Magical Numbers

Figures:

The ancient Egyptians used special symbols, known as pictographs, writing down numbers than 3000 years ago. Later, the Romans developed a system of numbers used letters from their alphabet instead of the special symbols. Today we can use the figures are based on Hindu-Arabic system. We can write a song using a combination of up to 10 different symbols (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). The ancient Egyptians developed several systems to keep track of what was bought and sold.

What is the number? We use numbers every day and tend to take them for granted. But where did the idea numbers occur? Each culture develops ideas for counting and numbers, separately or are these thoughts only occurred in some cultures, and then spread, for example through trade? Regner intuitive or did it occur to solve specific problems? Some of the oldest evidence of counting so far discovered from ancient artifacts belonging to groups of hunters and gatherers. For example, a wolf bone, dated around 30,000 BC, discovered in a series of notches cut into what appears to represent an overview of some sort.

Tally Systems and Number of Words:

Census seems to be among the earliest methods to record quantities and is found in many cultures. But is it really counts? For example, using an inventory system to keep track of a flock of sheep. A small stone can be placed in a pile for each sheep as it is let out to pasture in the morning and then a stone can be removed from the pile again for each sheep as they gathered at night. Any left over pebbles suggests that some sheep were missing. This is really only a direct comparison between two sets of objects, stones and sheep. It gives no idea of the actual number of sheep in the flock. Another idea is sometimes found at an early developmental stage of a culture is that different numbers of words used depending on the context. For example, there may be a word for four persons and a further four stone. At one point, an abstract idea about numbers and develop the concept, for example, "threeness" where a group of three fish and three stone believed to have something in common, is incorporated into the system.

The Pictographic Notation:

The pictographic notation the number is given by the repetition of a symbol representing the object. For example, "five men" should be represented by the symbol for "male" five times. Many cultures throughout history have represented the number of repetitions of a horizontal or vertical line, 1 at a stroke, 2 in two strokes, etc. In various forms of speech, the number five is expressed by the word "hand" or "hand - ready" , and ten of "two hands" or "two hands ready." In some South American language "all fingers" means ten, "all fingers and toes," 20 "fingers and toes for two, but" 40, "one of the other page, "six," one foot ", 11" two feet ", 12, and so on.

Hieroglyphs Systems:

Hieroglyphs number systems consisted of repetitions of a single unit, with the use of hieroglyphs (pictures or symbols) for the higher figure. These systems are often adopted the principle of propagation, where repetitions were too many for practical use. Numbers in the heart of mathematics. Just as our understanding of the natural world has evolved, so has our understanding of the number system. The number of systems defined in terms of sets. These sets are infinite in extent, with each subsequent set of expanding list of the previous one.

Integers:

Integer form the most basic number set. It is the counting numbers 1, 2, 3, and so on. Initially only positive integers were considered. Inclusion of negative integers is the first expansion of the number of sets. Surprisingly, although the idea of an empty or zero, which has existed for centuries, using a zero as a placeholder is a relatively new invention that was introduced by the Arabs in the Middle Ages. An important characteristic of integers is that every positive integer can be incorporated into a product of primes, and there is only one way to do this factoring. This is known as the basic arithmetic. For example, 536 = 2 x 2 x 2 x 67th This number is the only way to express 536 as a product of primes. Generally, an integer mathematically symbolized by the letter Z. If only the positive integers under consideration, the symbol N is often used.

Rational numbers:

Rational numbers was the first real expansion of the integer set. The rational numbers, which consists of a number that can be expressed as the ratio of two integers, as ½ or ¼. It can easily be seen that the integers are included in the rational number indicated by the ratio of a given integer to one. Overall, rational numbers written as p / q where p and q are both integers.Note that rational numbers can be represented by decimal numbers. Virtually all decimals are bounded in length, or have a repeating pattern is a representation of a rational number. Set of all rational numbers is usually indicated by the symbol Q.

Real Numbers:

While rational numbers allowed ratios to be expressed easily, they can not express every number. The most obvious examples can be found in geometry. Consider a square whose sides are all one unit long. So the distance across the diamond can be determined by Pythagoras' phrase, a2 b2 = c2, where a and b are the lengths of the two sides and c is the distance over diagonally. In this case, C2 = 12 12 = 2 rational numbers, but c itself can not be expressed as a simple fact. Also the ratio of the circumference of a circle to its diameter is not expressed as a simple fact. The only way to express these numbers was by extending the rational number set to include tracks these new numbers, known as irrational numbers. Overall, these numbers are represented by unique symbols, such as ASOR. With regard to decimal notation, irrational numbers can only be approximated because they are formed, and infinitely long string of decimals that never form a repeating pattern. The set of real numbers, which includes both rational and irrational numbers are usually indicated by the symbol R.

Complex numbers:

Inclusion of irrational numbers in the number that greatly expanded the range of numbers that could be used to describe something, but even the real number which does not cover everything.By definition, all numbers a radical in the real number is a positive number or zero. In other words, the figures are not defined as a real number. To include negative numbers under the radical sign, complex numbers were introduced. These are the numbers 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 indicated by the symbol C. We will study complex numbers in more detail the latter course.

Note that. Therefore, we prefer to solve a physical problem, if we worked only with complex numbers. But we can often find a solution through one of the subsets. A physical analogy is the solution for the proposal of a car down a highway. Ideally, this problem should be solved using the full machinery of special relativity, because we know that classical mechanics is a subset of special relativity. But the difference between the results of special relativity and classical mechanics is so small in this case that of course is overkill and considerably more work to use the theory of relativity.