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 arithmétic (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