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# arithmetic

Arithmetic is ” the science of numbers and operations on sets of numbers. Arithmetic is understood to include problems on the origin and development of the concept of a number, methods and means of calculation, the study of operations on numbers of different kinds, as well as analysis of the axiomatic structure of number sets and the properties of numbers.” The four basic operations are addition, subtraction, multiplication and division; in the absence of parentheses these are performed according to the rules of operator precedence. From multiplication follows exponentiation.

Most civilizations usually develop arithmetic before any other branches of mathematics, and in modern times children are usually taught arithmetic first (though in some curricula they might be taught set theory first).

Numeral systems of ancient civilizations often took letter symbols for numbers. Boyer called these letter symbols one-to-one ciphered numerals. The ancient Greeks, for example, used letters in Ionian and Dorian alphabets to represent numbers, as did the Romans in one alphabet. These systems were additive in one sense (that is, the value of the overall “word” was merely the sum of the values of the individual symbols). To convert rational numbers into a ciphered numeration system, Fibonacci implemented seven rules, one being a LCM multiple. For example, four of Fibonacci’s rules were used by earlier cultures. One was an Egyptian scribe. In 1650 BCE Ahmes converted $\frac{n}{p}$ by using large multiples. About 200 years earlier, a student converted $\frac{n}{pq}$ by using small LCM multiples.

It was not just the invention of zero as a symbol for that integers between 1 and -1 allowed Hindu-Arabic numerals to significant expand man’s ability to perform arithmetic operations. Significantly larger numbers came into use after zero became a placeholder in the new additive-exponential system. An algorithm written in Hindu-Arabic numbers before 1600 AD was also added. Modern decimals apply few lessons learned during the 3,200 year life of Egyptian fractions. One exception is aliquot parts, used in the fundamental theorem of arithmetic, and higher arithmetic.

The next great expansion of arithmetic power occurred with the invention of calculators and computers in the 20th Century, with both kinds of devices performing arithmetic in a binary algorithm. The algorithm, stated in a cursive form, had been used prior to 2,000 BCE.

As number theory generalizes the principles of arithmetic to all numbers or all numbers of a given kind, it is sometimes called the “higher arithmetic.”

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## Comments

## Also called "arithmetic"

Number theory is also sometimes just called "arithmetic," i.e., as in Serre's A Course in Arithmetic.

## Re: Also called "arithmetic"

This isn't a very widespread use of the word, though, and I don't think even Serre intended the two to be synonymous. I always took it more in the sense that arithmetic was the study of elementary (as in fundamental, not as in easy) properties of rational numbers, be it the standard rational numbers or the p-adic ones. Usually if one uses the word "arithmetic" with respect to modern number theory, it's used as an adjective, as in "arithmetic algebraic geometry."

Cam

## Re: Also called "arithmetic"

Surely "arithmetic" is the study of the algebraic properties which arise as a result of a ring or field not being algebraically closed. Hence the common use of phrases such as: "We work over an algebraically closed field so that issues of arithmetic do not arise".

Conversely "number theory" is not confined to algebraic properties but is normally rstricted to (ground) rings and fields of transcendence degree 0.

## Re: Also called "arithmetic"

Back in the days of Euclid and Diophantus, "arithmetic"

referred to the study of properties of natural numbers

and formed part of the quadrivium, the mathematical part

of the curriculum throughout the Middle Ages. It was only

much later when the quadrivium faded away and textbooks

on the rudiments of the subject for little kids came out

that the semantic shift happened. To get some idea how

recent this is, remember that Gauss used the term "Arithmetic"

and, as pointed out, this usage is still found with Serre

and Hardy (who had to relent in the title of his book

because of fear of popular misunderstanding but who uses the

term inside of it). In order to avoid confusion with simple

exercises in adding and multiplying numbers, the term

"higher arithmetic" is used for the study of more advanced

properties of the integers.

## Re: Also called "arithmetic"

Arithmetic in the form that medievals and Euclid used was born 1,700 years earlier. Ancient Near East scribes generally converted rational numbers by three-steps, obtaining optimal or elegant unit fraction series. Knowledge of aliquot parts of denominators, the second-step, was written in additive numerators, was well-known to Euclid.

Additional properties of Euclid' arithmetic were erased by the arrival of decimals, 400 years ago. Today, rational numbers are rounded-off in ways that may be making Archimedes, Euclid and Diophantus to roll-over in their graves.

## Re: Also called "arithmetic"

What exactly to you mean "normally restricted to (ground) rings and fields of transcendence degree 0?" If you're considering a field as a "ground" field, to speak of its transcendence degree makes no sense as it implies you have another "ground" field in mind; and it is certainly not true that number theory is restricted to algebraic extensions of ground fields (for instance the fields \mathbb{C}(t) and \mathbb{F}_p(t) are of interest in number theory).

## Re: Also called "arithmetic"

Even more to the point, arithemeticians (a.k.a. number theorists)

are certainly interested in transcendental numbers. For instance,

consider the formula expressing pi as product involving prime

numbers, formulas for prime numbers by truncating powers of

real numbers, questions about the transcendance of logarithm

and exponents as well as various constants which appear in

analysis, the recent interest in periods, continued fractions of

transcendental numbers, etc. A good part of analytic number

theory and Diophantine approximation deals with such issues

regarding transcendental numbers, so I would say that the real

number field is not just something useful in the study of

algebraic numbers but a primary object of study in higher arithmetic.

## Re: Also called "arithmetic"

Sorry, by transcendence degree 0 I meant over the subring additively generated by 1.

None the less your first example shows that I was talking complete rubbish as if you take \mathbb{C} as your ground ring for

\mathbb{C}(t) then \mathbb{C} contains transcendental numbers such as pi.

I still think my interpretation of "arithmetic" is the conventional one though.

## Re: Also called "arithmetic"

I figured that was the reason the term "higher arithmetic" arose in the first place, to avoid confusion. I can cite three different books titled "Higher Arithmetic" that are about number theory. For that reason I thought at least a remark was necessary.

P.S. Thanks to everyone else who weighed in on this.

## Re: Also called "arithmetic"

Oh I totally agree it's worth a remark. I wasn't suggesting anything was wrong with your entry, I was just making a comment. Go number theory.