A method of converting rational numbers, especially infinitely looping decimals, into fractions

Since the number of decimal places is infinite, it is obviously impossible to write ...... tenths, hundredths, thousandthsof the number. In fact, the difficulty of cyclic decimal fractions lies in the infinite number of decimal places. So I'll start here and find a way to "cut off" the "big tail" of the infinitely looping decimals.

The strategy is to use the method of multiplication to expand the infinite loop decimal by ten, hundred, or thousand......foldMake the enlarged infinite loop decimal exactly the same as the "big tail" of the original infinite loop decimal, and then subtract the two numbers, and the "big tail" will be cut off! Let's look at two examples: Summing into fractions.

Think 1: 100 1) i.e. 99 = 47 then Think 2: 10-1) i.e. 9 Then It follows that the pure cyclic decimal fraction, its decimal part can be written as a fraction like this:

The minimum number of digits of a pure cyclic decimal is a few of the number of cyclic nodes, and the denominator is a number composed of several 9s; A molecule is the number of a cyclic node in a pure cyclic decimal. Sum into fractions. Thought 1:

Use to get: So, think 2: Use to get:

So,

Just remember 1 2; 1/3;1/4;1/6;1/7;1/8;1/9;These numbers make up the number after the decimal point; Plus the number before the decimal point is what is sought.

There are n-bit loops.

The loop part n-bit 9) + the integer part is reduced to the end.

Non-circular part + circular part.

Not necessarily, because pi is an irrational number, but it can be reduced into fractions, because it is equal to the circumference of the circle divided by the straight meridian of the circle, so the proposition is a false proposition.

You can multiply the numerator and denominator of the fraction by 2 or 5 at the same time, divide it by 10, and add a decimal place to the fraction of the second case. For example, 1 6 = 5 30 = 3 30 + 2 30 = is a decimal plus 10 n of a pure cyclic decimal, which is a mixed cyclic decimal.

The mixed cycle rewrites the mixed cycle decimal into a fraction, and the numerator is the number composed of the numbers formed by the non-cyclic part and the first cyclic section, minus the difference between the number composed of the non-cyclic part. The first digits of the denominator are 9, the last digits are 0, the number of 9 is the same as the number of the loop section, and the number of 0 is the same as the number of the non-loop part. For example:

Proposition: Fractions do not appear infinitely cyclic decimals.

Proof: We can look at this problem from the process of integer division:

If there is an infinite non-cyclic decimal, which can be expressed as the simplest fraction p q, then divide p by q, which is inexhaustible, and the resulting decimal is infinitely acyclical.

Let's look at the process of division in the context of integer division.

When dividing to a certain digit, the quotient k and the remainder is r. This remainder must be finite (e.g., within 10, or within 100, or within 1000. Determined by the conditions of q)

Then this remainder can no longer appear in the following division (once it does, the result goes back into the loop. ）

But the remainder is finite, and its upper limit is also finite, such as within 10, then the appearance of the remainder is nothing more than these 10 numbers, that is, it is impossible to appear infinitely different remainders.

So, the score is bound to go into the loop.

The proposition proves that fractions do not appear infinitely cyclic decimals.

So, fractions must be reduced to finite decimals or infinite looping decimals.

The denominator only contains the factors of 2 and 5, and does not contain other factors, such as: 7/2, 6/5, 9/10, 19/40, etc.

But like 5 out of 6 you can't get it, because there is a 3 in addition to 2, but only to a finite decimal.

It must be a rational fraction with a root number, and the rest are finite decimals or infinite cyclic decimals.

Summary. A fraction must be reduced to a finite decimal or an infinite loop decimal, which is the correct expression.

A fraction must be reduced to a finite decimal or an infinite loop decimal.

A fraction must be reduced to a finite decimal or an infinite loop decimal, which is the correct expression.

Because the expression of the fraction must find a definite point on the number line, and then the finite decimal and the infinite cyclic decimal are both deterministic electricity on the number line, so the two can be converted. <>

Why can't infinite non-cyclic decimals be expressed as fractions? It is because infinite non-cyclic decimal is a point that cannot be determined on the number line.

All definite points on the number line can be expressed as fractions.

The number on the number line is determined + uncertain.

Children, you are still confused, communicate with the teacher. <>

I'm going to explain to you the most essential reason, and it may be a little difficult for you to understand.

Problem solving ideas: first calculate their quotient, find out the number that the decimal part of the quotient repeats in turn, that is, the cycle section, abbreviated notation: write the cycle section again, and write a small dot on the top of the first and last digits of the cycle section

Key Cons Jane. Review: Butan.

Test points of this question: the interaction between decimals and fractions; Cyclic decimals and their classification Test Center Comment Pants: This question examines how to represent cyclic decimals in a simple form, and the key is to find the number of the loop, i.e., the loop section

The method of decimalizing fractions in an infinite loop:

1. The method of pure cyclic decimal, e.g., cyclical) = (ab 99), and finally simplified. Here are some examples:

loop) = 3 9 = 1 3;

Cycle or Spike) = 7 9;

loop) = 81 99 = 9 11;

Cycle) = 1 and 206 999

2. Mixed circulation decimal method, e.g., circulation) = (abc a) 990Finally, simplify. Here are some examples:

loop) = (51 5) 90 = 46 90 = 23 45;

circulation) = (2954 29) 9900 = 13 44;Family pure.

Cycle) = 1 again (4189 4) 9990 = 1 again 4185 9990 = 1 again 31 74.

Decimals can be divided into two categories: finite decimals and infinitesimal decimals, which in turn are divided into two categories: infinitely cyclic decimal and infinitely non-cyclic decimals.

1. Definition of infinite cyclic decimals: the decimal infinite decimal number of the previous one or a section of numbers that begins to continuously appear after a certain digit after the decimal point. Such as232323…etc., the number that is repeated is called a circular section.

The abbreviation for infinite loop decimal is to omit all the digits after the first loop and add a small dot above the first and last two digits of the reserved loop section. For example, the abbreviation is, (pronounced "two point one six, six cycles"). In the classification of numbers, infinitely cyclic decimal numbers belong to rational numbers.

1. To see how many decimal places it is, add a few 0s to the end of 1 as the denominator;

2. Remove the decimal point from the decimal point and make the molecule;

3. The offer points that can be contracted.

For example, two small mountain Li numbers - add 2 zeros after 1 as the denominator (that is, 100) - remove the decimal point to make the numerator (that is, 25).

The score is 125 out of 100 – about 1 out of 4

cycle) is 23 99

circulation) is 23 990

23 can be any number, the three-digit number (234 cycle) denominator is more than 9, and the 0 before the cycle is replaced by a two-digit denominator, and there are 2 more zeros after the two-digit denominator

If it is not 0 before the cycle, multiply by the denominator and add the value of the cycle. For example, the denominator is 990, so the fraction is 211 990

See if that helps.

That's right, there seems to be a formula for transformation.

Step 1Divide the infinite loop decimal into 2 parts, and take the example you gave, divide it into these 2 parts.

Step 2This is a method of dividing these two parts into fractions. Let it be a first, then there is:

10a=1000a=

1000a-10a=45

990a=45

a=45/990=1/22

So step 3Adding the two parts together gives the result of the infinite loop decimal fraction: 3 10 + 1 22 = 66 220 + 10 220 = 76 220 = 19 55

So it's the same way to solve it.

1) Divide first.

2) Design. 1000a=

100000a=

100000a-1000a=12

99000a=12

a=12/99000=1/8250

Yes, because fractions are rational numbers. A fraction is either a finite decimal or an infinite cyclic decimal, and an infinite non-cyclic decimal like etc. cannot be replaced by a fraction.

The fraction is the ratio of an integer a to a positive integer b that is not equal to the integer.

When speaking in everyday language, fractions describe parts of a certain size, such as half, five-eighth, three-quarter. Numerators and denominators are also used for uncommon fractions, including compound fractions, complex fractions, and mixed numbers.

A fraction indicates that a number is a fraction of another number, or the ratio of one event to all events. The unit "1" is divided into several parts, and the number of such parts or parts is called a fraction. The numerator is on top and the denominator is on the bottom.

Fractions are rational numbers, yes and infinite non-cyclic decimal numbers are irrational numbers.