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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
4 years, 1 month ago

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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. Leonardo Cidrão · 4 years ago

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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. Mursalin Habib · 4 years ago

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@Mursalin Habib 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. Calvin Lin Staff · 4 years ago

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@Calvin Lin 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. Mursalin Habib · 4 years ago

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I think yes....as a real no. Can be written as r+i0.... Where r is the real part of the no. Dipanjan Mondal · 4 years ago

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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. Noel Quirol · 4 years ago

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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. Leonardo Cidrão · 4 years ago

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@Leonardo Cidrão 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. Calvin Lin Staff · 4 years ago

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@Calvin Lin 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. Tim Vermeulen · 4 years ago

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@Calvin Lin 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". Leonardo Cidrão · 4 years ago

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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. Tanishq Aggarwal · 4 years, 1 month ago

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