Repeating decimal is divided into two categories: mixed repeating decimal and pure repeating decimal. Mixed repeating decimal can be *10^n (n is the non-cyclic number of digits after the decimal point), so repeating decimal to fractions can eventually be transformed by pure repeating decimal.edited by geometric seriesInfinite loop Decimal, first find its circulation section (that is, the number of loops), and then expand it into a geometric series, find out the first n subparagraphs, take the limit, simplify. For example: 0.333333 ... The link is 3 0.33333.....=3*10^ ( -1) +3*10^ ( -2) +......+3^10 (-N) + ... The first n entries are: 0.3 (0.1) ^ (n))/(1-0.1) when n tends to infinity (0.1) ^ (n) =0 so 0.3333......=0.3/0.9=1/3 Note: The meaning of the m^n is the n-th side of M. Again such as: 0.999999 ..... The link is 9 0.9999.....=9*10^ ( -1) +9*10^ ( -2) +......+9*10^ (-N) + ... The first n and is: {0.9*[1-(0.1) ^n]}/(1-0.1) when n tends to infinity (0.1) ^n=0 therefore: 0.99999.....=0.9/0.9=1Solution Equation Method EditingThe fractional fraction of infinite loop can be divided into two kinds of cases, pure repeating decimal, mixed repeating decimaldecimal fraction Pure Repeating decimal
Example:0.1111 ... 1 of the loop, we can set this decimal number to X, can get: 10x-x=1.1111......-0.1111 ... 9x=1x=1/9
Example:0.999999.......=1 set x=0.9999999......10x-x=9.999999.....-0.999999.....9x=9x=1 on this aspect, can also use the limit of knowledge to prove, here is not repeat.
Example: Will infinite loop decimal 0.26 (•) Turn into fractions: solve the problem: known infinite loop fractional 0.26 (•), will be known infinite loop fractional 0.26 (•) The unknown score is set to X, which is 0.26 (•) =x--1, making 100x=100 (0.26+0.0026), 100x=26+0.26 (•) --2, (2-type) infinite loop fractional 0.26 (•) Replacement for X: 100x=26+x,100x-x=26,99x= 26,x=26/99,∴x=0.26 (•) =26/99, i.e.: 0.26 (•) =26/99
Example: Will infinite loop decimal 0.123 (•) Turn into fractions: solve the problem: known infinite loop fractional 0.123 (•), will be known infinite loop fractional 0.123 (•) The unknown score is set to X, which is 0.123 (•) = X--1, 1000x=1000 (0.123+0.000123 (•)), 1000x=123+0.123 (•) --2, (2-type) infinite loop fractional 0.123 (•) Replacement for X: 1000x=123+x,1000x-x=123, 999 x=123,x=123/999,x=41/333,∴x=0.123 (•) =41/333, i.e.: 0.123 (•) =41/333
inductiveIn order to formulate, we can say this: x 10∧b-x, where B is the number of bits that follow the link. This is suitable for all pure repeating decimalMixed Repeating decimal
Example:0.12111 ... 1 cycle, again, we set this decimal number to X, can be: 1000x-100x=121.111......-12.111 ... 900x=109x=109/900
Example: Will infinite loop decimal 0.123 (·) Into fractions: puzzle solving: Known infinite loop fractional: 0.123 (·), will known infinite loop fractional 0.123 (·) The unknown score is set to X,∴x=0.123 (·) --1 type, (1-type) multiply by 10 at the same time: 10x=1.23 (·) --2 type, (2-type)-(1-type): 9x=1.11,x =1.11/9,x =0.37/3,x =37/300,∴x=0.123 (·) =37/300, i.e.: 0.123 (·) =37/300
inductiveThe formula is: X 10∧ (A+c)-x 10∧a, where a is the number of digits before the circular section of the decimal point, and C represents the number of cyclic nodes. With decimals also applicable!!differencePure repeating decimal and mixed repeating decimal have differences in the formula of fractions, but theoretically x 10∧ (a+c)-x 10∧a applies to all circular decimals. Because Infinite does not repeating decimal (irrational number) No male ratio, so infinite not repeating decimal (irrational number) can not be converted into fractional form, that can not be expressed as the form of n/m, ....Set Formula Method Editor Pure CycleWith 9 to do the denominator, there are several cycles of 9, such as 0.3, 3 of the cycle is 9 3,0.654,654 cycle is 999 654, the 0.9,9 cycle is 9 9 (1), and so on.Mixed CycleLet's take a look at some examples
Example:0.228˙ the mixed repeating decimal into fractions:
Solution:0.228˙=[(228/1000) +8/9000)]=228/(900+100) +8/9000=[(228/900)-(228/9000)]+ (8/9000) = (228/900) +[(8/9000)-(228/ 9000)]= (228/900)-(22/900) = (228-22)/900=206/900=103/450;
Example:Mix repeating decimal 0.123˙68˙ into fractions: solution: 0.123˙68˙= (0.12368+0.00000˙68˙) = (12368/100000) + (68/9900000) =[(12368/99000)-(12368/ 990000)]+ (68/9900000) = (12368/99000) +[(68/9900000)-(12368/9900000)]= (12368/99000)-(12300/9900000) = (12368-123)/ 99000
FormulaWith 9 and 0 to do the denominator, first there is a loop section has a few numbers on a few 9, and then there are several not joined the number of the loop to add a few 0, and then the second cycle section of the fractional part of the fraction of the number of parts of the non-cyclic part of the difference between the molecules, For example, 0.43, 3 of the cycle, there is a number did not join the cycle, just 9 after adding a 0 to do the denominator, and then 43 minus 4 to do the numerator, 90 per cent of the 39,0.145,5 cycle with 9 2 0 to do the denominator, and 145 minus 14 to do the numerator, 900 131,0.549,49 cycle, With 99 after adding 1 0 to do the denominator, 549 minus 5 to do molecules, and finally 990 per 545, and so on, can numerator to simplify.Other Fractional edits
1, the finite fraction into fractions:The first number of the denominator is 1 followed by the number of 0,0 is the same as the number of decimal places, the numerator is to take the finite fraction as an integer, the number to the right of the decimal point as an integer as a molecule, but does not include the right of the decimal point is very bit, percentile, thousand percentile, ... On the 0, can numerator to simplify, for example: will 0.678 into fractions, namely 678/1000=339/500,0.1681=1681/10000,0.087=87/1000,0.0078=78/10000=39/5000, ... ; 2
, with decimals (mixed decimal) into fractions:For example: The 2.18 into fractions, solution: Because 2.18=2+0.18, so, 2.18=2+0.18=2+ (18/100) =2+ (9/50) =109/50, 3.1415 into fractions, ∵3.1415=3+0.1415,∴3.1415=3+ (1415/10000) =3+ (283/2000) =6283/2000, and so on, and so on, can numerator must be simplified; 3
, negative fractional into fractions of the law, the method is the same as above:For example:-0. ˙186˙=-186/999=-62/333,-0.0˙87˙=-87/990=-29/330,-0.5678=-5678/10000=-2839/5000, and so on and so on, can numerator must be converted to the simplest fraction. The above from Baidu Encyclopedia,
Infinite Loop Fractional fraction,
