$ a positive number but not a strictly positive number, but on the Dutch IMO 2013 paper (problem 6) they say "[…], and let $N$ be the number of ordered pairs $(x,y)$ of (strictly) positive integers such that […]". The word 'strictly' is not mentioned on the English paper. Note that Belgians living in the northern part of Belgium speak Dutch.

- 6 years, 1 month ago

in our school we used to define a complex number sa the superset of real no.s .. that is R is a subset of C.

- 6 years, 2 months ago