Number order in standard form. Standard form of a positive number

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 is known, decimals You can multiply not only among each other, but also by ordinary integers (see lesson “”). Special interest represents 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.

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

For the second expression: 0.00005 1000.

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

Last expression: 8.0034 100.

  1. Significant parts: 8.0034 → 80034 (shift right by 4 digits); 100 → 1 (shift left by 2 digits);
  2. Multiply: 80,034 · 1 = 80,034;
  3. 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:

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

  1. 2.73 · 10 = 2.73 · 10 1 = 27.3;
  2. 25.008: 10 = 25.008: 10 1 = 25.008 · 10 −1 = 2.5008;
  3. 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 = ...

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

  1. 8.25 · 10 4 = 82,500;
  2. 3.6 10−2 = 0.036;
  3. 1.075 · 10 6 = 1,075,000;
  4. 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:

  1. Write out significant part the original number and place a decimal point after the first significant digit;
  2. 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;
  3. 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:

  1. 9280;
  2. 125,05;
  3. 0,0081;
  4. 17 000 000;
  5. 1,00005.
  1. 9280 → 9.28. Shift the decimal point 3 places to the left, the number decreased (obviously 9.28< 9280). Результат: 9,28 · 10 3 ;
  2. 125.05 → 1.2505. Shift - 2 digits to the left, the number has decreased (1.2505< 125,05). Результат: 1,2505 · 10 2 ;
  3. 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 ;
  4. 17000000 → 1.7. The shift is 7 digits to the left, the number has decreased. Result: 1.7 · 10 7 ;
  5. 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:

  1. Compare powers of ten. The largest number will be the one with this degree greater;
  2. If the degrees are the same, we begin to compare the significant figures - as in ordinary decimal fractions. Comparison is underway from left to right, from most significant to least significant. The largest number will be the one in which the next digit is larger;
  3. 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:

  1. 8.0382 10 6 and 1.099 10 25;
  2. 1.76 · 10 3 and 2.5 · 10 −4 ;
  3. 2.215 · 10 11 and 2.64 · 10 11 ;
  4. −1.3975 · 10 3 and −3.28 · 10 4 ;
  5. −1.0015 · 10 −8 and −1.001498 · 10 −8 .
  1. 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 ;
  2. 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 ;
  3. 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 ;
  4. −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 ;
  5. −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 .

















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Lesson type: a lesson in explaining and initially consolidating new knowledge.

Equipment: route sheet(MR) ( Annex 1 ); technical equipment of the lesson - computer, projector for demonstrating presentations, screen. Computer presentation in Microsoft PowerPoint.

DURING THE CLASSES

I. Organization of the beginning of the lesson

Hello! Please check availability handouts on your desk and your readiness for the lesson.

II. Communicating the topic, purpose and objectives of the lesson

– Before starting to study a 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? (SCREEN)
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. Today we will talk about the standard form of numbers. So that's the topic of today's lesson.

III.Updating students' knowledge. Preparation for active educational and cognitive activity at the main stage of the lesson

– Let’s make a lesson plan:

  1. Repetition
  2. Determination of powers of a number;
  3. Determining the power of a number with a negative exponent;
  4. Properties of degree;
  5. Definition of the standard type of number;
  6. Actions with numbers written in standard form;
  7. Application.

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.

– 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.)

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 factor before the power of 10 so that it is convenient to REMEMBER the number and present it?

IV. Learning new knowledge

– Please open your textbooks, paragraph 35, and find the definition of the standard type of number and write it down on the route sheets.
– The standard form of a number is a notation of the form A 10n, 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 )

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