Transition from 16 to 10 number system. Hex code

Lesson type: lesson – consolidation of what has been learned. (summarizing)

Type: combined lesson.

Goal: To generalize and apply knowledge about the methods and methods of number translations to solve the problem. Development of cognitive interest and creative activity of students.

Lesson objectives:

Educational: deepening, generalizing and systematizing techniques for converting numbers from one number system to another.
Educational: development of cognitive interest, logical thinking.
Developmental: development of algorithmic thinking, memory, attentiveness.

Lesson progress:

1. Organizational moment (3 min).
2. Checking homework:

a) Theory: Calculator (3 min);
b) Practice: checking the work history at the PC (7 min).

3. The “8-2-16” principle

a) theory: the essence of the principle, examples (10 min);
b) practice: complete a practical task (using cards) (15 min).

4. Recording homework (2 min).
5. Summing up.

1. Organizational moment.
2. Checking homework:

a) Go through the rows and look (superficially - whether there is or not) the recordings of the solutions to the exercises. Invite students to check their homework on their own using a PC. For this we use standard application Windows OS – Calculator.

Write on the board and in your notebook:

Launch: Start – Programs – Accessories – Calculator

Team: Type – Engineering.

With this program you can convert numbers written in binary, octal, decimal and hexadecimal coordinate systems. Have designations:

Hex (Hexadecimal) - hexadecimal

Dec (Decimal) - decimal

Oct (Octal) - octal

Bin (Binary) – binary.

Figure 1

Number translation algorithm:

For example, convert the number 19F 16 =X 10.

1. Set the switch to the Hex position (by clicking on it with the left mouse button).
2. Enter the number using the mouse or keyboard (Latin letters).
3. Set the switch to the Dec position - we get the answer.
4. Check the correctness in your notebook and put +.

b) Students sit down at the computers and perform a self-test.

1. We have learned how to convert numbers from one system to another (in writing or using the Calculator program), and now let's look at transfer methods that do not require any calculations from us. Let's call it the “8-2-16 Principle”.

a) I distribute cards with tables on the table:

In octal number system eight digits: 0, 1, 2, 3, 4, 5, 6, 7. Conversion from this system to binary is quite simple. It is enough to make a table of triads (three digits each).

When converting an octal number to binary, replace each octal digit with the corresponding triad from the table (see examples in the card).

For the reverse operation, that is, to convert from binary to octal, the binary number is divided into triads (from right to left), then each group is replaced by one octal digit.

Similarly, we convert from hexadecimal to binary systems and vice versa.

b) I suggest that the guys compete with each other “Who is faster” to consolidate their skills; in addition to speed, attentiveness and accuracy play a big role here.

• Let's write the numbers in the octal number system so that there are 17 of them: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20 (in this number In the series after the number 7, the digit is exceeded since the number 8 does not exist, we move from the units category to the tens category and so on). It is no coincidence that we needed these numbers, because we will consider the coordinate plane for the octal number system. You will be given the coordinates of the drawing in the binary coordinate system, and the drawing must be done in the octal system. Connect the points in the order they appear.
• I distribute cards with coordinates (2-4 options) and show the first point (arbitrary) with an example (on the board: by writing out the coordinates and showing them on the coordinate plane). Examples of tables with coordinates:

Option 1.

Option 2.

• The first 2-3 people who complete the task correctly (the picture matches the original) receive a grade of “5”.

Examples of drawings - answers:

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Figure 2

Figure 3

1. For homework, please draw a picture in hexadecimal system notation, write the coordinates in a table in the binary system.
2. So we looked at several ways to translate numbers: general and particular. Some of them required you to be able to solve problems using mathematical methods, others with the use of a computer, and others with the use of triads and tetrads. Thus, we repeated the topic “Translation of numbers into various systems dead reckoning” and prepared for test work. Good luck. Goodbye!

Literature used:

1. Encyclopedia for children. Volume 22. Computer Science/Chapter. ed. E. A. Khlebalina, leading scientific ed. A.G. Leonov. - M.: Avanta+, 2003. – 624 p.: ill.
2. Efimova O., Morozov V., Ugrinovich N. Course computer technology with the basics of computer science. Tutorial for high school. –M.: LLC “AST Publishing House”; ABF, 2000. – 432 pp.: ill.

Those taking the Unified State Exam and more...

It is strange that in computer science lessons in schools they usually show students the most complex and inconvenient way to convert numbers from one system to another. This method consists of sequentially dividing the original number by the base and collecting the remainders from the division in reverse order.

For example, you need to convert the number 810 10 to binary:

We write the result in reverse order from bottom to top. It turns out 81010 = 11001010102

If you need to convert to the binary system, quite big numbers, then the division ladder takes on the size of a multi-story building. And how can you collect all the ones and zeros and not miss a single one?

IN Unified State Exam program in computer science includes several tasks related to the translation of numbers from one system to another. Typically, this is a conversion between octal and hexadecimal systems and binary. These are sections A1, B11. But there are also problems with other number systems, such as in section B7.

To begin with, let us recall two tables that would be good to know by heart for those who choose computer science as their future profession.

Table of powers of number 2:

Table of binary numbers from 0 to 15 with hexadecimal representation:

Integer conversion

So, let's start by converting directly to the binary system. Let's take the same number 810 10. We need to decompose this number into terms equal to powers of two.

1. We are looking for the power of two closest to 810 and not exceeding it. This is 2 9 = 512.
2. Subtract 512 from 810, we get 298.
3. Repeat steps 1 and 2 until there are no 1s or 0s left.
4. We got it like this: 810 = 512 + 256 + 32 + 8 + 2 = 2 9 + 2 8 + 2 5 + 2 3 + 2 1.

Method 1: Arrange 1 according to the ranks of the indicators of the terms. In our example, these are 9, 8, 5, 3 and 1. The remaining places will contain zeros. So, we got the binary representation of the number 810 10 = 1100101010 2. Units are placed in 9th, 8th, 5th, 3rd and 1st places, counting from right to left from zero.

Method 2: Let's write the terms as powers of two under each other, starting with the largest.

810 =

Now let's add these steps together, like folding a fan: 1100101010.

That's it. Along the way, the problem of “how many units in binary notation number 810?

The answer is as many as there are terms (powers of two) in this representation. 810 has 5 of them.

Now the example is simpler.

Let's convert the number 63 to the 5-ary number system. The closest power of 5 to 63 is 25 (square 5). A cube (125) will already be a lot. That is, 63 lies between the square of 5 and the cube. Then we will select the coefficient for 5 2. This is 2.

We get 63 10 = 50 + 13 = 50 + 10 + 3 = 2 * 5 2 + 2 * 5 + 3 = 223 5.

And, finally, very easy translations between 8 and hexadecimal systems. Since their base is a power of two, the translation is done automatically, simply by replacing the numbers with their binary representation. For the octal system, each digit is replaced by three binary digits, and for the hexadecimal system, four. In this case, all leading zeros are required, except for the most significant digit.

Let's convert the number 547 8 to binary.

One more, for example 7D6A 16.

Let's convert the number 7368 to the hexadecimal system. First, write the numbers in triplets, and then divide them into quadruples from the end: 736 8 = 111 011 110 = 1 1101 1110 = 1DE 16. Let's convert the number C25 16 to the octal system. First, we write the numbers in fours, and then divide them into threes from the end: C25 16 = 1100 0010 0101 = 110 000 100 101 = 6045 8. Now let's look at converting back to decimal. It is not difficult, the main thing is not to make mistakes in the calculations. We expand the number into a polynomial with powers of the base and coefficients for them. Then we multiply and add everything. E68 16 = 14 * 16 2 + 6 * 16 + 8 = 3688. 732 8 = 7 * 8 2 + 3*8 + 2 = 474 .

Converting Negative Numbers

Here you need to take into account that the number will be presented in additional code. To convert a number into additional code, you need to know the final size of the number, that is, what we want to fit it into - in a byte, in two bytes, in four. The most significant digit of a number means the sign. If there is 0, then the number is positive, if 1, then it is negative. On the left, the number is supplemented with a sign digit. We do not consider unsigned numbers; they are always positive, and the most significant bit in them is used as information.

For translation negative number in binary's complement code you need to convert a positive number to binary, then change the zeros to ones and the ones to zeros. Then add 1 to the result.

So, let's convert the number -79 to the binary system. The number will take us one byte.

We convert 79 to the binary system, 79 = 1001111. We add zeros on the left to the size of the byte, 8 bits, we get 01001111. We change 1 to 0 and 0 to 1. We get 10110000. We add 1 to the result, we get the answer 10110001. Along the way, we answer the Unified State Exam question “how many units are in the binary representation of the number -79?” The answer is 4.

Adding 1 to the inverse of a number eliminates the difference between the representations +0 = 00000000 and -0 = 11111111. In two's complement code they will be written the same as 00000000.

Converting fractional numbers

Fractional numbers are converted in the reverse way of dividing whole numbers by the base, which we looked at at the very beginning. That is, using sequential multiplication by a new base with the collection of whole parts. The integer parts obtained during multiplication are collected, but do not participate in the following operations. Only fractions are multiplied. If the original number is greater than 1, then the integer and fractional parts are translated separately and then glued together.

Let's convert the number 0.6752 to the binary system.

The process can be continued for a long time until we get all the zeros in the fractional part or the required accuracy is achieved. Let's stop at the 6th sign for now.

It turns out 0.6752 = 0.101011.

If the number was 5.6752, then in binary it will be 101.101011.

Hexadecimal number system(also known as hexadecimal code) is a positional number system with an integer base of 16. The term hex (pronounced hex, short for English hexadecimal) is also sometimes used in the literature. The digits of this number system are usually used in Arabic numerals 0-9, as well as the first characters of the Latin alphabet A-F. The letters correspond to the following decimal values:

• * A -10;
• *B—11;
• *C—12;
• * D -13;
• * E - 14;
• * F - 15.

Thus, ten Arabic numerals, coupled with six Latin letters, make up the sixteen digits of the system.

By the way, on our website you can convert any text into decimal, hexadecimal, binary code using the online code calculator.

Application. Hex code widely used in low-level programming as well as in various computer reference documents. The popularity of the system is justified by architectural solutions modern computers: They have a byte (consisting of eight bits) as the minimum unit of information - and the value of a byte is conveniently written using two hexadecimal digits. The byte value can range from #00 to #FF (0 to 255 in decimal notation) - in other words, using hexadecimal code, you can write any state of the byte, while there are no “extra” digits not used in the recording.

Encoded Unicode Four hexadecimal digits are used to record the character number. The RGB color notation (Red, Green, Blue) also often uses hexadecimal code (for example, #FF0000 is a bright red color notation).

A method for writing hexadecimal code.

Mathematical way of writing. In mathematical notation, the base of the system is written in decimal form as a subscript to the right of the number. Decimal notation the numbers 3032 can be written as 3032 10; in hexadecimal, this number will have the notation BD8 16.

In the syntax of programming languages. Syntax different languages programming sets the format for writing a number using hexadecimal code:

* The syntax of some varieties of assembly language uses the Latin letter “h”, which is placed to the right of the number, for example: 20Dh. If the number starts with Latin letter, then a zero is placed in front of it, for example: 0A0Bh. This is done in order to distinguish values ​​using constants from constants. hexadecimal code;

* Other types of assembler, as well as Pascal (and its variants such as Delphi) and some Basic dialects, use the "$" prefix: $A15;

* In language HTML markup, as well as in cascade CSS files, to specify the color in RGB format with hexadecimal notation, the prefix “#” is used: #00DC00.

How to convert hexadecimal code to another system?

Convert from hexadecimal to decimal. To perform a conversion operation from the hexadecimal system to the decimal system, you need to represent the original number as the sum of the products of the digits in the digits of the hexadecimal number and the power of the base.

For example, you need to translate the hexadecimal number A14: it has three digits. Using the rule, we write it as a sum of powers with a base of 16:

A14 16 = 10.16 2 + 1.16 1 + 4.16 0 = 10.256 + 1.16 + 4.1 = 2560 + 16 + 4 = 2580 10

Converting numbers from binary to hexadecimal and vice versa.

A notebook table is used for translation. To convert a number from the binary to the decimal system, you need to split it into separate tetrads from right to left, and then, using the table, replace each tetrad with the corresponding hexadecimal digit. Moreover, if the number of digits is not a multiple of four, then it is necessary to add the corresponding number of zeros to the right of the number in order for the total number binary digits became a multiple of four.

Table of notebooks for translation.

To convert from hexadecimal to binary, you need to perform the reverse operation: replace each digit with a tetrad from the table.

Example conversion from hexadecimal to binary: A5E 16 = 1010 0101 1110 = 101001011110 2

Example conversion from binary to hexadecimal: 111100111 2 = 0001 1110 0111 = 1E7 16

In this example, the number of digits in the original binary number was not equal to four (9), so non-significant zeros were added - the total number of digits became 12.

Automatic translation. Quick transfer from hexadecimal number system to one of three popular systems(binary, octal and decimal), as well as reverse translation, can be performed using standard calculator supplied with Windows OS. Open the calculator, select View -> Programmer from the menu. IN this mode you can set the number system used in at the moment(see menu on the left: Hex, Dec, Oct, Bin). In this case, changing the current number system automatically produces a translation.

The result has already been received!

Number systems

There are positional and non-positional number systems. The Arabic number system, which we use in everyday life, is positional, but the Roman number system is not. In positional number systems, the position of a number uniquely determines the magnitude of the number. Let's consider this using the example of the number 6372 in the decimal number system. Let's number this number from right to left starting from zero:

Then the number 6372 can be represented as follows:

6372=6000+300+70+2 =6·10 3 +3·10 2 +7·10 1 +2·10 0 .

The number 10 defines the number system (in in this case this is 10). The values ​​of the position of a given number are taken as powers.

Consider the real decimal number 1287.923. Let's number it starting from zero, position of the number from the decimal point to the left and right:

Then the number 1287.923 can be represented as:

1287.923 =1000+200+80 +7+0.9+0.02+0.003 = 1·10 3 +2·10 2 +8·10 1 +7·10 0 +9·10 -1 +2·10 -2 +3· 10 -3.

In general, the formula can be represented as follows:

C n s n +C n-1 · s n-1 +...+C 1 · s 1 +C 0 ·s 0 +D -1 ·s -1 +D -2 ·s -2 +...+D -k ·s -k

where C n is an integer in position n, D -k - fractional number in position (-k), s- number system.

A few words about number systems. A number in the decimal number system consists of many digits (0,1,2,3,4,5,6,7,8,9), in the octal number system it consists of many digits (0,1, 2,3,4,5,6,7), in the binary number system - from a set of digits (0,1), in the hexadecimal number system - from a set of digits (0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E,F), where A,B,C,D,E,F correspond to the numbers 10,11,12,13,14,15. In the table Tab.1 numbers are presented in different systems Reckoning.

Converting numbers from one number system to another

To convert numbers from one number system to another, the easiest way is to first convert the number to the decimal number system, and then convert from the decimal number system to the required number system.

Converting numbers from any number system to the decimal number system

Using formula (1), you can convert numbers from any number system to the decimal number system.

Example 1. Convert the number 1011101.001 from binary number system (SS) to decimal SS. Solution:

1 ·2 6 +0 ·2 5 + 1 ·2 4 + 1 ·2 3 + 1 ·2 2 + 0 ·2 1 + 1 ·2 0 + 0 ·2 -1 + 0 ·2 -2 + 1 ·2 -3 =64+16+8+4+1+1/8=93.125

Example2. Convert the number 1011101.001 from octal number system (SS) to decimal SS. Solution:

Example 3 . Convert the number AB572.CDF from hexadecimal number system to decimal SS. Solution:

Here A-replaced by 10, B- at 11, C- at 12, F- by 15.

Converting numbers from the decimal number system to another number system

To convert numbers from the decimal number system to another number system, you need to convert the integer part of the number and the fractional part of the number separately.

The integer part of a number is converted from decimal SS to another number system by sequentially dividing the integer part of the number by the base of the number system (for binary SS - by 2, for 8-ary SS - by 8, for 16-ary SS - by 16, etc. ) until a whole residue is obtained, less than the base CC.

Example 4 . Let's convert the number 159 from decimal SS to binary SS:

As can be seen from Fig. 1, the number 159 when divided by 2 gives the quotient 79 and remainder 1. Further, the number 79 when divided by 2 gives the quotient 39 and remainder 1, etc. As a result, constructing a number from division remainders (from right to left), we obtain a number in binary SS: 10011111 . Therefore we can write:

159 10 =10011111 2 .

Example 5 . Let's convert the number 615 from decimal SS to octal SS.

When converting a number from decimal SS to octal SS, you need to sequentially divide the number by 8 until you get an integer remainder less than 8. As a result, constructing a number from division remainders (from right to left) we get a number in octal SS: 1147 (See Fig. 2). Therefore we can write:

615 10 =1147 8 .

Example 6 . Let's convert the number 19673 from the decimal number system to hexadecimal SS.

As can be seen from Figure 3, by successively dividing the number 19673 by 16, the remainders are 4, 12, 13, 9. In the hexadecimal number system, the number 12 corresponds to C, the number 13 - D. Therefore, our hexadecimal number- this is 4CD9.

To convert proper decimal fractions ( real number with a zero integer part) into a number system with base s, it is necessary to sequentially multiply this number by s until the fractional part is pure zero, or we obtain the required number of digits. If, during multiplication, a number with an integer part other than zero is obtained, then this integer part is not taken into account (they are sequentially included in the result).

Let's look at the above with examples.

Example 7 . Let's convert the number 0.214 from the decimal number system to binary SS.

As can be seen from Fig. 4, the number 0.214 is sequentially multiplied by 2. If the result of multiplication is a number with an integer part other than zero, then the integer part is written separately (to the left of the number), and the number is written with a zero integer part. If the multiplication results in a number with a zero integer part, then a zero is written to the left of it. The multiplication process continues until the fractional part reaches a pure zero or we obtain the required number of digits. Writing bold numbers (Fig. 4) from top to bottom we get the required number in the binary number system: 0. 0011011 .

Therefore we can write:

0.214 10 =0.0011011 2 .

Example 8 . Let's convert the number 0.125 from the decimal number system to binary SS.

To convert the number 0.125 from decimal SS to binary, this number is sequentially multiplied by 2. In the third stage, the result is 0. Consequently, the following result is obtained:

0.125 10 =0.001 2 .

Example 9 . Let's convert the number 0.214 from the decimal number system to hexadecimal SS.

Following examples 4 and 5, we get the numbers 3, 6, 12, 8, 11, 4. But in hexadecimal SS, the numbers 12 and 11 correspond to the numbers C and B. Therefore, we have:

0.214 10 =0.36C8B4 16 .

Example 10 . Let's convert the number 0.512 from the decimal number system to octal SS.

Received:

0.512 10 =0.406111 8 .

Example 11 . Let's convert the number 159.125 from the decimal number system to binary SS. To do this, we translate separately the integer part of the number (Example 4) and the fractional part of the number (Example 8). Further combining these results we get:

159.125 10 =10011111.001 2 .

Example 12 . Let's convert the number 19673.214 from the decimal number system to hexadecimal SS. To do this, we translate separately the integer part of the number (Example 6) and the fractional part of the number (Example 9). Further, combining these results we obtain.

Converting numbers from the 8th number system to the 16th. 568?2E16.

Picture 19 from the presentation “Translation of number systems” for mathematics lessons on the topic “Types of number systems”

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Types of number systems

"Binary system" - 1, 2, 4, 8, 16, 32, 64, 128,... Converting integer decimal numbers to binary code. Any decimal number can be represented as the sum of the terms of a series: Wilhelm Gottfried Leibniz (1646-1716). Let's convert the number 121 to the binary number system. Binary number system. Method 1 – difference method.

“Examples of number systems” - Roman number system. CCC. Discharges. 11. 1999 =. Numbers: 123, 45678, 1010011, CXL Numbers: 0, 1, 2, … 4 3 2 1 0. M M. = 1644. – 10. 5. I, V, X, L, … IX. 6. = 1·24 + 0·23 + 0·22 + 1·21 + 1·20 = 16 + 2 + 1 = 19. Topic 2. Binary number system.

“Positional and non-positional number systems” - All number representation systems are divided into positional and non-positional. Any positional number system is characterized by a base. Therefore, positional number systems are predominantly used. An expanded form of writing numbers in the positional number system. Number systems. In practice, the abbreviated notation of numbers is used: A= anan-1 ... a1a0a-1... a-m.

“Different number systems” - Summing up the lesson, homework. Position systems Reckoning. Alphabetic number systems. Lesson is over, goodbye! Practical task: Write in Roman numerals: 29, 57, 128, 1024. Learn theoretical material. The SS alphabet is the digits used to write numbers. Get the correct equalities (you are allowed to move 1 stick): VII – V = XI; IX – V = VI.

“Writing numbers in number systems” - The contents of any file are presented in this form. The Roman system is fundamentally not much different from the Egyptian one. Decimal system. Number systems. Alphabetic systems were more advanced non-positional number systems. Binary system. The symbols used to represent a number are numbers from 0 to 9.

“Number systems lesson” - How does a computer work? Lesson 7. Binary arithmetic (16 ss). Lesson 1. 2cc: 0, 1 8cc: 0, 1, 2, 3, 4, 5, 6, 7 10cc: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 16cc: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A , B, C, D, E, F. What number system does the computer use? The clock operates in duodecimal SS. 111, 555. The computer operates in the binary number system.

There are a total of 13 presentations in the topic