A Brief History of Numbers
Numbers lie at the heart of mathematics. Just as our understanding of the natural world has evolved, so has our understanding of the number system. Number systems are defined in terms of sets. These sets are infinite in extent, with each subsequent set expanding the enumeration of the previous one.
Integers form the most basic number set. These are the counting numbers, 1, 2, 3, and so on. Initially only positive integers were considered. The inclusion of negative integers represents the first expansion of the number set. Surprisingly enough, while the concept of an empty, or null, set has been around for centuries, the use of a zero as a placeholder is a relatively recent invention, first introduced by the Arabs during the Middle Ages.
One important property of integer numbers is that every positive integer can be factored into a product of prime numbers, and there is only one way to do this factoring. This is known as the Fundamental Theorem of Arithmetic. For example, the number 536 = 2 x 2 x 2 x 67. This is the only way of expressing 536 as a product of primes.
In general, an integer number is symbolized mathematically by the letter Z. If only the positive integers are being considered, the symbol N is often used.
Rational numbers were the first true expansion of the integer number set. The rational number set consists of any number that can be expressed as the ratio of two integers, such as ½ or ¼. It can easily be seen that integers are included within the rational number set as the ratio of a specific integer to one. In general, rational number can be written as p/q, where p and q are both integer numbers. Note that rational numbers can be represented by decimal numbers. Indeed, any decimal number that is finite in length or has a repeating pattern is another representation of a rational number. The set of all rational numbers is usually denoted by the symbol Q.
While rational numbers allowed for ratios to be expressed easily, they can't express every number. The most obvious examples can be found in geometry. Consider a square whose sides are all one unit long. Then the distance across the diagonal can be determined by Pythagorian's theorem, a2 + b2 = c2, where a and b are the lengths of the two sides and c is the distance across the diagonal. In this case, c2 = 12 +12 = 2 is a rational number, but c itself cannot be expressed as a simple ratio. Similarly, the ratio of the circumference of a circle to its diameter cannot be expressed as a simple ratio. The only way to express these numbers was by expanding the rational number set to include numbers these new numbers, known as irrational numbers. In general, these numbers are represented by unique symbols, such asor. In terms of decimal notation, irrational numbers can only be approximated, since they are formed by and infinitely long string of decimals that never forms a repeating pattern. The set of real numbers, which includes both rational and irrational numbers, is usually denoted by the symbol R.
The inclusion of irrational numbers into the number set greatly expanded the range of numbers that could be used to describe something, but even the real number set doesn't cover everything. By definition, all of the numbers under a radical in the real number set is a positive number or zero. In other words, the numberis not defined as a real number. In order to include negative numbers under the radical sign, complex numbers were introduced. These are those numbers that can be written in the form z = a +ib, where a and b are real numbers and i is defined as. The set of complex numbers is usually denoted by the symbol C. We will study complex numbers in more detail latter in the course.
Note that. Thus, we could ideally solve any physical problem if we worked only with complex numbers. However, we can frequently find a solution using one of the subsets. A physical anology is solving for the motion of a car down a highway. Ideally, this problem should be solved using the full machinery of special relativity, since we know that classical mechanics is a subset of special relativity. However, the difference between the results produced by special relativity and classical mechanics is so small in this case that it is obviously overkill and considerably more work to use relativity theory.