The formula for the detailed form of the number of the number. As from the twisted form of the decimal number, go to its deployed form. Translation of decimal numbers to other number systems

Notation

Notation - this is a way of image numbers and the corresponding rules of action on numbers. A variety of number systems that existed earlier and which are used in our time can be divided into non-aposition and positional. Signs used when recording numbersCalled numbers.

AT non-phase surgery systems the value of the number does not depend on the position among.

An example of a non-transposition number system is the Roman system (Roman numbers). In the Roman system, Latin letters are used as numbers:

Example 1. The number CCXXXII is folded from two hundred, three dozen and two units and is equal to two thirty two.

In Roman numbers, the numbers are recorded from left to right in descending order. In this case, their values \u200b\u200bare added. If the left of the smaller digit is recorded, and the right is large, their values \u200b\u200bare deducted.

Example 2.

Vi \u003d 5 + 1 \u003d 6; IV \u003d 5 - 1 \u003d 4.

Example 3.

Mcmxcviii \u003d 1000 + (-100 + 1000) +

+ (–10 + 100) + 5 + 1 + 1 + 1 = 1998.

AT positional viewing systems the value indicated by the number in the number of numbers depends on its position. The number of numbers used is called the base of the positioning system.

The number system used in modern mathematics is positional decimal system. Its foundation is ten, because The recording of any numbers is made using ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The positional nature of this system is easy to understand by the example of any multi-valued number. For example, among the 333 first three means three hundred, the second is three dozen, the third is three units.

To write numbers in the position system with the base n. Must have alphabet of n. figures. Usually for this when n. < 10 используют n. first arabic numbers, and when n. \u003e 10 to ten Arabic figures add letters. Here are examples of alphabets of several systems:

If you want to specify the base of the system to which the number relates, it is attributed to the lower index to this number. For example:

101101 2, 3671 8, 3B8F 16.

In the level of the number with the base q. (q.- Official Number) Units of discharges serve sequential degrees of numbers q.. q. Units of any discharge form the unit of the next discharge. To write a number in q.- Official Number System Required q. different signs (numbers) depicting numbers 0, 1, ..., q. - 1. Record number q. at q.- Official number system has the form 10.

The detailed form of the number of numbers

Let AQ. - the number in the system with the base q., aI - numbers of this number system present in the number records A., n. + 1 - the number of discharges of the whole number, m. - The number of discharges of the fractional part of the number:

Deployed form of the number BUT called recording in the form:

For example, for decimal numbers:

The following examples provide the detailed form of hexadecimal and binary numbers:

In any number system, its base is recorded as 10.

If all the terms in the unfailed form of the non-deficent number in the decimal system and calculate the resulting expression according to the rules of decimal arithmetic, then the number in the decimal system is equal to this. According to this principle, a transfer is made from the non-followed system in decimal. For example, the translation into a decimal system written above the numbers is made like this:

How from the twisted form of a decimal number to go to his expanded form?

Answer

Consider the decimal number 14351.1. Its twisted form of recording is so habitant that we do not notice how in the mind they go to an expanded record, multiplying the numbers of the number on the "weight" of the discharges and folding the products obtained:

1 · 10 4 + 4 · 10 3 + 3 · 10 2 + 5 · 10 1 + 1 · 10 0 + 1 · 10 -1.

Switching from the folded form to the deployed

1. Look at the number given to you and determine the number of its numbers.

Example:
Write 5827 in the deployment.

Read the number out loud: five thousand eight hundred twenty-seven.

Please note that there are four digits in this number. As a result, the deployed form will contain four terms.

2. Rewrite the number in the form of the amount of its numbers, leaving between them some distance to multiply each digit to some number (more than this).

Example:
5827 Rewrite:

3. Numbers are located in certain positions that match (right to left) units, dozens, hundreds, thousands and so on. Determine the name of the position and its value for each digit (right to left).

Example:
Since in this number four digits, then you need to determine the names of four positions (right to left).

7 corresponds to units (value \u003d 1 \u003d 10 0).
2 corresponds to dozens (value \u003d 10 \u003d 10 1).
8 corresponds to hundreds (value \u003d 100 \u003d 10 2).
5 corresponds to thousands (value \u003d 1000 \u003d 10 3).

4. Multiply each digit of this number to the value corresponding to it.

Example:
5 · 10 3 + 8 · 10 2 + 2 · 10 1 + 7 · 10 0

The basis of the positioning system is called an integer Q, which is erected into a degree.

The basis of the positioning system is called the sequence of numbers, each of which determines the quantitative equivalent (weight) of the character depending on its place in the number code.

Basis of the decimal number system: ... 10 N., 10 N. –1 ,…, 10 1 , 10 0 , 10 –1 , …, 10 – M. ,…

Basis of an arbitrary positioning system: ... q N., q N. –1 , …, q. 1 , q. 0 , q. –1 , …, q. – M., …

The base in any system is depicted as 10, but has a different quantitative value. It shows how many times the quantitative value of the number changes when it moves to the adjacent position. There are many positions, since for the base of the number system, any number, not less than 2 can be taken.

The name of the number system corresponds to its base (decimal, binary, five-handed, etc.).

In the level of the number with the base q. (q.- Official Number) Units of discharges serve sequential degrees of numbers q, in other words, q. Units of any discharge form the unit of the next discharge.

To write numbers in q.- Official Number System Required q. different signs (numbers) depicting numbers 0, 1, ..., q. – 1.

Consequently, the basis of the positional number system is equal to the number of characters (characters) in its alphabet. Record number q. at q.- Official number system has the form 10.

Example 1.Octal number system.

Base: q \u003d. 8.

Alphabet: 0, 1, 2, 3, 4, 5, 6 and 7.

Numbers: for example, 45023,152 8; 751.001 8.

Example 2.Patrachetic number system .

Base: q. = 5.

Alphabet: 0, 1, 2, 3 and 4.

Numbers: for example, 20304 5; 324.03 5.

Example 3.Hex number system.

Base: q \u003d 16.

Alphabet: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Here only ten digits from sixteen have a generally accepted designation 0-9. To record the remaining symbols of the alphabet (10, 11, 12, 13, 14 and 15), the first five letters of the Latin alphabet are commonly used.

Numbers: for example, B5C3,1A2 16; 355,0FA01 8.

In a positional number system, any real number can be represented as follows:

A Q. = ±( a N. -1 × q N. –1 + a N. -2 × q N. –2 +…+ a. 0 × q. 0 + a. -1 × q. –1 + a. -2 × q. –2 +…+ a. –m. × q -m.), (1) or ±.

Here BUT - number; q - radix;
a I.- figures belonging to the alphabet of this number system; p - the number of integers of the number; t - The number of fractional discharge numbers.

The decomposition of the number according to formula (1) is called an expanded form of recording . Otherwise this form is called polynomial or power.

Example 1.Decimal number BUT 10 \u003d 5867.91 by formula (1) seems to be as follows:

A. 10 \u003d 5 × 10 3 + 8 × 10 2 + 6 × 10 1 + 7 × 10 0 + 9 × 10 -1 + 1 × 10 -2.

Example 2.Formula (1) for an octal surcharge system has the form:

A. 8 \u003d ± ( a N. -1 × 8. N. –1 + a N. -2 × 8. N. –2 +…+ a. 0 × 8 0 + a. -1 × 8 -1 + a. -2 × 8 -2 + ... + a -M.× 8 - m.),

where a I. - Figures 0-7.

The octal number A 8 \u003d 7064.3 in the form (1) will be recorded as follows:

A. 8 \u003d 7 × 8 3 + 0 × 8 2 + 6 × 8 1 + 4 × 8 0 + 3 × 8 -1.

Example 3.Pattyapic number BUT 5 \u003d 2430.21 by formula (1) will be recorded as follows:

BUT 5 = 2 × 5 3 + 4 × 5 2 + 3 × 5 "+ 0 × 5 ° + 2 × 5 -1 + 1 × 5 -2.

Calculates this expression, you can get a decimal equivalent of the specified five-day number: 365.44 10.

Example 4.In a hexadecimal number system, recording 3 AF 16 means:

3AF 16 \u003d 3 × 16 2 + 10 × 16 1 + 15 × 16 0 \u003d 768 + 160 + 15 \u003d 943 10.