Counting unit: One (one), ten, hundred, thousand, ten thousandCounting unit。One is the basic unit of counting.
Decimal notation: 10 ones are 10, 10 tens are 100... 10 10 billion are 100 billion, and the progression rate between two adjacent counting units is 10.This counting method is called decimal counting method.
When measuring objects, we often getintegerThe number of.So the ancients inventeddecimalTo complement integers.Decimal isDecimal fractionA special form of expression of.The dot in the decimal is calleddecimal point, which is the integer part of a decimal and the decimal partDividing lineThe part to the left of the decimal point is the integer part, and the part to the right of the decimal point is the decimal part.Integer partA decimal that is zero is calledpure decimal, and decimals whose integer part is not zero are calledwhole number with a decimal。For example, 0.3 is a pure decimal, and 3.1 is a decimal.
The basic nature of decimals is that adding or removing zero at the end of decimals keeps the size of decimals unchanged.
Real sumNumber axisThe points on are one-to-one.In other words, real numbers can represent anyline segmentAnd the same line segment has only one length.[1]
Starting from a certain digit after the decimal point, repeatedly displaying the previous digit or a section of digitdecimal systemInfinite decimals.For example, 2.1666..., 35.232323..., the repeated number or a section of number is calledCircular section。The abbreviation of circular decimal is to omit all the numbers after the first circular section, and add a dot above the first and last two digits of the reserved circular section.For example, 2.166... is abbreviated as
(read as "2:16, six cycles"), 0.34103103... 103... abbreviated as
(read as "zero point three four one zero three, one zero three cycle").In the classification of numbers, infinite recurring decimals belong toRational number。
Some decimals are infinite but do not loop.as
Value
2.12459537621Irrational number。Irrational numbers are not like recurring decimals. Every number is repeated, but they also belong to infinite decimals.[1]
Size comparison
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Like integers, the decimal counting units are also arranged in a certain order, and their positions are called decimaldigit。The numerical order is decile, percentile, thousandth, ten thousandth, one hundred thousandth, one millionth.
decimalSize comparison: first look at the integer part. If the integer part is larger, the number will be larger;If the integer part is the same, it depends on the decimal place. If the decimal place is larger, the number will be larger;If the decile is the same, it depends on the percentile. If the percentile is larger, the number will be larger.and so on.
Move the decimal point one place, two places and three places to the right, and the decimal value will be expanded to 10 times, 100 times and 1000 times of the original number.Move the decimal point one place, two places and three places to the left, and the decimal value will be reduced to one tenth, one hundredth and one thousandth of the original number.For example, to expand 7.4 to 10 times the original number, just move the decimal point of 7.4 to the right by one place, that is, 74;To reduce 3.08 to one percent of the original number, just move the 3.08 decimal point 2 places to the left, that is, 0.0308 (note that when the number of decimal places is not enough, you need to add a corresponding "0" in front).[2]
conversion
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Conversion of decimals to fractions
The number that can write the ratio of two integers is calledRational number。Integers and fractions are rational numbers.Rational numbers can be divided intoPositive rational number, 0, andNegative rational number。For example, 3, -98.11, 5.72727272......, 7/22, etc. are all rational numbers.Atdecimal systemIn decimal representation system, rational number can be expressed asterminating decimal Or unlimitedRecurring decimalThe number of.This definition is also applicable under other carry systems (such as binary).
AllfractionCan be expressed asdecimal, division of decimal by infinite non recurring decimal（Irrational number）All can be expressed as fractions.
If the denominator of a simplest fraction contains no prime factors other than 2 and 5, the fraction can be reduced to a finite fraction;If the denominator contains prime factors other than 2 and 5, the fraction cannot be converted into a finite decimal, but can be converted into an infinite decimal.[3]
Correlation calculation
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Example 1: Convert the following fractions into decimals.
Solution: All known fractions can be expressed as decimals.The above fractions are the simplest fractions. If the denominator contains no other prime factors except 2 and 5, the fraction can be reduced to a finite fraction, otherwise it is an infinite fraction.The results are as follows.
Example 2: Judge whether the following decimals can be converted into fractions.
Solution: All known decimals can be expressed as fractions except infinite non cyclic decimals (irrational numbers).It can be seen that
Cannot be converted into fractions, 0.2
And 0.51 can be converted into fractions.I.e
。[3]
Teaching application
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In the process of teaching, let students experience the process of inquiry, which will help students master new knowledge.Mathematics learning process contains mathematical thoughts and methods with more intellectual value than knowledge.In teaching, teachers should pay attention to students' learning process, and fully respect students' understanding level and existing knowledge and experience.Let students calculatefractionturndecimal(Keep three decimal places except for countless decimal places), guide students to observe and compare, encourage students to guess boldly and verify.In this way, it provides students with space and time for independent inquiry.In the process of verifying their own conjectures, students' thinking is active, and they can find that all scores can be expressed as decimals through careful observation and independent thinking.If the denominator of a simplest fraction contains no prime factors other than 2 and 5, the fraction can be reduced to a finite fraction, otherwise it is an infinite fraction.conversely,decimalturnintegerIn, all decimals can be expressed as fractions except infinite non recurring decimals (irrational numbers).In the whole process of inquiry, fully mobilize the enthusiasm and initiative of students to learn. Through the process of knowledge inquiry, students find and understand the knowledge they have learned, thus mastering a learning method of "guess verification".[1]
