Calculus vs calculation


Hi, sorry if this not the right place to post such question, but I'm Italian, and I want to know from English-speaking people:

What's the difference between calculus and calculation?
What's the right plural for both?

I ask this because two dictionaries I consulted give different definitions and different plurals!

"Ask the source, so."

Thanks in advance.

Antonio Maschio


Hi Antonio,

I'm not a native speaker either but let me explain what I believe is correct.


[pl. calculations]

A calculation deals with numbers, they are summed, multiplied, etc. A calculator is a tool to do such operations. In business terms, a calculation is done in advance to know if a businiss will be profitable, or afterwards, to check if it was. The TVM formulas built into the business calculators are typical applications for calculations.


[no plural]

Calculus and Algebra are two main areas of math. Both deal with formulas more than with numbers. Calculus deals with the properties of functions, their derivatives and integrals. A central term is the limit of some term where one of the parameters goes towards infinity or zero. Two simple examples follow.

lim --- (A)
x->oo x

lim ---
x->0 x (B)

(A) The more you increase x, the smaller the quotient 1/x will be. But how big the value for x may be, the result of 1/x is never zero. Calculus now deals with the case that x is actually bigger than any number and defines the limit to be zero.

(B) If you divide x2 by x the result is always x, isn't it? The answer is no, because x can be any real value, including zero. The problem: 02/0 cannot be computed that way! The term is undefined for x=0 but calculus can still deal with it. As long as x is not equal to zero, x2/ x can be simplified to just the term x. Now make x as small as you like. The limit is zero.



Very very clear!


(Europeans are not second to anyone!)

-- Antonio


These two definitions are totally correct. However, there are two further definitions for "calculus". It also can be loosely used to mean "calculation", and most often, if used, in the sense of a plan, or scheme, or plot: "all the events that came about fitted well with his devious calculus on this operation"... come to think of it, "calculation" sometimes also has this meaning!

But it ultimately derives from its original Latin meaning of a stone or rock, which was used in ancient times to help one in... calculating! I believe some medical people might sometimes still refer to a hard mass as a calculus.


Right, calculus has this meaning too (hard mass in the medical sense) even in Italian.

But do you think I can use both term equivalently, in the sense of "series of operation with numbers of variables, formulas or whatever"?

-- Antonio


No, they are not equivalent. It depends on what you are trying to say. There are connotative layers of meaning as well as what is in the dictionary.

And don't forget the word "compute" as opposed to "calculate." These words are close and often interchangeable; yet one might be better suited than the other in certain circumstances.


Yeah, no one ever got complimented or reprimanded over dirty or illegal computations, but for their "cold calculation" or "shifty calculus" (although that could mean keying in an integral or differential equation on your HP calculator)!


Don't be a native English speaker either, but IMO "calculus" corresponds to "Analysis" in German, at least to the part dealing with differentiation and integration. There is also an English word "analysis". It seems "calculus" is a part of mathematical "analysis" in English, too.


for an american child, his or her first exposure to that word, "Calculus" is likely to be some time in middle school when he learns that after he learns Algebra and geometry, he will then have to take a course in analytical geometry (sometimes called trigonometry), and then "calculus." Most students regard this transition with trepidation. Introductions to calculus have been made to him in physics or analytical geometry: the concepts of asymtote and infinity was strange enough, but now he will have to understand the concepts of "infinetesimal" and "limit". And so "calculus" has a certain connotation in american language of transcendance, (and not merely of the transcendentals either ;-)and higher learning. It certainly has analysis as a part of it, but it is by no means a synonym. Sometimes the word "calculus" is used in literary contexts to get at the idea of a reasoned and difficult or rather non-apparetn thought process.

"Calculation" on the other hand is a word that one is exposed to vbery early, as in "something that a calculator does." It has a connotation of "mechanical" or uniform, straightforward, work with numbers, yet even "calculation" has an air of the intellectual or brainy about it, in some circumstances.

Here are some sentences that this native speadker of american english has produced on the fly:

[pre]"When the onboard computer failed, astronaut J.S. Bach calculated his trajectory by hand."

Note that it would have worked essentially the same had I said:

[pre]"When the onboard computer failed, astronaut J.S. Bach computed his trajectory by hand."

Edited: 6 Mar 2006, 5:13 p.m.


Yes, analysis is another branch of mathematics. Basically, (as far as I understand it), analysis is the math of formally proving calculus. So rather than taking integrals, you prove limits, and directly apply definitions of calculus to yield theoretical results. Analysis is mainly theory, while calculus is plugging and chugging, so to speak.


Hi Ben! I hear you are in beantown! Hope you like it there...we are not so far from there.

Have you figured out all the nuances of the veritable 15c yet?




Antonio --

There have been some good responses to your question, but I'll "chime in", anyhow...

Calculation is the process of performing mathematical computations using known quantities. It is what calculators do.

The Calculus are special analytical methods of symbolic mathematics, developed primarily by Newton and Leibniz, to determine the solutions to many kinds of computational problems. These methods involve use of infinitesimal quantities known as differentials.

Represented graphically, these problems include the slope at any point of a curved line (differential calculus), and the area over some interval between a straight axis and a curved line (integral calculus).

Of course, these two problems are inter-related by the Fundamental Theorem of Calculus.

Limit analysis is an area of symbolic mathematics that utilizes algebra and techniques of calculus (e.g., L'Hopital's Rule).

Wikipedia has some informative entries in English, Italian, and other languages, as well.

-- KS



the reason why I used the limit calculation as a calculus example is my own experience at school. In the physics class we were dealing with speed and acceleration before the topic of differentials had been discussed in the math class. So the physics teacher chimed in and explained what it means to compute the speed of a moving object in a moment of time by letting the time interval become infinitely small.



YEP, same thing happend with me. Honors Physics in 11th grade; took Calculus in 12th grade.


Thanks very much to everyone. Now the concept is clear to me (despite my *bad* dictionaries).

Hope I'll find a way to be equally useful for any question on your side.


-- Antonio


Hee hee hee (evil laugh).

I gave my son (teenager) a HP-39G; you know, the one with IR transmission capability. I can try to amplify its transmitter power and if you have one, too, so that on his next Italian test, you can beam over the answers. I think that calculator also has alphanumeric capability as well. Unfortunately, it may not have all the grammatical marks necessary for Italian! ;)

(I guess I fully understand why Europe only sells the HP-40G, which has a nice CAS, but has a disabled IR function.)

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