Is the smallest number 0 or 1?

This issue has been controversial for a long time. Let's take a look at the description of "how many digits" on page 98 of "A Book for Primary School Mathematics Teachers in Nine-year Compulsory Education for Six Years": "Usually, in natural numbers, numbers containing several digits are called how many digits. For example, "2" is a number containing a number, which is called a number; 30 "is a number with two digits, called two digits; 405 "is a three-digit number, called three digits ... but it should be noted: generally speaking, 0 is not a number.

Let's listen to the expert's explanation: in the theory of natural numbers, the definition of "several numbers" is as follows: "A number represented by only one significant number is called a number; A number represented by only two digits (the first digit on the left is a significant digit) is called a two-digit number ... Then, how many digits are there in a number (the first digit on the left is a significant digit), and this number is called a few digits.

The so-called maximum and minimum digits here are usually studied in the range of non-zero natural numbers. So there are nine numbers * * *, namely: 1, 2, 3, 4, 5, 6, 7, 8, 9.

0 is not the smallest number.

02

Why is 0 also a natural number?

The stipulation that "0 is also a natural number" in the textbook of curriculum standard subverts people's traditional understanding of natural numbers.

Here, Chen Changzhu, editor-in-chief of the textbook compilation group of the Central Institute of Education, said: All along, there have been different definitions of natural numbers internationally. Most countries, such as France, think that natural numbers start from 0, while China's textbooks always follow the view of the former Soviet Union that 0 is not a natural number. In 2000, when the Ministry of Education hosted a meeting on textbook adaptation, it was clearly put forward that 0 should be classified as a natural number. This revision is also in line with international practice.

From the teaching practice, it is also of positive practical significance to define "0" as "natural number".

"0" as the "income" of natural numbers

As we all know, sets in mathematics are divided into finite sets and infinite sets. A finite set is a set of finite elements, just like a group of students in a class. An infinite set is a set with infinite elements, such as a set of fractions. Because natural numbers have the property of "radix", it is natural to describe the number of elements in a finite set with natural numbers.

But in the finite set, there is one of the most important and basic sets, which is called empty set {} and the number of elements is 0. If 0 is not a natural number, then the number of elements in an empty set cannot be represented by a natural number. If "0" is regarded as a natural number, then this natural number can complete the task of describing the number of elements in a finite set. Here, from the perspective of "cardinal number of natural numbers", we see the benefits of taking "0" as a natural number.

Taking "0" as a natural number will not affect the "operation function" of natural numbers.

When "0" is added to the traditional set of natural numbers, all the "operation rules" remain unchanged. For example, any two natural numbers in a new set of natural numbers {0, 1, 2, …, n, …} can be added and multiplied, and the operation result is still a natural number. At the same time, the associative law and commutative law of addition and multiplication operations and the distributive law of multiplication will not be affected.

Therefore, it is natural for "0" to join the set of natural numbers, not just a man-made "regulation". It makes us better understand natural numbers and their functions, and at the same time makes us realize that we should not only know and remember the "definition" and "law" of mathematics, but also think about the mathematical significance behind "law" in teaching.

03

What is a valid number?-an invalid number.

A significant number is the accuracy of an approximation of a number. If the same divisor keeps more significant figures when choosing, it will be more accurate than keeping fewer significant figures.

Generally speaking, the approximate number is rounded to which place, that is, the approximate number is accurate to which place. At this time, all digits from the first non-zero number on the left to this number are called the significant digits of this number.

For example, the approximate number 0.00309 has three significant figures: 3, 0 and 9; 0.520 also has three valid words: 5, 2 and 0.

Three zeros on the left in 0.00309 and one zero on the left in 0.520 are called invalid numbers.

04

Is addition, subtraction, multiplication and division a reciprocal operation?

"Addition and subtraction are inverse operations, multiplication and division are inverse operations" seems to be the mantra of many teachers, but it is actually a misunderstanding. For example:

Addition "2+3 = 5", its inverse operations are "5-2 = 3" and "5-3 = 2".

So the inverse operation of addition is only subtraction;

If you subtract "5-2 = 3", the inverse calculation is "5-3 = 2" and "2+3 = 5".

Therefore, there are two operations for the inverse operation of subtraction: subtraction and addition.

To sum up, we can only say that subtraction is the inverse of addition, but we can't say that addition and subtraction are reciprocal.

In the same way, we can only say that division is the inverse of multiplication, but we can't say that multiplication and division are reciprocal.

05

Why not write "Time"?

When learning the application question "How many times is one number the other", many children will naturally ask such questions, such as: "The feeding group raised 12 chickens and 3 ducklings. How many chickens are there? " Why not write "times" after "12 ÷ 3 = 4"?

First of all, we must affirm students' doubts (students have a strong sense of problem-solving norms). But at the same time, it should be explained to students that when solving application problems, the unit name of the number is generally written after the number.

Such as: 12 "only"; Eight grams. Only with the unit name can a number accurately represent the quantity, size, length, weight and so on of an object. However, "times" is not a unit name, it represents the relationship between two quantities. For example, the above calculation result "4" indicates that there are four 3s in 12, which means that the number of chickens in 12 is four times that of three ducklings.

Therefore, don't write "times" in the formula to avoid confusion between "times" and the company name.

06

The difference between "times" and "multiples"

In the first period, we learned the concept of "times", and in the second period, we learned the concept of "times". So, are the words "multiple" and "multiple" the same thing? What's the difference between these two words?

"Times" refers to the quantitative relationship, which is based on the concept of multiplication and division. For example, boys 10, girls 30. Because "10×3=30" or "30÷ 10=3", we say that the number of girls (30) is three times that of boys (10). Bo Ning said that "times" is actually the quotient of two numbers (this quotient can be an integer, a decimal, a fraction and other expressions).

"Multiplication" refers to the relationship between numbers, which is based on the concept of divisibility. For example, 30 is divisible by 6, and 30 is a multiple of 6. It can be seen that "multiplicity" cannot exist independently (with specific directivity), and the form of logarithm has special requirements (it must be an integer).

At the same time, we also see that 30 is five times that of 6, because 6×5 = 30, "6×5" is five times that of 6. Therefore, from this perspective, the meaning of "duo" should be broader than that of "duo", and the latter can be regarded as a manifestation of the former under certain circumstances.

07

What is the difference between "hour" and "hour"? How to use "hour" and "hour"?

The first thing to be clear is that [small] time is not an international time unit. In 1984 "Order on Unification of Legal Units of Measurement" issued by the State Council, second is the basic unit of time, and non-international time units such as day (day), hour and minute are the auxiliary units.

(Note: The words in [] can be omitted without confusion).

In this way, the legal time units used in China are: day (day), hour (hour), minute and second.

Therefore, "time" can represent both time and time. Because the two different concepts of "time" and "moment" are easily confused, when the time unit "time" is actually applied, the current textbooks treat it like this:

7. 1 When calculating the length of continuous time, write the time unit "hours" in parentheses. For example, supermarket business hours: 2 1-9= 12 (hours). (The word "small" can be omitted here)

7.2 When expressing the length of time in language, in order to avoid confusing the two concepts of "time" and "moment", a word "small" is added before "time". For example, the business hours of supermarkets are 12 hours.

7.3 When the time is expressed in words, the word "hour" shall not appear. For example, the park leaves at 7: 30 every morning (instead of 7: 30).

08

Are "rewriting" and "ellipsis" the same?

Formally, this example puts the changes of "rewrite" and "omit" logarithms under the same requirement (that is, rewrite them as numbers in units of "100 million"). We really hope that editing is not intentional, because the essence of "rewriting" and "ellipsis" is completely different. Show in:

8. 1 has different uses.

The purpose of "rewriting" is to facilitate the reading and writing of large numbers, and "ellipsis" is an approximation of numbers.

8.2 Different methods

"Rewriting" here is to remove the 0 after the "billion" bit and write a word "billion", while "ellipsis" should not only find the "billion" bit, but also consider omitting the highest digit of the mantissa, and then find the approximate number by rounding.

8.3 Different symbols

"Rewriting" only changes the expression form of the number, and the size has not changed, so it is connected with "="; The "ellipsis" not only changes the form of the number, but also changes the size of the number, so it is connected with "≈".

09

Is "distance" the same as "distance"?

These two words are used interchangeably in many teachers' teaching languages, but they are not.

"Distance" refers to the length of the route from one place to another; And "distance" refers to the length of a straight line segment connecting two places.

The route a "journey" takes can be a curve, a straight line or a broken line.

Generally speaking, the "distance" between two places is greater than the "distance" between two places. Distance and distance are equal only if the route between the two places is straight.

Although teachers all know that this equation is valid, but our students have no corresponding knowledge reserves, so how to bypass the "limit" and find a proof method that primary school students can understand and accept.

10

Is the maximum scoring unit 1/2 or11?

Let's look at the meaning of decimal unit: divide the unit "1" into several parts to represent such a number.

Obviously, in the sense of score, the key is "score". Without "score", there is no "share".

Because the unit "1" is divided into at least two parts on average (if it is 1, there is no "score"), the unit of score obtained is 1/2, so 1/2 is the largest unit of score.

Although11can also be regarded as a score in a broad sense, it is not the kind of score that we usually know as opposed to an integer (generated on the basis of average score). So the maximum unit of the score should be 1/2.

1 1

Are numbers like 0/3, 0.2/3, 3/0.2 fractions?

The definition of fraction clearly tells us that dividing the unit "1" into several parts to represent such a number or part is called fraction. Among them, the number of divided shares is called the denominator of the fraction, and the number of shares to be expressed is called the numerator.

So the numerator and denominator of the fraction should be non-zero natural numbers. In this sense, the above figure is the form of a fraction, not the essence of a fraction, so it should not be regarded as a fraction.

Furthermore, when examining students' understanding of the meaning of "score", we should focus on the score in the usual sense and bring these changes into the scope of thinking, which is of little practical significance for training students' thinking, and will make the proposition of "score greater than 0" embarrassing.

12

Should the number 1/2 greater than 6 be "6+ 1/2" or "6+( 1+ 1/2)"?

To understand this problem, we must first understand the nature of "6". Obviously, the essence of "6" here is a number, not a quantity. Finding a number larger than 6 1/2 should belong to the category of "finding a number larger than 6". The "several" in the question are all specific numbers, and the "several" here can be both integers and numbers. So "1/2" here refers to the number of "1/2" based on 6, not "0/2 of 65438+6".

So the number "1/2 is greater than 6" should be "6+ 1/2".

Of course, if the topic is "1/2 is greater than 6", then the answer belongs to the latter.

13

Can attendance be calculated without multiplying by 100%?

Let's take a look at the understanding of similar problems in three different versions of textbooks: People's Education Edition, Beijing Normal University Edition and Soviet Education Edition.

Under the same curriculum standard, different textbooks give different understandings, which brings confusion to the instructor: can we not take the exam 100%? The author thinks that the result of finding "×× rate" must be percentage. Take attendance as an example, that is, what percentage should the actual attendance be?

If the formula is only written as: attendance rate = actual attendance/attendance, we say that this is only a fraction form (that is, to find the "fraction" of actual attendance to attendance), not a percentage.

Therefore, multiplying the formula by "100%" can not only keep the calculated value unchanged, but also ensure that the result form meets the percentage requirements. Therefore, "100%" should be multiplied in the formula for calculating attendance, germination rate, flour yield and qualified rate.

At the same time, it is suggested that the editors of all editions of textbooks should unify their thinking so as not to cause confusion to front-line teachers.

14

Are all angles less than 90 degrees acute?

According to the definition of the curriculum standard textbook, an angle less than 90 degrees is called an acute angle. The answer seems to be yes, but this leads to a new question: what is an angle of 0 degrees, and is it also an acute angle?

The definition of acute angle actually has an implicit premise, that is, all angles discussed in primary school mathematics are positive angles. Traditionally, we call the angle obtained by a ray rotating counterclockwise a positive angle and the angle obtained by a ray rotating clockwise a negative angle. When the light does not rotate, it is considered as a zero-degree angle. If the concept of angle is extended to any angle, it should be divided into positive angle, negative angle and zero angle.

So the strict definition of acute angle should be: an angle greater than 0 degrees and less than 90 degrees is called acute angle.

15

Is "3︰2" on the scoreboard of football match a mathematical comparison?

We can understand their differences from at least two aspects.

1. "3 ~ 2" in the ball game represents the score of both sides in the game, which is the "poor" ratio, that is, it means the poor relationship, one side gets 3 points, the other side gets 2 points, and the difference between the two sides is 1 minute; "3︰2" in mathematics means "3 ︰ 2", which is the ratio of "times" and the quotient is 1.5. In view of this, the "ratio" (actually the score) in ball games can be zero, while the "ratio" and its subsequent numbers (equivalent to divisor) in mathematics cannot be zero.

Secondly, "ratio" in mathematics can be simplified, such as "4: 2 = 2:1"; The same "4￠2" cannot be simplified in a ball game. If simplified, it will not reflect the actual scores of both sides in the game.