Number system. An example of a non-positional number systems

Number Systems - what is it? Even without knowing the answer to this question, each of us inevitably in life uses a number system and not know about it. So, in the plural! Not one, but several. Before examples of non-positional number systems, let us look at this issue, let's talk about the positional system also.

The Need to account

From ancient times people had need for the expense, that is intuitively realize that you need some way to Express a quantitative vision of things and events. The brain tells that you must use objects for counting. The most comfortable have always been the fingers, and this is understandable, because they are always available (with rare exceptions).

That had ancient members of the human race to bend his fingers in a literal sense - to designate the number of dead mammoths, for example. The names of these elements account have not been, and a visual picture matching.

Modern positional numeral systems

The number System is a method (way) of the demise of quantitative values, and values through specific characters (symbols or letters).

It is Necessary to understand a positional feature and positionnot in the account before you give examples of non-positional number systems. Positional number systems a lot. Now used in various fields of knowledge the following: binary (contains only two significant elements: 0 and 1), six (the number of digits - 6), octal (digits - 8), duodecimal (twelve characters), hex (includes sixteen characters). And each number of characters in the system starts from zero. Modern computer technology based on the use of binary codes - binary positional notation.

Recommended

"Knowledge is light and ignorance is darkness": the value, meaning and alternatives

There are some sayings that would seem to need no explanation, such as “teaching & ndash; light and ignorance – darkness”. But some still do not understand their meaning. But not only for such people is written by our article. I...

What was invented by Mendeleev for the army. The history and fate of the invention

D. I. Mendeleev was a brilliant Russian scientist-polymath, who made many important discoveries in various fields of science and technology. Many people know that he is the author of “Fundamentals of chemistry" and the periodic law of chem...

The origin of the Slavs. The influence of different cultures

Slavs (under this name), according to some researchers, appeared in the story only in 6 century ad. However, the language of nationality bears the archaic features of the Indo-European community. This, in turn, suggests that the origin of the Slavs h...

Decimal system

The positional feature is the presence in varying degrees of important positions, which are the digits of the number. This can best be demonstrated on the example of decimal number system. After all, we are accustomed to use since childhood. Characters in this system ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Take number 327. It has three digits: 3, 2, 7. Each is on its position (place). Seven occupies the position designated under a single value (one), two - tens and three hundreds. Since the number of three-digit, therefore, positions there are only three.

Based on the foregoing, this three-digit decimal number can be described as follows: three hundreds, two tens and seven units. The significance (importance) of position is counted from left to right, from a weak position (units) to stronger (hundreds).

We are very comfortable feel in the decimal positional number system. We have ten fingers, legs as well. Five plus five or so, thanks to the fingers, from childhood we easily imagine a dozen. That is why it is easy for children to learn the multiplication table of five and ten. And yet so easy to learn to count banknotes, which are often multiple (i.e. evenly divided) into five and ten.

Other positional numeral systems

To the surprise of many, I should say that not only in the decimal counting system our brains used to do some calculations. Still the mankind uses the sexagesimal and duodecimal systems of notation. That is, in such a system there is only six digits (in sexagesimal): 0, 1, 2, 3, 4, 5. They are twelve in duodecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a And b, where A - denotes the number 10, the number 11 (as the sign there should be one).

Judge for yourself. We believe time six, isn't it? One hour - sixty minutes (six dozen), one day is twenty-four hours (two times twelve), year - twelve months and so on... All the time intervals can easily fit into six and a duodecimal series. But we are so used to it that not even wonder at the timing.

Non-positional number system. Unary

You Must decide that it is - non-positional number system. This is a landmark system in which there are no positions for signs of the numbers, or the principle of "reading" the number of the position depends. It also has its own rules of entry, and calculations.

Give examples of non-positional number systems. Back to ancient times. People need account and came up with the most simple invention - nodules. Non-positional number system is nodular. One item (a bag of rice, a bull, haystack, etc.) counted, for example, when buying or selling and tying a knot in a string.

In the end on the rope was so many nodules, how many bags of rice bought (as an example). But it could also be notches on a wooden stick, on a stone slab, etc. This notation became known as nodular. She has a second name - unary, or one ("UNO" in Latin means "one").

It Becomes apparent that the number system - non-positional. After all, some of the positions can be discussed, when she (position) only one! Oddly enough, in some parts of the world are still in the course of a unary non-positional number system.

Also to non-positional systemsnotation include:

• Roman (for writing numbers using the letters of - Latin characters);
• Ancient Egyptian (similar to the Roman, was also used characters);
• Alphabetical (used letters of the alphabet);
• Babylonian (cuneiform used straight and inverted "wedge");
• Greek (also referred to as the alphabetic).

Roman numeral system

The Ancient Roman Empire and its science, was very progressive. The Romans have given the world many useful inventions of science and art, including its system of accounts. Two hundred years ago the Roman numbers are used to denote amounts in business documents (thus avoiding fakes).

The Roman numeral system - an example of non-positional number system, it is known to us now. Also the Roman system is actively used, but not for mathematics, but for targeted action. For example, with the help of Roman numbers to denote historical dates, ages, numbers of volumes, sections, and chapters in books. Often use Roman characters for the decoration of dials. As well as the Roman numeral system is an example of a non-positional number system.

The Romans marked the numbers by letters of the Latin alphabet. And the number they wrote down according to certain rules. There is a list of key symbols in the Roman numeral system, they recorded all the numbers without exception.

Rules of writing numbers

Required number was obtained by summing the signs (letters of the Latin alphabet) and the calculation of their amount. Consider, symbolically written signs in the Roman system and how they need to "read". List the main laws of the formation of the Roman numbers in non-positional notation.

1. The Number four - IV, consists of two symbols (I, V - one and five). It is obtained by subtracting the smaller from the larger sign, if he stands to the left. When a smaller sign is on the right, you need to add, then get the number of six - VI.
2. You Must put two of the same sign, standing nearby. Example: SS - 200 (C - 100), or XX - 20.
3. If the first character of the number is less than the second, then the third in this series can be a character whose value is still less than the first. To avoid confusion, here is an example: CDX - 410 (decimal).
4. Some large numbers can be represented in different ways, which is one of the disadvantages of the Roman system account. Examples include: MVM (the Roman system) = 1000 + (1000 - 5) = 1995 (decimal system) or MDVD = 1000 + 500 + (500 - 5) = 1995. And that's not all.

Methods arithmetic

Non-positional number system is sometimes complex set of rules of formation of numbers of processing (operations on them). Arithmetic operations in non-positional numeral systems can be difficult for modern people. I do not envy the ancient Roman mathematicians!

Example addition. Let's try to add two numbers: XIX + XXVI = XXXV, this job runs in two steps:

1. First - take and fold a smaller share numbers: IX + VI = XV (I, after V and I to X "undo" each other).
2. Second - stacking large shares of the two numbers is: X + XX = XXX.

Subtraction is performed is more complicated. Minuend the number you want to split into its constituent elements, and then in umanesimo and visitemos to reduce the duplicated characters. From number subtract 500 263

D - CCLXIII = CCCCLXXXXVIIIII - CCLXIII = CCXXXVII.

Multiplying Roman numbers. By the way, it should be mentioned that the Romans there were no signs arifmeticheskikh operations, they are just words meant to them.

The Multiplicand, the number to multiply it was necessary for each individual multiplier symbol, there were several pieces that needed to be folded. This method of producing the multiplication of polynomials.

As for dividing, this process in the Roman numeral system was and remains the most difficult. There was used the ancient Roman abacus - abacus. To work with people specially trained (and not every person was able of such a science to master).

About the shortcomings of non-positional systems

As mentioned above, in the non-positional numeral systems have their shortcomings and inconveniences in use. Unary simple simple enough for the account, but arithmetic and complex calculations it is not necessary at all.

In the Roman there are no unified rules for the formation of large numbers and confusion, and it is very difficult to compute. In addition, the largest number that could record the ancient Romans using their method, it was 100000.