What Is Arithmetic?

On might ask whether the first five years of the curriculum even include such a thing as mathematical instruction. The subject taught in those years is what used justly to be designated not math, but "arithmetic."

H. M. Enzensberger
Drawbridge Up: Mathematics? A Cultural Anathema,
A K Peters, 1999, p. 35


Arithmetic, then, means dealing logically with certain facts that we know, about number, with a view to arriving at knowledge which as yet we do not possess. ...

Remember that arithmetic requires nothing from you except absolute honesty and patient work.

Mary Everest Boole
Philosophy And Fun Of Algebra,
London: C. W. Daniel, LTD, 1909

Arithmetic is a branch of mathematics that deals with properties of the counting (and also whole) numbers and fractions and the basic operations applied to these numbers. As a matter of fact, as a noun in the above sense, the word is used quite seldom. In the early grades, when numbers are the main object of study, the subject is often designated as mathematics. The mathematics appellation sticks around until much later when it paradoxically becomes Algebra I. The latter is usually associated with the use of letters as place holders for generic or unknown numbers.

One explanation for the common avoidance of the word arithmetic stems from the fact that, besides learning numbers and how to deal with them, students are often taught about shapes and the skill of measurements, which takes the subject somewhat beyond the purview of arithmetic. However, the common meaning of "Mental Math" is the skill of carrying out basic calculations in one's head without recourse to paper, pencil, or other ancillary devices. The misnaming of the subject is entrenched in mathematics education literature. The titles Children's Mathematics, Children Doing Mathematics and Children's Mathematical Development (the first is so-so, the second is good, the third is excellent) are characteristic in the field, while Children's Arithmetic and Young Children Reinvent Arithmetic(two classics) are not.

The etymology of the word is interesting [Words, pp. 28-29]:

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 must once have 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 term is a fixed number apart from adjacent terms, just as the counting numbers of arithmetic are equally spaced. Interestingly enough, 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.

The distinction between arithmetic and algebra that underscores the use of letters is valid on an elementary level. Elementary arithmetic does not use symbols other than numerals and those of basic operations and equality. Elementary algebra, a step ahead of arithmetic, does use letters for formulating and solving problems and to annunciate properties of the arithmetic operations in a general form. (However, in Higher Arithmetic, which is another name for Number Theory, letters are used extensively as they are in the rest of mathematics.)

The commutative law that in arithmetic may be expressed as

The result of adding one number to another does not change if the order of addition is reversed.

is expressed in algebra in a much more succint form:

a + b = b + a.

While the algebraic form is visually more appealing, the same fact can still be conveyed in arithmetic classes and inculcated through practice and exercises. From a remarkable book by Liping Ma we learn that elementary arithmetic can be and is being taught differently. In China, the transition from arithmetic to algebra is mostly painless, while in the US it causes a major problem to the majority of students.

Letters are not necessarily needed for solving even word problems. Many striking examples could be found in the very first transcripts that survived from the second millennia B.C. Consider just one example [Fauvel, p. 16]: Problem 25 from the Rhind papyrus (1650 B.C, a copy of a document written probably two centuries earlier.)

A quantity and its 1/2 added together become 16. What is the quantity?

While in algebra the problem is reformulated as 3/2·x = 16 and solved as x = 16·2/3 = 32/3 = 10 2/3, the papyrus documents a letterless solution:

Assume 2. 
\12
\21
Total3

As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number.

13
26
412
2/32
1/31
Total5 1/3
15 1/3
210 2/3

Algebraic, i.e. general, facts, however expressed, are a great arithmetic tool. This is nowhere more apparent than in explaining and devising rapid math tricks. I would also add that arithmetic is more concerned with obtaining/calculating the end result, whereas in algebra it is more important to formulate and apply the rules for doing that.

The four elementary operations addition, subtraction, multiplication and division are commonly referred to as the four arithmetic operations, even though the terms apply to operations on numbers other than integer, rational, or decimal and on mathematical objects of completely different kinds. In this context, the word arithmetic is used as adjective. Similarly, as an adjective it appears in the term arithmetic sequence (or arithmetic progression.)

References

  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

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