In mathematics, we have some numerical sets, such as Naturals, Integers and Rationals. The natural numbers are formed by the numbers 0, 1, 2, 3, 4, 5, 6... Integers are composed of the natural numbers and their negative version, that is, …, -2, -1, 0, 1, 2, 3... Rational numbers, on the other hand, are all those numbers originating from a division, remembering that every division can be expressed through a fraction, for example, 1 ÷ 2 = ½. We can then separate the rational numbers into three classifications:
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Exact division – 8 ÷ 2 = 4
10 ÷ 5 = 2
9 ÷ 3 = 3
Finite decimals - 1 ÷ 2 = 0.5
5 ÷ 4 = 1,25
9 ÷ 5 = 1,8
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Periodic tenth - 3 ÷ 9 = 0.33333...
21 ÷ 99 = 0,21212121...
100 ÷ 999 = 0,100100100...
All decimal numbers that have infinitely many decimal places, with a repeating number sequence, are called periodic tithe. The number that is repeated is called time course. In the examples cited above, 0,33333..., 0,21212121... and 0.100100100..., the periods are, respectively, 3, 21 and 11.
But given the periodic decimal, do you know how to find the fraction that gave rise to it? We have a handy device that quickly indicates the fraction whose division generated the periodic tithe, also known as generating fraction. Let's look at some cases:
0,444444...
In this case, we have a period periodic decimal 4 and with the integer part null, that is, before the comma there is only 0. As our period only has a digit, let's divide it by 9. Our generating fraction will look like this:
0,444444... = time course = 4
9 9
In the case of 0.32332232..., the period has two digits, therefore, to find your fraction, we will divide the period by 99:
0,323232...= time course = 32
99 99
And so on.
See another example: 0, 100100100100...
In that case, the period is 100, number formed by three digits, so it should be divided by 999.
0,10010010 = time course = 100
999 999
Another case occurs when we have an equal periodic decimal 0,254444... In this periodic tithe, there is a period 4 and a non-periodic part after the comma, the 25. If we consider the non-periodic part, followed by the period, we will have: 254. From this value, we will subtract the non-periodic part: 254 – 25 = 229. To divide the 229, we need to analyze our tithe: for each digit of the period, we put the 9, and for each digit of the non-periodic part, we fill it with 0. Getting the following:
0,254444... = 254 –25 = 229
900 900
Let's look at other examples:
0,31252525... = 3125 – 31 = 3094
9900 9900
0,411222... = 4112 – 411 = 3701
9000 9000
0,0291291291... = 0291 – 0 = 291
9990 9990
Finally, we have the case where the number that appears before the comma is not zero, that is, when there is an integer part in the periodic decimal. In this case, we must separate the integer part from the decimal part. For example, in the case of 1,4444..., we must write it as 1 + 0,4444... We transform the decimal part into a fraction using the proper method, just like we did in the first example. Look:
0,444444... = time course = 4
9 9
Just add this fraction with the whole part:
Therefore, 13/9 is the generating fraction of 1.4444...
By Amanda Gonçalves
Graduated in Mathematics
Take the opportunity to check out our video lesson on the subject:
