Lo and behold, I was unable to find an adequate definition of arithmetic until I looked for its meaning since its discovery, or the etymology of arithmetic. I found the following etymological definition by S Schwartzman (Schartzman, 1994) to be very clear as a working definition of the word, with the editing I have done on sentences 3 and 4 and to which I have added additional information gleaned from the excellent teaching method of NJ Wildberger, author of Divine Proportions: Rational Trigonometry to Universal Geometry
arithmetic (noun, adjective): from the Greek arithmos “number”, from the Indo-European root ar- “to fit together.” A related borrowing from Greek is aristocrat, presumably a person in whom the best qualities are fitted together. Arithmetic has been conceived of as fitting things together or arranging or counting them. An arithm√©tic (notice the stress on the third syllable) series is one in which each space on a grid or web is a fixed number or segment, apart from adjacent spaces, just as the counting numbers or natural numbers of arithmetic are equally spaced.
The same Indo-European root found in arithmetic appears in native English word read, since when you read you have to fit the sounds together into words. So of the so-called three R’s — reading, (w)riting, and (a)rithmetic — two of them are etymologically related. Because arithmetic is a foreign word, English speakers have sometimes misconstrued it. In the 14th and 15th centuries it was known in England by the Latin-like name ars metrik “the metric art,” out of confusion with metric. It has similarly been called arithmetric.
From The Words of Mathematics by S. Schwartzman, sentence 3 and 4 are updated/corrected/made clearer to reflect the work of A/Professor NJ Wildberger, author of Divine Proportions: Rational Trigonometry to Universal Geometry
To which definition of Arithmetic I would also add:
Definition of arithmetic: The shapes and patterns on a gridline that express volume or quantity. By expressing numbers on a web or grid pattern we see that fixed number or segment can be seen as natural numbers, equally spaced revealing volume and patterns (overlaid). One of these basic patterns is a rectangle, which shows the composite of multiple points and segment portions. With the grid we are able to see the first function of mathematics that we study: addition. By adding these natural numbers or grid lines, squares and rectangles, we can perform the first basic function of arithmetic: addition. We can then learn by observation, the pattern the following additional functions: multiplication and the opposite function of each: subtraction and division.
Great thinkers and sublime educators are the source of my information. References included below.
1. J. Fauvel and J. Gray, The History of Mathematics: A Reader, The Open University, 1987
2. Liping Ma, Knowing and Teaching Elementary Mathematics, Lea, 1999
3. S. Schwartzman, The Words of Mathematics, MAA, 1994
4. NJ Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry 2005 Wild Egg
5. Professor Max Tegmark Cosmologist, MIT, See BBC Documentary http://www.youtube.com/watch?v=E_ExaP41tR8
6. http://www.cut-the-knot.org/WhatIs/WhatIsArithmetic.shtml