Knowledge points of integer fractions and percentages

Knowledge points of integer fractions and percentages

1. Organize knowledge points of decimals, fractions, integers and percentages (1) decimals.

The meaning of 1 decimal divides the integer 1 into 10, 100, 1000 ... one tenth, percentage, one thousandth ... can be expressed in decimal.

One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...

Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.

In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10. 2 Classification of decimals

Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals.

With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are both finite decimals with decimals: the digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.

Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 14 15926 ... infinite acyclic decimal: the decimal part of a number with irregular digits is called infinite acyclic decimal. For example: ∈

Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal. For example: 3.555 … 0.0333 …12.15438+009 …

The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, the period of 3.99 ... is "9", and the period of 0.5454 ... is "54". Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal. For example: 3.111.5656 ...

Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222 ...0.03333 ... When writing a cyclic decimal, you only need to write a cyclic segment for the cyclic part of the decimal, and point a point at the first and last bit of this cyclic segment. If there is only one number in the circle, just click a point on it. For example: 3.777 ... Jane writing 0.5302302 ... Jane writing.

(2) scores

1 significance of the score

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.

In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.

Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.

2 Classification of scores

True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction. 3 Reduction and comprehensive score

Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.

The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.

Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.

(3) Integer

Meaning of 1. Integer: Both natural numbers and 0 are integers.

2 natural numbers: when we count objects, 1, 2, 3 ... the numbers used to represent the number of objects are called natural numbers.

There is no object, which is represented by 0. 0 is also a natural number.

Counting units: one, ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, one hundred million ... are all counting units. The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method.

4 digits: Counting units are arranged in a certain order, and their positions are called digits.

(4) Percentage 1 indicates that one number is the percentage of another number, which is called percentage, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.

Although the denominator of the percentage is 100, the numerator can be greater than 100, for example, 200% means twice the original number. For example, a company's net profit last year was 6.5438+0.2 million yuan, and this year's net profit was 6.5438+0.2 million yuan, which can be expressed as "this year's net profit increased by 20% compared with last year" or "this year's net profit increased by 654.38+0.20% compared with last year", but this writing method is rarely used. Percentages are sometimes misleading. Many people think that an increase in percentage will be offset by a decrease in the same percentage. For example, a 50% increase in 100 equals 100+50, that is, 150. And a 50% reduction from 150 is 150-75, which is equal to 75. The final result is less than the original number 100.

2. Organize the knowledge of decimals, fractions, percentages and ratios. Decimal: Fractions with denominators of ten, hundreds and thousands, expressed in digital form.

For example, 0.2 is 2/ 10 and 0.57 is 57/ 100. Percentage: A fraction whose denominator is 100.

Usually indicated by a percent sign. Often used for comparison, it does not represent the actual quantity.

The former term of the ratio is equivalent to the numerator of the fraction and the dividend of the division, while the latter term is equivalent to the denominator of the fraction and the divisor of the division, and the ratio is equivalent to the quotient of the fractional value of the fraction and the division. There are both connections and differences between the law of quotient invariance and the basic properties of fractions.

According to the relationship between division and fraction, the quotient of dividing two numbers can be expressed by fraction. The numerator of a fraction is equivalent to the dividend in division, the denominator of a fraction is equivalent to the divisor (except 0), and the fractional value is equivalent to the quotient.

The constant law of quotient and the basic properties of fraction have the same effect in problem-solving evaluation. However, they are different. First, they are in different forms, one is the division form and the other is the fraction form. Secondly, the result obtained by applying the law of constant quotient can be integer or decimal, while the result obtained by using the basic properties of fraction can only be expressed in the form of fraction.

3. What are the main knowledge points about decimal percentage and fraction? Let me summarize the following points for you: 1. Decimal component number: there are several decimal points, so writing a few zeros after 1 as the denominator and removing the decimal point after the original decimal point as the numerator can reduce the number of quotation points.

2. Fractions become decimals: numerator divided by denominator. Those that are divisible are converted into finite decimals, and some that are not divisible are converted into finite decimals. Generally three decimal places are reserved.

3. Decimal percentage: Just move the decimal point to the right by two places, followed by hundreds of semicolons. 4. Decimal percentage: Decimal percentage, just remove the percent sign and move the decimal point two places to the left.

5. Convert fractions into percentages: usually, first convert fractions into decimals (three decimal places are usually reserved when they are not used up), and then convert decimals into percentages. 6. Decimal percentage: First, rewrite the percentage into a component number and make a quotation that can be converted into the simplest fraction.

4. What do you know about fractions and percentages?

(1) score

1, the meaning of the score

Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.

In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.

Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.

2. Classification of scores

True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.

3. Subtraction and total score

Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.

The denominator of a numerator is a fraction of a prime number, which is called simplest fraction.

Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.

(2) Percentage

A number indicating that one number is a percentage of another number is called a percentage, also called a percentage or a percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.

Differences and connections:

Percent is another way to express a score. Percent is a fraction whose denominator is 100. Percentages cannot replace units, and fractions cannot replace units when expressing fractions, but they can replace units when expressing quantities.

5. How to read and write integers, decimals, fractions and percentages and how to compare sizes. I'll make it up for you in the sixth grade ~ sister, I'm also your compatriot! ~ Numbers like 0. 1.2.3 are called integers, and the number of integers is infinite. There is no minimum integer and no maximum integer. A natural number is part of an integer. How to write an integer: from high to low, any number without a counting unit is written as 0. How to read: from high to low, read at each level. This number is generally graded first, then looked at, and then looked at several levels. The method of comparing the sizes of two integers depends on their digits. If the number of digits is different, the number with more digits will be larger. If the number of digits is the same, the number with more digits on the same digit will be larger. Decimal reading: when reading decimals, read the integer parts from left to right according to the integer reading method. The decimal part reads the numbers on each digit from high to low, even if it is a continuous 0, it should be read in turn. The decimal part is written from left to right, and the integer part is written as the integer part (the integer part is written as' 0'). The decimal point is written in the lower right corner of the unit, and the decimal part writes the numbers on each digit in turn from high to low. Comparison of decimals: Look at their integers first. Integer parts are the same, and the number with the largest tenth digit is the largest; The number of decimal places is unchanged, and the number of decimal places is unchanged. And so on. Meaning of fraction: divide the unit "1" into several parts on average, and the number representing one or several parts is called fraction ~ fractional unit: divide the unit into several parts to represent such a fractional unit is called this fraction. When writing a fraction, write the fractional line first and then the denominator. Write the fractions again. The integer part should be aligned with the fractional line and the distance should be compact. In the column type, the fractional line should be aligned with the middle of two horizontal lines in the "=" symbol. How to read fractions: read the denominator of fractions first, then read "fractions", and finally read the numerator. When reading with fractions, read the integer part first and add "you" in the middle. How to compare scores: (1) The denominator is the same, and the larger the numerator, the greater the score; If the numerator is the same, the score with small denominator will be large; The numerator and denominator are different, so divide them into fractions with the same denominator or numerator, and then compare them. (2) The scores of different integer parts are larger. Reading method of percentage: read the percent sign first, and then read the number before the percent sign. For example, 35% is read as: 35%. Writing method of percentage: percentage is usually not written in the form of fraction, but "%"is added after the original molecule. Write the percent sign again. Method of percentage size comparison: it is not necessary to compare sizes in the form of component numbers or fractions. In fact, you can compare directly, which number is larger, which percentage is larger. Thank you! My answer is over, please accept my answer, but I answered with my eyes open in the middle of the night.

6. Relevant knowledge about integers, fractions, decimals and percentages can be rounded to decimals: integer.0 (there can be any number of zeros); Rounding fraction: integer/1 rounding percentage: integer multiplied by 100 plus% decimal can hardly be rounded to integer; (You can only use all zeros after the decimal point, just delete the zero and decimal point, and the others can only be equal to integers. Decimal fraction: remove the zero after the decimal point and the decimal point/1+N zeros (n is a few decimal places after the original decimal point, just add a few zeros after 1).

Finally, they are usually assigned to the simplest scores. Decimal percentage: Decimal multiplied by 100 plus% Decimal can hardly be converted into integer; (Only numerator can be an integer multiple of denominator, others can only be equal to integers) Fractional decimal: numerator divided by denominator (some can directly calculate the result, some can not directly calculate the result, but the result can be expressed by a cycle, and some can not calculate the result, such as irregular cycle).

Fraction percentage: divide the fraction into decimals first, and then into percentages. Percentages can hardly be converted into integers; (Only the percentage is an integer multiple of 100, and others can only be equal to integers. ) Percent is decimal: divide the value by 100 (basically, the decimal point is shifted to the left by two places). First, percentage the decimal point, and then change it from decimal to fraction.

7. Percent knowledge sorts out what percentage means that the percentage of one number is another number is called percentage. Percent is also called percentage or percentage. The meanings of percentage and fraction are not exactly the same.

Fraction can represent the fraction of a number in the unit "1" or a number, but percentage cannot represent a number. Therefore, percentages cannot be expressed in units.

The significance score of percentage can represent a score or a number. When representing a number, you can have the name of the unit of measurement (this is the main difference between the two); When expressing scores, you can't talk to any company name.

Percentage only represents percentage, and it cannot be followed by any company name. Percent means that one number is a percentage of another. Percent is also called percentage or percentage. Percentages are usually not written in the form of fractions, but are represented by the symbol "%"(called percent sign). If it is written as 4 1%, 1% is 0.0 1. Because the denominator of the percentage is 100. Percentage is widely used in industrial and agricultural production, science and technology and various experiments, especially in investigation, statistics, analysis and comparison. Although the denominator of the percentage is 100, the numerator can be greater than 100, for example, 200% means twice the original number.

For example, a company's net profit last year was 6.5438+0.2 million yuan, and this year's net profit was 6.5438+0.2 million yuan, which can be expressed as "this year's net profit increased by 20% compared with last year" or "this year's net profit increased by 654.38+0.20% compared with last year", but this writing method is rarely used. Percentages are sometimes misleading. Many people think that an increase in percentage will be offset by a decrease in the same percentage. For example, a 50% increase in 100 equals 100+50, that is, 150.

And a 50% reduction from 150 is 150-75, which is equal to 75. The final result is less than the original number 100.

The numerator of percentage can also be decimal. The formation of the concept of percentage should be based on students' real life or examples in industrial and agricultural production. For example, there are 100 students in senior one, among whom 47 are girls, accounting for 47% of the whole grade and 47% of the writing. For example, there are 200 senior two students, including girls 100, accounting for 50% of the whole grade. The number of students in two grades is "standard quantity", while the number of girls is "comparative quantity". In the teaching of percentage application problems, we should grasp the quantitative relationship of = percentage (percentage) for analysis. The application of percentage in daily life will report the weather situation and precipitation probability of the night and tomorrow in the daily TV weather forecast program, prompting everyone to prepare in advance, just like the precipitation probability of tonight is 20%.

20% and 10% are clear and concise. With the rapid development of science and technology, every middle-aged person is now equipped with various styles of mobile phones.

Glenn Wilson, a psychologist at Royal College, University of London, has proved that reading short messages with his head down all the time will lead to low work efficiency and slow down the brain reaction ability of staff, and the IQ of people who read short messages often will drop by 10%, which once again proves in percentage form that although mobile phones provide convenience for people, they are very harmful to human health. This is the information about percentage that I found in my life.

I believe that as long as you observe carefully, you will also find that percentage is everywhere in your life. 80% of China is the largest producer of energy-saving lamps in the world, but 80% of its products are exported and the domestic usage is seriously low.

47. 1% are undergraduate and junior college students for 200 1 year, and 47. 1% of the contracted college students have a monthly salary below 1500 yuan. 85.53% An online survey shows that 85.53% of netizens have never read famous books in recent years.

In addition, 8.58% netizens have never read a famous book in the past ten years, and 6.75% netizens said that they have never read a famous book. The percentage application problem has the following three calculation problems: ① Find the percentage of one number to another, for example, find the percentage of 45 to 225, that is, =20%. ② Find the percentage of a number. For example, find 75% of 2.2, that is 2.2 * 75% = 1.65. Find this number, namely 165÷75%=220. This means that one number is a percentage of another. Percent is also called percentage or percentage. Percentages are usually not written in the form of fractions, but are represented by the symbol "%"(called percent sign). For example, write 4 1% and 1. Convenient comparison. Therefore, percentage is widely used in industrial and agricultural production, science and technology and various experiments. Especially in surveys, statistics, analysis and comparison, percentages are often used. Expansion and improvement: 1. A number indicating that one number is a thousandth of another number is called thousandth rate, and thousandth rate is also called thousandth rate.

Like a percentage, one thousandth has a micron. 2. The internal relationship between percentage and fraction: both can express the multiple relationship of two quantities.

3. The difference between percentage and fraction: (1) has different meanings, and percentage only indicates the multiple relationship between two numbers, and it can't bring the company name; The score can represent a specific number or the relationship between two numbers, and the specific number can be represented by the company name. (2) Percentages can be integers or decimals; The numerator of a fraction cannot be a decimal, but only a natural number other than 0; Percentages cannot be reduced, but fractions are generally divided into the simplest fractions by reduction.

(3) Any percentage can be written as a fraction with a denominator of 100, and fractions with a denominator of 100 do not all have the meaning of percentage. (4) Different application ranges, percentage reproduction and life span are often used for investigation, statistics, analysis and comparison, while scores are often used when the results in calculation and measurement are less than integers.

8. Knowledge of decimals, integers, percentages and ratios. Decimal knowledge summary

When measuring objects, we often get numbers that are not integers, so the ancients invented decimals to supplement integers. Decimal is a special form of decimal. All fractions can be expressed as decimals, except infinite acyclic decimals, all decimals can express the number of components. Irrational numbers are infinitely cyclic decimals.

According to the decimal bit value principle, the decimal part is written in the form without denominator, which is called

Do decimals. The point in the decimal is called the decimal point, which is the dividing line between the integer part and the decimal part of a decimal. The left part of the decimal point is an integer part and the right part of the decimal point is a decimal part. Decimals with zero integer parts are called pure decimals, and decimals with non-zero integer parts are called decimals. For example, 0.3 is a pure decimal and 3. 1 is a decimal.

Like integers, the counting units of decimals are arranged in a certain order, and their positions are called decimals.

Numbers. The numerical sequence is as follows:

There are two ways to read decimals: one is to read fractions, and the whole part with decimals is to read integers; small

The digital part is read by fractions. For example, 0.38 is pronounced as 38%, and 14.56 is pronounced as 14 and 56%. In another way, the integer part is still read as an integer, the decimal point is read as a "dot", and the decimal part reads the numbers on each digit in sequence. For example, 0.45 is read as 0.45; 56.032 is pronounced as 56.032.

The comparison method of decimal size is basically the same as that of integer, that is, the numbers on the same digit are compared in turn from the high position.

Therefore, to compare the sizes of two decimals, first look at their integer parts, and the larger the integer part, the larger the number; If the integer parts are the same, the one with the largest number in the tenth place is larger; If the deciles are the same, the percentile is larger;

Because decimals are decimal fractions, they have the following properties: ① Add or remove zero at the end of decimals, and the size of decimals.

No change. For example; 2.4=2.400,0.060=0.06.② When the decimal point is shifted to the right by one, two and three places respectively, the decimal size will change, and the decimal value will be enlarged by 10 times, 1000 times and1000 times respectively. ...

Times; If the decimal point is shifted to the left by one place, two places and three places respectively, the decimal value will be reduced by 10 times, 100 times and 1000 times respectively. For example, expand 7.4 10 times to be 74, and expand 100 times to be 740. ..

Infinitely circulating decimals can only be expressed by fractions, such as 1/7, and all decimals can be expressed by fractions. Fractions are divided into finite decimals such as 1/5, infinite cyclic decimals such as 1/7, and infinite cyclic decimals such as 1/3.

Rationalnumber: a number that can be accurately expressed as the ratio of two integers.

For example, 3, -98. 1 1, 5.72 ... and 7/22 are rational numbers.

Integers and fractions are rational numbers. Rational numbers can also be divided into positive rational numbers, 0 and negative rational numbers.

In the decimal representation system of numbers, rational numbers can be expressed as finite fractions or infinite cyclic fractions. This definition also applies to other decimals (such as binary). Encyclopedia of China (Mathematics)

Therefore, there is no contradiction.

Add "0" or remove "0" at the end of the decimal, and the size of the decimal remains the same. This is the so-called nature of the decimal.

Decimal times integer:

Convert decimal multiplication into integer multiplication calculation.

First, expand the decimal into an integer, multiply by the integer, and the product will be reduced by as many times as the factor is expanded.

The decimal places of the product are related to the decimal places of the multiplicand. If the multiplicand has several decimal places, so does the product. Because to convert decimal multiplication into integer multiplication, the product will be amplified as many times as the multiplicand. So how many times must the product be reduced?

Calculate decimal multiplication with integers. First, calculate the product according to the calculation method of integer multiplication, and then see how many decimal places the multiplicand has. Count a few from the right of the product and point to the decimal point.

A decimal, starting from somewhere in the decimal part, and one or several numbers are repeated in turn. This decimal is called a cyclic decimal.

Circular part: the decimal part of a circular decimal, which is a number that appears repeatedly in turn.

It is called the cyclic part of this cyclic decimal. For example: 0.33 ... The loop part is "3"

2. 14242 ... The cyclic part is "42"

Pure cyclic decimal: the cyclic part starts from the first position of the decimal part.

Mixed cycle decimal: the cycle part does not start from the first position of the decimal part. (For example:

Blackboard book)

Simple notation: When writing cyclic decimals, for simplicity, only the cyclic part of decimals is written.

The first cycle part. If there is only one number in the loop, add a dot to this number; If there are multiple numbers in the loop part, please add a dot to the first and last numbers in this loop part.