Reason it is a notion with a large number of meanings. In this case, we are interested in highlighting its use in the field of mathematics, where the reason is the two-digit quotient.
The mathematical reason, therefore, it is a link between two quantities that are comparable to each other. It is about what results when one of the magnitudes or quantities is divided or subtracted by another. Ratios, therefore, can be expressed as fractions or as decimal numbers.
Example of mathematical ratio
Let’s see a example. The reason for 24 Come in 6 is equal to 4. This means that if we divide 24 on 6, we will get 4 as a mathematical reason.
24/6 = 4 (In other words: 6 x 4 = 24).
We can affirm, following the same example, that 24 have 4 times 6.
What is a geometric ratio
It should be noted that, on many occasions, a distinction is made between geometric ratio and arithmetic ratio. The geometric ratio assumes the quotient of a geometric progression and consists of comparing, as we did in the previous example, two quantities from their quotient (determining how many times one is present in the other).
Since the geometric ratio is part of the concept of geometric progression, it is also necessary to explain its meaning: it is a sequence in which each element can be obtained through the multiplication of the previous one by a reason (a constant that is also known by the name of progression factor). The use of the word is generally preferred progression for those sequences that have a well-defined beginning and ending, while succession it is usually used for the cases of infinite terms.
A geometric progression could be the following: 4, 12, 36, 108, 324. In this case, the mathematical ratio (or geometric, to be more precise) is 3, since it is the number by which it is necessary to multiply each element to obtain the next one. The equation To be able to quickly access any element of this progression, it has on one side the unknown with the order number (n) that we want to find as a subscript and on the other side, the first of the terms multiplied by the ratio raised to the n minus 1.
Let’s see an example based on the previous geometric progression, to check the effectiveness of said equation when looking for the value of any of its elements: if we consider that 4 is the first, the value of the fifth we can find out by multiplying 4 by 3 (the mathematical reason for this progression) raised to 4 (that is, to the order number of the element we want to know, 5, minus 1); 3 raised to 4 gives us 81, which multiplied by 4 returns us 324.
The arithmetic ratio, another type of mathematical ratio
The arithmetic ratio, on the other hand, is the difference which exists in an arithmetic sequence. In this case, the mathematical ratio is the difference between the two figures (that is, the result of the subtraction). The reason 8-3, in this sense, is 5.
A arithmetic progressionUnlike a geometric one, it is used to describe a numerical sequence in which each pair of successive terms has the same difference as any other, since to obtain one, a constant must be added to the previous one. This constant is known as progression difference or distance. Taking the example from the previous paragraph, if the mathematical ratio is 5, a possible progression could be 3, 8, 13, 18 and 23.
In both the geometry ratio and the arithmetic ratio, in short, we work with the link between two terms that are successive, known as antecedent and consequent.