# How common is the use of the term "primitive" to mean "antiderivative"?

I don't know if this should actually be asked on the English stackexchange. It seemed like I would find better answers here.

I have all but finished an undergraduate degree in mathematics in the United States, but I have never once heard the term "primitive" to mean "antiderivative" until recently, when someone from Europe pointed it out to me. According to him, it's a common term there. So I was wondering if people could give me an idea of how common this term is, and where. I know for sure that if someone says "primitive" to a math student in the US, that the student won't know what he is talking about. Does the reverse hold for "antiderivative" (or the also common "integral") elsewhere?

• Spivak uses the term in his Calculus, which I think is quite popular in the US. Jun 26 '11 at 3:49
• In Romania, from where I am from, the term primitive is used. I thought at first that the term primitive is not used in English, but as mentioned, there are books in English which use it. Jun 26 '11 at 11:11
• I'm thinking that the standard term was "primitive", but at some point an American textbook writer invented the term "antiderivative", which has gradually become the most popular in calculus textbooks. Jun 26 '11 at 12:46
• Re: last question. I think antiderivative is pretty self-explanatory (even if I find the word a bit ugly and unnecessarily complicated). Even if I had never heard it before, I would immediately have thought of an integral. (my native language is German and there Stammfunktion is the common term).
– t.b.
Jun 27 '11 at 2:15
• Here in Argentina (spanish language) the term "primitiva" is used, almost exclusively. Jun 27 '11 at 2:42

Apostol's "Calculus" volume 1 uses "primitive" in that sense 41 times, whereas "antiderivitive" is only used 4 times. Two of those are in the main text, always as part of an "or antiderivative" after the term "primitive", and the other two are in the index.

• Upvoted for cunning use of ctrl-F.
– barf
Jun 26 '11 at 11:59

In complex analysis the antiderivative is often refered to as the primitive.

• Ah, that may be. I've yet to take complex analysis (I will have it next semester). Jun 26 '11 at 5:58
• I checked the book I will be using, "Complex Variables, Second Edition" by Ablowitz and Fokas, and "antiderivative" is in the index twice, but not "primitive". Furthermore, I used Google Books to search for the word "primitive" within the book, and it found no results (but four results for "andiderivative"). So I think you are wrong about it being used universally in complex analysis. Jun 27 '11 at 4:09
• i said often used. not universally. see ahlfors or conway Jun 29 '11 at 3:21
• I've since taken two semesters of complex analysis and my professor never used the term (anti-derivatives aren't even that important in the field anyway). Jun 17 '12 at 3:07

I think this is due to Richard Courant. In his 2 volume book Differential and Integral Calculus, he uses the term 'primitive' to mean antiderivative.

• It's certainly a lot older than Courant. See jeff560.tripod.com/mathword.html which traces "primitive" back to Lagrange in 1797. For "anti-derivative" the first reference they have is 1903. Jun 26 '11 at 14:25

I can just say that "the primitive function" (primitivní funkce) is the only official name of this object in Czech. Also, I think that the most standard English term is not "antiderivative" but rather "indefinite integral".

http://en.wikipedia.org/wiki/Antiderivative

• "Integral" is used more often (ambiguously), but the rigorous usage always uses "antiderivative". Rigorously, I think the difference might be that "indefinite integral" refers the the family of all possible antiderivatives (F(x) + C), whereas "antiderivative" refers to any one of those (F(x) is "an antiderivative" of f(x)). Jun 26 '11 at 5:57
• Though I don't think it's exactly standard, I like the use (saw this in Apostol, I think?) of "indefinite integral" to specifically mean $F(x)=\int_a^x f(t)dt$ for some f and a. Of course this turns out to be the antiderivative when f is continuous, but... Jun 26 '11 at 13:53

In the Dutch language, the antiderivative is known as the "primitieve".

• I know it as "primitieve". Jun 26 '11 at 10:04
• While we're at what it's called in other languages: In Danish it's called "stamfunktion" as in "f stems from F" Jun 26 '11 at 10:28
• In French it's only called "primitive" too, there's no equivalent for "antiderivative". Jun 26 '11 at 10:53
• @Jonas, You are right, that is emberassing Jun 26 '11 at 21:56
• @kahen: (Almost) the same in German: Stammfunktion
– t.b.
Jun 26 '11 at 22:36