Monday, September 2, 2013

Mathematics -- Part One: Arithmetic

The term “arithmetic” comes from the Greek word "ἀριθμός," pronounced “arithmos” which means "number." It is the oldest and most elementary branch of mathematics. It is used for counting, balancing checkbooks, and advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. More precisely, this is “elementary arithmetic” in contrast with the so called “higher” or “advanced” arithmetic, which is about number theory.

Besides the basic operations of addition, etc., there are other concepts such as equality and various representations called “notation.” That’s the way that numbers, fractions, exponents, etc. are expressed. This notation has been invented and evolved over the centuries and is a common language in the world. Whether English or French or Chinese or Russian or African or from India, the notation and the symbols for the modern ten digits and the mathematical operations are the same. They may add a slash to the seven in Europe or a stroke through the zero in the Army, but in our world of multiple languages, the language of mathematics is universal. That, in itself, is something to ponder philosophically.

Virtually all ancient cultures understood arithmetic and basic calculations encountered in agriculture and store keeping. Arithmetic is closely related to the system for representing numbers and basic counting. Early math history is more about the advancement of notation with more modern studies of arithmetic focused on proving the correctness of results. The ancients were happy if the math fit common sense, but we now have a complex system of axioms and proofs at the heart of all math including arithmetic.

Various mechanical devices such as the abacus or the more modern adding machine and even digital computers have been developed to “crunch numbers.” These replaced the use of simple fingers for counting which share the term “digits” with these numerical symbols. Some cultures had a base of 60 and even 360, values we still see on clocks and compasses, but most gravitated toward the decimal or “ten-based” system although other concepts such as "dozen" still exist.

As the science of arithmetic advanced, new number types were discovered in addition to the counting numbers from the negative numbers to zero to fractional numbers and even imaginary numbers. The simplest set of numbers is called the counting numbers or “Natural Numbers.” They start with 1 and increase by adding 1 to each preceding number: 1, 2, 3, etc. Formally, these counting numbers are called the Positive Integers. Add zero and the negative values and you have the whole of the Integers.

Negative numbers first appeared as the result of subtraction and were used to represent financial concepts such as debt. Zero was actually a little harder to develop since the concept of nothing was challenging to early societies, but zero soon entered the lexicon. Note that the Zero is neither Positive nor Negative, but represents the boundary between the two sets when shown on the “Number Line.”

Fractional numbers also were an early development, often stated as a fraction with a numerator or “top” and a denominator or “bottom.” They were the natural result of division, and — in fact — a fraction is one way to represent division. The division symbol (÷) is also recognizable as an abstraction of a fraction with the two dots representing the numerator and denominator.

All the numbers that can be formed by making fractions of Integers were called by the Greeks the “Rational” Numbers. We often think of these as “decimal” numbers in modern math where the “decimal point” separates the whole number part from the fractional part.

When the Greeks first leaned there were fractional numbers that weren’t constructed from Integer fractions they were shocked. Legend says that a member of the Pythagorean school who showed that the square root of 2 was not such a number was thrown into the sea to drown for his heresy. Once they became convinced of the existence of such numbers, then the Greeks began calling values like the square root of two “Irrational.” We later leaned that some natural constants such as pi are also Irrational. Therefore, the numbers that could be formed from fractions of Integers were called the Rationals. Those that could not be formed from a division of Integers were called Irrational.

Not only can’t Irrational Numbers be represented as a fraction of Integers, but as decimals they never repeat or terminate. Rationals do. For example, 1/2 = 0.5 and 1/3 = 0.333… to infinity. Note that you could consider 1/2 as repeating without end if you view it as 0.5000…. The numbers in pi, on the other hand, never repeat no matter how many digits you expand the decimal: 3.1415926535897932385…

Now days we consider the Rational numbers plus the Irrational numbers as a set called the “Real” numbers. All the Real numbers can be represented as a point on a number line. That’s sort of like a ruler or scale drawn on paper. Zero is in the center and the positive numbers extend to the right (toward infinity) and the negative numbers extend to the left (toward negative infinity). You can imagine both 2 and 3 as well as 2.5 or 1/3 and even the square root of 2 (about 1.4) as points on the number line or scale.

Mathematicians also considered what appeared to be impossible numbers such as the square root of minus 1, a number that would seem to not exist, and therefore is called “Imaginary.” The square root of minus one is a number that, when squared, equals minus one. But all numbers, when squared or multiplied by themselves, produce positive numbers. Therefore, it would appear that the square root of -1 does not exist except in the imagination of the mathematician. The square root of minus one is often represented by the lowercase letter "i," usually in italics: i. By definition, that is we simply declare it to be true, i = √-1.

Imaginary numbers can’t be represented on a number line, although they can be represented using a Cartesian graph. In fact, a major use of Complex Numbers is to represent multidimensional values and Cartesian coordinates.

“Complex Numbers” consist of the set of Reals and the set of Imaginary numbers combined. They are written as the sum of the real component and the imaginary component: R ± i I.

So, in terms of membership, the largest collection is the Complex Numbers made up of Real + Imaginary. The Reals are made up of Rational plus Irrational Numbers, and the Rational Numbers include both Integers and Fractional numbers. Finally, the Integers include the Natural Numbers plus the Negative Integers and Zero.

It seems like many of these problem numbers came from square roots, and you could question if a square root is a basic arithmetic operation, but square roots are a natural extension of the inverse of multiplication, and multiplication is just accumulated addition, so it is among the basic operations. (An inverse is an operation that “undoes” another operation. Subtraction is the inverse of addition and division is the inverse of multiplication. Repeated multiplication is called exponentiation or “powers” and roots are an inverse of that. ((There is actually another inverse of exponentiation called logarithms. It is part of arithmetic too.)) )

So you can argue that all the trouble started with simple addition and its inverse. Arithmetic is just numbers and these simple and extended operations on numbers. Seems more complicated than just 2 + 2, but it actually grows naturally out of that basic equation or formula.

Usually the purpose of all forms of higher math is to, eventually, get back to arithmetic and numbers. (Not always true, but often true.) Ultimately, arithmetic and numbers give us the answer we usually seek. Saying math is all about numbers is a bit over simplified because geometry, also a part of math, isn’t about numbers directly, although we now realize geometry and algebra can be combined; something called “analytic geometry.” So, one can argue that it all started with numbers. 1, 2, 3, … infinity.

(By the way, infinity is not a number. It is a concept and a destination and a limit, but it is not a number.)

Next we will continue to study numbers and arithmetic, resulting in the modern concept of “sets” and the following article titled “Mathematics -- Part Two: Sets.”

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