C 46 is the standard form of number. Standard form of a positive number – Knowledge Hypermarket

Any decimal fraction can be written as a ,bc ... · 10 k . Such records are often found in scientific calculations. It is believed that working with them is even more convenient than with ordinary decimal notation.

Today we will learn how to convert any decimal fraction to this form. At the same time, we will make sure that such an entry is already “overkill”, and in most cases it does not provide any advantages.

First, a little repetition. As you know, decimal fractions can be multiplied not only among themselves, but also by ordinary integers (see lesson “”). Of particular interest is multiplication by powers of ten. Take a look:

Task. Find the value of the expression: 25.81 10; 0.00005 1000; 8.0034 100.

Multiplication is performed according to the standard scheme, with the significant part being allocated for each factor. Let's briefly describe these steps:

For the first expression: 25.81 10.

Significant parts: 25.81 → 2581 (shift right by 2 digits); 10 → 1 (shift left by 1 digit);
Multiply: 2581 · 1 = 2581;
Total shift: right by 2 − 1 = 1 digit. We perform a reverse shift: 2581 → 258.1.

For the second expression: 0.00005 1000.

Significant parts: 0.00005 → 5 (shift right by 5 digits); 1000 → 1 (shift left by 3 digits);
Multiply: 5 · 1 = 5;
Total shift: right by 5 − 3 = 2 digits. We perform the reverse shift: 5 → .05 = 0.05.

Last expression: 8.0034 100.

Significant parts: 8.0034 → 80034 (shift right by 4 digits); 100 → 1 (shift left by 2 digits);
Multiply: 80,034 · 1 = 80,034;
Total shift: right by 4 − 2 = 2 digits. We perform a reverse shift: 80,034 → 800.34.

Let's rewrite the original examples a little and compare them with the answers:

25.81 · 10 1 = 258.1;
0.00005 10 3 = 0.05;
8.0034 · 10 2 = 800.34.

What's happening? It turns out that multiplying a decimal fraction by the number 10 k (where k > 0) is equivalent to shifting the decimal point to the right by k places. To the right - because the number is increasing.

Likewise, multiplying by 10 −k (where k > 0) is equivalent to dividing by 10 k, i.e. shift by k digits to the left, which leads to a decrease in number. Take a look at the examples:

Task. Find the value of the expression: 2.73 10; 25.008:10; 1.447: 100;

In all expressions, the second number is a power of ten, so we have:

2.73 · 10 = 2.73 · 10 1 = 27.3;
25.008: 10 = 25.008: 10 1 = 25.008 · 10 −1 = 2.5008;
1.447: 100 = 1.447: 10 2 = 1.447 10 −2 = .01447 = 0.01447.

It follows that the same decimal fraction can be written infinite number ways. For example: 137.25 = 13.725 10 1 = 1.3725 10 2 = 0.13725 10 3 = ...

The standard form of a number is expressions of the form a ,bc ... · 10 k , where a , b , c , ... are ordinary numbers, and a ≠ 0. The number k is an integer.

8.25 · 10 4 = 82,500;
3.6 10−2 = 0.036;
1.075 · 10 6 = 1,075,000;
9.8 10−6 = 0.0000098.

For each number written in standard form, the corresponding decimal fraction is indicated next to it.

Switch to standard view

The algorithm for transitioning from an ordinary decimal fraction to a standard form is very simple. But before you use it, be sure to review what the significant part of a number is (see the lesson “Multiplying and dividing decimals”). So, the algorithm:

Write out significant part the original number and place a decimal point after the first significant digit;
Find the resulting shift, i.e. How many places has the decimal point moved compared to the original fraction? Let this be the number k;
Compare the significant part that we wrote down in the first step with the original number. If the significant part (including the decimal point) is less than the original number, add a factor of 10 k. If more, add a factor of 10 −k. This expression will be the standard view.

Task. Write the number in standard form:

9280;

125,05;

0,0081;

17 000 000;

1,00005.

9280 → 9.28. Shift the decimal point 3 places to the left, the number decreased (obviously 9.28< 9280). Результат: 9,28 · 10 3 ;
125.05 → 1.2505. Shift - 2 digits to the left, the number has decreased (1.2505< 125,05). Результат: 1,2505 · 10 2 ;
0.0081 → 8.1. This time the shift was to the right by 3 digits, so the number increased (8.1 > 0.0081). Result: 8.1 · 10 −3 ;
17000000 → 1.7. The shift is 7 digits to the left, the number has decreased. Result: 1.7 · 10 7 ;
1.00005 → 1.00005. There is no shift, so k = 0. Result: 1.00005 · 10 0 (this happens too!).

As you can see, not only decimal fractions are represented in standard form, but also ordinary integers. For example: 812,000 = 8.12 · 10 5 ; 6,500,000 = 6.5 10 6.

When to use standard notation

In theory, standard number notation should make fractional calculations even easier. But in practice, a noticeable gain is obtained only when performing a comparison operation. Because comparing numbers written in standard form is done like this:

Compare powers of ten. The largest number will be the one with this degree greater;
If the degrees are the same, we begin to compare the significant figures - as in ordinary decimal fractions. The comparison goes from left to right, from the most significant to the least significant. The largest number will be the one in which the next digit is larger;
If the powers of ten are equal, and all the digits are the same, then the fractions themselves are also equal.

Of course, all this is true only for positive numbers. For negative numbers, all signs are reversed.

A remarkable property of fractions written in standard form is that any number of zeros can be assigned to their significant part - both on the left and on the right. A similar rule exists for other decimal fractions (see lesson “ Decimals”), but they have their own limitations.

Task. Compare the numbers:

8.0382 10 6 and 1.099 10 25;

1.76 · 10 3 and 2.5 · 10 −4 ;

2.215 · 10 11 and 2.64 · 10 11 ;

−1.3975 · 10 3 and −3.28 · 10 4 ;

−1.0015 · 10 −8 and −1.001498 · 10 −8 .

8.0382 10 6 and 1.099 10 25. Both numbers are positive, and the first has a lower degree of ten than the second (6< 25). Значит, 8,0382 · 10 6 < 1,099 · 10 25 ;
1.76 · 10 3 and 2.5 · 10 −4. The numbers are again positive, and the degree of ten for the first of them is greater than for the second (3 > −4). Therefore, 1.76 · 10 3 > 2.5 · 10 −4 ;
2.215 10 11 and 2.64 10 11. The numbers are positive, the powers of ten are the same. We look at the significant part: the first digits also coincide (2 = 2). The difference starts at the second digit: 2< 6, поэтому 2,215 · 10 11 < 2,64 · 10 11 ;
−1.3975 · 10 3 and −3.28 · 10 4 . These are negative numbers. The first has a degree of ten less (3< 4), поэтому (в силу отрицательности) само число будет больше: −1,3975 · 10 3 >−3.28 · 10 4 ;
−1.0015 · 10 −8 and −1.001498 · 10 −8 . Negative numbers again, and the powers of ten are the same. The first 4 digits of the significant part are also the same (1001 = 1001). At the 5th digit the difference begins, namely: 5 > 4. Since the original numbers are negative, we conclude: −1.0015 10 −8< −1,001498 · 10 −8 .

Lesson topic:

STANDARD TYPE OF NUMBER

Lesson objectives:

Cognitive:

1. Familiarize students with writing numbers in standard form and use the resulting values when solving problems. Establish interdisciplinary connections.

2.Show ways to write large and small numbers.

3. Develop the ability to synthesize and generalize acquired knowledge.

4.Show the significance of the topic in the study of related disciplines.

5. To develop students’ cognitive interest in the subject.

Developmental:

develop in students thinking, speech, memory, the ability to highlight the main thing, and continue to develop the ability to analyze.

Educational:

bring up general culture, activity, independence, ability to communicate, patriotism.

Lesson type:

lesson of explanation and primary consolidation of new knowledge.

Equipment:

route sheet,

technical equipment lesson - computers,

computer presentation in Microsoft PowerPoint.

Teaching methods:

according to the source of acquired knowledge - verbal, practical, visual;

according to the level of cognitive activity - problematic, partially search.

Lesson format: workshop lesson.

“The road will be mastered by the one who walks...!”

DURING THE CLASSES:

Organization of the beginning of the lesson

Hello! Please check your readiness for the lesson.

And now let’s turn to the epigraph of our lesson “The one who walks will master the road...!”

What do these words mean?

Each of you will receive a route sheet in which you will record your work and evaluate it at the end of the lesson.

(route sheets are distributed)

Slide No. 1

Vitamins, minerals, products.

(Task No. 1 on ML)

The correct answers are recorded on back side boards.

Self-test. Slide No. 2-3

We collect points.

II Message of the topic and purpose of the lesson

Slide No. 4

– Before you start studying new topic, complete the tasks on the first page of the route sheet (check on the screen). If you completed the tasks correctly, then you should receive the word - STANDARD.
What is a standard? Where have you come across this word? What does it mean?

(The very first task on ML - table)

Slide No. 5

Standard (from English - standard) A sample, standard, model with which similar objects and processes are compared. (Universal encyclopedic Dictionary). That is, when they talk about a standard, it is easier for people to imagine what they are talking about we're talking about. Today we will talk about the standard form of numbers. So, this is the topic of today's lesson.

Slide number 6

Updating students' knowledge.

Preparation for active educational and cognitive activity at the main stage of the lesson

In the world around us we encounter very large and very small numbers. We already know how to write large and small numbers using powers.

IV.Assimilation of new knowledge

Slides No. 7-8

– Is it convenient to write numbers in this form? Why? (Take up a lot of space, waste a lot of time, and are difficult to remember.)
– What do you think was the way out of this situation? (Write numbers using powers.)

(Task No. 3 on ML)

Use of the concept makes the expression more concise and compact.

Degrees are especially often used when writing large numbers. Such numbers are written using powers with a base of 10. For example:

10 -1 = 0,1

10 0 = 1

10 1 = 10

10 2 = 100

10 3 = 1000

!!! The base 10 exponent shows how many zeros should be written after the number 1.

For example, radius globe, approximately equal to 6.37 million m, is written as 6.37 10 6 m.

The power of 10 6 is equal to 1,000,000 therefore:

6.37 10 6 m = 6,370,000 m

In addition, writing numbers using degrees is used to write natural numbers in the form

4 835 = 4 1000 + 8 100 + 3 10 + 5 = 4 10 3 + 8 10 2 + 3 10 + 5

!!! Every number greater than 10 can be written in standard form:
a 10 n , where 1 ≤ a ≤ 10 and n is a natural number.

This notation is called the standard form of a number.

Slide No. 9

Write the mass of the Earth using powers. 598 10 25 g. Now write down the mass of the hydrogen atom. 17 10 –20 Is it possible to write these numbers differently using powers? Try it! 59.8 10 26, 5.98 10 27; 0.598 10 28 ; 5980 10 24.
17 10 –20 ; 1,7 10 –19 ; 0,17 10 –18 ; 170 10 –21 ;

– All results are correct. But can we talk about standard recording? What should I do? (Agree on a single recording of numbers.)
– Try to discuss with your neighbor what kind of record should be a single, standard one?
– What should be the multiplier before the power of 10, so that it is convenient to REMEMBER the number and represent it?

– Please open slide number 10

And textbooks p. 11 p. 104, find the definition of the standard type of number and write it down on the route sheets.

– Standard type of number called a record of the formA 10 n , where 1< A < 10, n – целое. n – называют порядком числа.

– In standard form you can write any positive number!!!
Why? (By definition. Since the first factor is a number belonging to the interval from )