Hello netizens, the editor-in-chief of Wenxun Baitong. Two days ago, a child sent me a private message, saying that he had just learned rational numbers. Tell him that rational numbers include integers and fractions, and that fractions can be converted to either finite decimals or infinitely looping decimals. He wanted to ask me, on what basis does a fraction have to be converted into a finite decimal or an infinite loop decimal? Is the inverse proposition of this question, a finite decimal or an infinite cyclic decimal also necessarily convertible into a fraction? Solving this problem is not as simple as everyone thinks. Today, let's take a look at how fractions and decimals are converted to each other.
First, let's look at how to convert decimals into fractions. For finite decimals, converting to fractions is intuitive, e.g. 04 can be expressed as 4 10 or simplified to 2 5;0.65 can be expressed as 65 100 and can also be reduced. But when it comes to infinitely looping decimals, how do you convert them? For example, if a decimal x is denoted as 03333...So what fraction is this number equal to? Obviously, it is equal to 1 3. But how do we do that?
There is an important knowledge point here, which is the continuous fraction. Consecutive fractions are a method of representing infinitely cyclic decimals, and a concrete representation of fractions can be obtained by derivation. For the above example 03333...We can express it as 0+1 (3+1 (3+1 (3+....)where "...Indicates an infinite loop. We can denote the part in parentheses as x, then the original can be expressed as 0+1 x. Next, let's calculate the value of x, which has x=3+1 x, and we get x=3 2. Therefore, we can change 03333...It is expressed as 3 9, which is 1 3. The same method can be used to convert other infinitely looping decimal numbers into fractions.
Next, let's take a look at how to convert fractions to decimals. For finite fractions, the process is relatively simple and only requires division. For example, 2 5 can be expressed as 04, while 3 8 can be expressed as 0375。However, when there are other quality factors in the denominator, it is necessary to pass the score. For example, 5 7 can be expressed as (a 10 + b) 10, where a and b are two integers. If we multiply 5 7 by 10, we get 50 70 = a + b 10, and then we divide 50 and 70 to get 5 7 = a + 3 70. Therefore, 5 7 can be represented as 071428571...i.e. cyclic decimals.
So can any fraction be expressed as a decimal? The answer is no. We know that a fraction can be expressed as a cyclic decimal if and only if its denominator contains only two factors, 2 and 5. For example, 1 4 can be represented as 025, while 1 5 can be represented as 02。But when the denominator also contains other quality factors, the situation becomes complicated. At this point, we need to convert the fraction into a decimal by means of a common fraction. However, there are some fractions that cannot be expressed as finite or cyclic decimal regardless of the method, such as 1 3.
In general, the conversion between fractions and decimals seems simple, but in fact it involves many complex knowledge points. By understanding concepts such as continuous fractions, common scores, etc., we can better grasp these transformation methods. Hopefully, this article will provide readers with some useful knowledge and give you a deeper understanding of fractions and decimals in mathematics.
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