Wednesday, October 12, 2005

Mathematics and Arithmetic

Contagion asked, "What exactly is the difference between mathematics and arithmetic." There's a short answer and a longer answer. Since I live to ramble on about math, this is the longer answer.

Arithmetic is simply the calculation/computational discipline. In terms of my particular interest in mathematical cognition, I consider arithmetic one of the building blocks necessary to mathematics.

Mathworld (a really great online source for short answers to math questions and just fun reading) defines arithmetic as "the branch of mathematics dealing with integers or, more generally, numerical computation. Arithmetical operations include addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, and subtraction. " (

Balancing your checkbook is arithmetic. Having a degree in mathematics does not mean that I'm any better (probably significantly worse) at that than the average person.

Mathematics is more difficult for me to define because it covers many areas so back to Mathworld for a definition. "Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined." ( Things like logic, which is essential to proofs, figure prominently in the study of mathematics.

Mathworld's main index lists the following topics and serves as a fairly good example of what is included in mathematics: Algebra, Applied Mathematics, Calculus and Analysis, Discrete Mathematics, Foundations of Mathematics, Geometry, History and Terminology, Number Theory (my personal favorite), Probability and Statistics, Recreational Mathematics, and Topology. No arithmetic. The foundations of mathematics section includes axioms, logic, set theory, and theorem proving. Again, no arithmetic.

If you look at the history of mathematics, different maths were developed to serve the needs of people. All mathematics through calculus were application driven and strongly tied to arithmetical calculations. Early in the 19th century, math went crazy (probably too much math happy juice) and new maths were developed for their own sake. It happens that many of these "pure" maths have applications in today's world, but the applications came later and are only of interest to applied mathematicians (or scientists, engineers, etc.) using them.

Have I put you to sleep yet? LOL If not, here's one of my favorite math quotes - A mathematician is a machine for converting coffee into theorems.


Contagion said...

Now to get to the "unwanted" part of my being the unwanted stump.

By the definitions you listed Arithmetic is mathematics, However mathematics is not necessarily arithmetic... so now we are into philosphy.

Anyway, I think I'm more confused then before.j/k :)

MathCogIdiocy said...

Well - arithmetic is pretty much just addition, subtraction, multiplication and division. Your basic calculating functions.

Mathematics looks at things in more depth, tries to descibe "reality" through equations, and requires rigorous proof. Mathematics is more a set of logical arguments rather than a set of computations.

If you're feeling masochistic, check out the Clay Mathematics Institute's site ( and see some of the things that mathematicians do.

Harvey said...

So... are imaginary numbers arithmetic or mathematics?

MathCogIdiocy said...

Harvey - imaginary numbers grew out of the need to define the square root of -1. They're a constuct of mathematics but are used in the everyday calculations of engineers and scientists. So, I'd call them a sort of hybrid of arithmetic and mathematics. (Now there's a political non-answer for you.)

Math Student said...

This really helped!!! All other sites were really confusing. Thanks 4 your help!!!