Whether you are studying linear algebra or some major math class, there will always be that need to simplify operations and equations. The reason this is done is to have a much easier reading of the result, no matter what field you are in. Therefore, the fact of having to simplifiedtor fractions in math It is an advantage and a help at the calculation level.
The curious thing about all this is that those who are beginning their learning in calculus do not know what this is, much less how to do it or why. For your luck, we will explain as detailed as possible how you can apply the simplification of fractions, regardless of whether they are simple, mixed or very large in quantity. Also, you will meet something called equivalent and complex fractionsBut we’ll leave that for later.
Definition and concept of simplifying fractions
It is a mathematical operationa great help that can be applied in most cases. On the one hand, we have the fraction term and on the other, the simplification term. The first is attributed to that algebraic expression in which there is a numerator and a denominator. That is, two numbers in which there is a division in between. This applies to independent terms, equations of different degrees, variables, numbers, among others.
On the other hand, we have the simplification part, which is a way of to be able to reduce algebraic fractions to their minimum expression. Of course, there are several aspects to take into account such as:
The numerator and denominator are divisible by each other.
Respect the positive or negative signs that the numbers have.
That like terms are grouped correctly.
It is necessary that the division is not indeterminatethat is, that the divisor number is greater or less than zero, but never zero.
That the mathematical operations are solved to be able to opt for simplification.
With all this, we can say that the simplification of fractions, whether they are complex, equivalent or mixed, is to take them to their minimum expression. All this taking into account the five points that we have just detailed in the list. On a practical level, there are mathematical properties, formulas, and other aspects that influence the simplification of a fraction. Some properties of mathematics and algebra help to apply theorems and equations that facilitate this whole process.
What is simplifying algebraic fractions?
In summary, fraction simplification is nothing more than finding a number that is capable of dividing both the numerator and the denominator. This mathematical operation can be perfectly applied to the field of physics, chemistry and other scientific fields. Now, the simplification of fractions is not only limited to the numerical aspect, but also applies to variables. The latter is known as alphanumeric expressionsthat is, they make the combination of numbers and variables or letters.
Of course, the latter applies mainly when you’re working with equations and polynomials that contain variables. However, in learning situations and more complex math problemss, you usually work clearly with letters and independent variables. It is thanks to the simplification that numbers begin to appear as the fraction or equation is reduced as the case may be.
What is simplifying equivalent fractions?
if i never asked you which is an equivalent fraction, now we explain it to you. They are fractions that have the same value, no matter how big or different their numerators and denominators are. To make it easier to understand, look at the following fractional examples:
½ = 0.5
2/4 = 0.5
3/6 = 0.5
We put the result of each fraction so that you can visualize it better. The first thing we notice is that the result is still the same, but the numbers or fractions are different at first glance. In these cases, there are two ways to solve algebraic expressions that we have shown you The first one is to take a calculator and do the division. The second and the one that best suits this article is to reduce the fraction 2/4 and 3/6 to its minimum expression.
If you do this, you will notice that you reach a point where you can no longer simplify. That is to say, that what is known as irreducible fraction. This means that there is no number that is capable of dividing the numerator and denominator and that is the same. Therefore, there is no number that divides 1 and 2.
So, equivalent fractions are those that regardless of the numbers used to represent a fraction, will always have or produce the same result. For example, ½ is equivalent to writing 30/60. They are completely different numbers and with a big difference, but when solved, their result is the same: 0.5.
What is simplifying complex fractions?
For this case, the situation changes and its complexity increases a little. Now you no longer work with a single fraction, but with two. The curious and interesting thing about complex fractions is that it is a division between two fractions one above the other. The good news is that there are ways to simplify this algebraic expression that is so common in math and linear algebra.
In certain schools and academies the conventional simplification method known as the “double C” is often used. This consists of multiplying the numerator of the first fraction with the denominator of the second fraction. Once this is done, now multiply the denominator of the first fraction with the numerator of the second fraction. This will give us a much simpler fraction of reduce to its minimum expression. Of course, this will largely depend on whether it is possible to do such a simplification. Remember that you have to look for a number that is divisible between both numbers of the algebraic expression.
Now, there is a slightly easier way at the visual level of give a solution to this type of fraction. The idea is simple and it is enough to take the complex fraction and rewrite it in such a way that it looks like a multiplication of fractions. However, the detail is that the second fraction will have to be invested. That is, move the numerator down and the denominator up. For example, if you have ½ and 9/5 as the second fraction, when you convert it to a linear expression, it will look like this: ½ * 5/9. What remains from this point is to multiply numerator with numerator and denominator with denominator.
What is the use of simplifying fractions?
There is no formula as such that allows you make a simplification of factors. This is something that applies as long as the right conditions exist. Be that as it may, it is a mathematical operation that helps us to achieve the minimum expression of something and, at the same time, make it easier to manipulate and understand. Imagine for a moment that you have the following fraction: 35,500/70,000. If you look closely, these are rather large numbers that may surprise some. But it is thanks to simplification, that We can write that same expression as 0.5 or failing that, like ½ if what you want is to maintain the structure of a fraction.
This is something that not only applies to numbers, but also to equations where there are variables and terms raised to the power, exponentials, or roots of different indices. The simplification is not only limited to the fractional aspect, but also to more complex algebraic expressions. In a certain way, there are theorems, equations and formulas that we can use as the case may be. An example of this is when we are in the presence of a perfect square trinomial. If you meet the right conditions, these three terms can be reduced to just 2, but raised to a power squared.
It must be taken into account that all simplification process it must be carried out maintaining the equivalent terms on both sides of the equality. Whether you divide, multiply, or raise the numerator to a power, I have to do the same for the denominator. This for the simple reason that, by not doing so, we would be altering the balance or balance, so to speak.
That is, if you put a (-1) on the right side of the equality, you have to do the same on the left side of it. This could be considered as a slightly more complex form of simplification that leads to other mathematical operations.
Examples of simplifying a fraction in mathematics
At this point, it is necessary to know some very specific examples on the simplification of fractions in mathematics. However, everything will depend on its complexity, since not all of them are solved in the same way. For this reason, we have categorized some specific factors for each of the different types of fractions that exist. Among these, there are the following that we will mention.
How to simplify simple fractions?
Simple fractions are those that are a single term. The vast majority have a single element in its numerator and denominator. However, for slightly more complex cases, we may find ourselves with a fraction in which there is a perfect square binomial at the top and three terms that divide it at the bottom. Starting with the simplest, we will find the following examples:
For the first case, what we can do is find a number that is capable of dividing both the numerator and the denominator and that is the same. In this example, the number 5 is used as dividing factor for simplification. This gives us the following fraction as a result: ¼. What has been done is to divide both numbers by 5. At this point, an irreducible fraction has been reached, since there is no way to continue simplifying.
Something similar happens with the following two examples, but with one difference, that instead of remaining as a fraction, the result remains as an integer. The same concept applies find a number that can divide both numbers. For example two, the result is 5, while for the third exercise, 50 would remain.
How to simplify mixed fractions?
This type of fractions is characterized by presenting a fraction like any other, with the difference that there is one more term that multiplies or divides it. For example, we can have the expression 2* (¼). This is considered a mixed fraction and there are two easy ways to solve them, at least taking the example we just gave you as a reference.
The first way is to make a multiplication of the whole number (2) by the numerator of the fraction. After this, we proceed to carry out the simplification that we have already explained throughout the article. Now, the second way is perhaps the easiest and most used. The idea is to identify if the number or term that multiplies is a multiple or can be divided by the denominator. For example, two is a multiple of the number four. Therefore, when this compatibility exists, what we can do is divide the denominator and the term that is multiplying by two. To make it easier for you to see, it is like doing the following:
Basically, you would be applying a double C, only you will do it directly. The two that multiplies is divided by itself and to maintain equivalence, divide the fraction by two too. This gives you the result ½. If you make a comparison with both methods, possibly the first one is easier to execute. However, the second becomes much more feasible, it just requires practice to be able to understand and execute it more quickly.
How to simplify large fractions?
This is nothing more than reducing the terms in a fraction as much as possible until reaching something smaller and easier to manipulate. In theory, the same bases that we have already explained apply, with the difference that it is something that is applied mainly in complex equations where there are many terms that are adding, subtracting or are raised to a certain power. The mathematical theoretical basis remains the same.