One of the ways in which we can express a fragmented number is by means of a generating fraction. Where it can be periodic or exact depending on the parts that it has. The corresponding process to calculate it is essential in order to obtain the exact results that these operations must give.
These in turn are very helpful in mathematics from its simplest form. This is why it is essential to delve into this matter, which is of the utmost importance. So, read on to understand a little more about this.
Concept and meaning of a generating fraction
When we talk about the generating line, we are inquiring about decimal numbers. Where the portion will result in that same sign. With respect to an exact part, it is quite simple, since the numerator is going to be the number without the fraction. The denominator of the unit will be followed by several zeros according to decimal data who owned the broken amount.
The main function of the generating fraction is to be able to reach express a decimal number. This by means of a fraction that is not reducible, which means that it does not have divisors. If we are facing an irreducible, in which the numerator and denominator do not have factors in common. It is when the portion, in this case, cannot be simplified to pass it to its smallest figures.
In order to find it, we must place the numerator, but without the decimal point. While, with the denominator, the unit will be followed by several zeros that will be the same fractions of the fraction. In order to do this, the part must be passed to a decimal number to simplify so that we can continue calculating. upcoming operations.
It is part of the wide world of exercises that carry with them the decimal system. Where its numerator is going to be a figure, but without any fraction. Its denominator will be the unit preceded by the same amount of zeros as the fractional numbers that it had in the decimal figure. Once this generating part is obtained, we can simplify it. An example of this type of fracture is if we have 6/8, which is reducible, which would be equivalent to 3/4, which is not irreducible.
What is a generating fraction used for?
They are a tool that is used in many different ways. Although there are no specific methods to solve some problems in certain situations. When we have a decimal number, either exact or periodic, we can express it in several ways. The first step is to find its generating part.
What are the types of generating fractions?
exist three main types of generating fractions: exact, pure and mixed. Let’s look at each of them in more detail.
- Generating fraction of an exact decimal number: Your goal is to convert an exact quantity into a generating fraction. To do this, we write the figure without the comma in the numerator, and in the denominator we place a one followed by zeros according to the number of decimal places.
- Generating fraction of a pure repeating decimal number: In this case, we write the numerator without the comma or the integer part. In the denominator, we put as many nines as there are digits in the period, which is the repeating part of the number. It is important to distinguish between pure and mixed decimals.
- Generating fraction of a mixed periodic decimal number: In the numerator, we put the difference between the number without a comma and the periodic part, not including the integer part. In the denominator, we use as many nines as there are digits in the period, followed by zeros in the same amount as the non-recurring decimal part.
By applying these steps, we can obtain the generating fraction of each type decimal number.
What is an unlimited generating fraction?
An unlimited generating fraction is one that represents a non-repeating decimal number in a finite pattern, that is, its decimal part is infinite and not periodic. In other words, it cannot be expressed as an exact fraction or as a generating fraction with a repeating period.
These unlimited generating fractions are represented using ellipsis above the decimal part that does not repeat, indicating that that part is infinite. For example, the unlimited generating fraction of the decimal number 0.123456789…, would be written as 0.1234.
What is an irreducible generating fraction?
An irreducible generating fraction is one that cannot be further simplified, that is, it is in its simplest or reduced form. In a generating fraction, it is considered irreducible when the numerator and denominator have no common factor other than 1.
For example, if we have the generating fraction 4/8, we can simplify it by dividing both the numerator as the denominator times its greatest common factor, which in this case is 4. Doing so gives us the irreducible fraction 1/2.
It is important to find the generating fraction in its irreducible form because it represents the simplest relationship between numerator and denominator, which facilitates their understanding and mathematical calculations.
How do you get the generating fraction of a decimal?
To obtain the generating fraction of a decimal number, follow the next steps:
- Identifies the type of decimal number: Determine if the decimal number is exact, pure periodic, or mixed periodic.
Generating fraction of an exact decimal number: If the decimal number is exact, that is, it has no repeating decimal part, the generating fraction is simply the fractional form of the number. You can convert it to a fraction by putting the decimal number as the numerator and a denominator of 1. For example, if you have the decimal number 0.75, its generating fraction would be 75/100, which can be simplified to 3/4.
Generating fraction of a pure repeating decimal number: If the decimal number has a pure repeating decimal part, that is, a repeating sequence of digits, follow these steps. to. Put the number formed by the repeating digits (the period) in the numerator. b. In the denominator, put as many nines as there are digits in the period. c. Simplify the fraction if possible.
For example, yes you have the repeating decimal number pure 0.333…, the 3 repeats itself infinitely. The generating fraction would be 1/3, since you put 3 in the numerator and 9 in the denominator, and then simplify.
- Generating fraction of a mixed periodic decimal number: If the decimal number has a mixed recurring decimal part, that is, a recurring decimal part followed by a nonrecurring decimal part, follow these steps. to. Subtract the total decimal number minus the non-periodic part and get the result. b. Place the result obtained in the numerator. c. In the denominator, put as many 9’s as there are digits in the period, followed by as many 0’s as there are digits in the non-periodic part. d. Simplify the fraction if possible.
For example, if you have the mixed repeating decimal number 2.64(135), where 135 is the period and 64 is the non-periodic part, follow these steps. Subtract 2.64135 – 2.64 = 0.00135. The generating fraction would be 135/99900, which can be simplified.
Remember that simplify the generating fraction, if possible, it involves dividing both the numerator and denominator by their greatest common factor to get the simplest form of the fraction.
Examples of a Generating Fraction
If we have the generating fraction of a pure decimal digit which is 2.46, the result is going to be partition 123/50.
Another example is the generating line 3.23232323… whose representation will be 320/99.
If it is the case of finding one with the pure periodic decimal 0.428571428571428571428571428571… Where its period is 428571, we present it in this way 3/7.
The generating part of 5.061212121212… is the fraction 8351/1650
Within the system of generating fractions, we can observe different classes that are divided according to their purpose and the processes they involve. These classes encompass most of the decimal numbers in the decimal system, since the spanning fractions are closely related with them. It is not possible to speak of a generating fraction without taking into account the decimal numbers that they represent. Both things go hand in hand with the aim of simplifying them.
It is important to highlight the importance of the generating fractions to represent decimal numbers exact or periodic, depending on the parts they have. It is also relevant to mention that generating fractions are very useful tools in mathematics in everyday life.
It is essential to remember the difference between pure and mixed decimal numbers, and also pay attention to the simplest details, like the ellipses that indicate that the decimal numbers that follow are infinite.