Perhaps the most convenient thing, before approaching an explanation on how it is and how it should be done the Geometric interpretation of the sum of integers, is to revise some definitions, which will allow to understand this procedure within its just mathematical context.
In this sense, it may also be convenient to delimit this theoretical revision to two specific notions: the first of them, the very definition of Whole Numbers, since these are the elements on which the sum is made, and the subsequent geometric interpretation. It will also be necessary to take a moment to review the concept of Numerical Line, as it is the graph on which the geometric representation of the sum of integers is made. Next, each one of these definitions:
In this way, we can begin to say that Mathematics has defined Whole Numbers as those numerical elements through which exact quantities are given written expression, or integers, as its name indicates. Likewise, this discipline has indicated integers as those numbers that make up the set Z, and which are made up of three different types of elements:
- Positive Integer Numbers: these numbers constitute at the same time the set of natural Numbers (N) so they can be used to count the elements of a set, or even assign them a specific position or number, allowing them to be ordered. They are located on the Number Line, to the right of zero, from where they extend in that direction to infinity. They are also used to indicate or express in writing exact or whole quantities. They have a positive sign, which however sometimes is not written down, being taken for granted.
- Negative integers: on the other hand, negative integers are interpreted as negative inverses of positive integers. Consequently, they are located on the Numerical Line to the left of zero, point from where they extend -in the opposite sense to the positive ones- towards infinity. They have a negative sign, which must be noted at all times, to distinguish it from its positive inverse. These numbers are used to express in writing the absence or lack of a specific quantity.
- Zero: finally, zero is also considered part of the set Z, or the set of integers. It is located on the Number Line in the center, serving as the point of origin for both positive and negative numbers. However, zero is not considered a number as such, but a symbol by means of which Mathematics manages to give expression to the absence of full quantity. For this reason they do not have a sign, neither positive nor negative.
In the second instance, it will also be necessary to throw lights on the definition of Numerical Line, which has been understood as a one-dimensional graph -that is, that only represents one dimension– on which the whole numbers are represented. In this sense, the Numerical Line will also have the quality of being graduated.
In the center of it, the zero will be located, and from it, to equal spaces, the positive integers will be placed, to the right, and those of negative sign to the left. In this way, whenever you want to locate an integer, geometrically, you will make use of the number line, because it is the one-dimensional mathematical graph that allows it.
Geometric interpretation of the sum of integers
Once each of these definitions has been reviewed, it may then certainly be much simpler to approach an explanation of how any geometrical interpretation of the sum of all kinds of integers should be made. In this order of ideas, it is necessary to begin by remembering that the Sum can be defined as the mathematical operation, by means of which two or more specific numbers combine their values, are the end of obtaining a total.
Likewise, it is necessary to point out that the geometrical interpretation of this operation between integers has been explained in a general way by the different mathematical sources, such as the procedure by means of which the meaning is written down in the Numerical Line or what really happens at the mathematical level when two integers are added together. In other words, how the operation begins at a specific point on the Number Line, and according to the number with which it adds, it will advance to the left or right of this graph, towards the destination point, which will represent the total of the operation.
However, taking into account that integers can be both positive and negative, it will be necessary to address each of the specific cases that occur with respect to the sum of these elements, and how their respective geometric interpretation should be made. Next, each one of them:
If two positive integers are added together
In case the numbers involved in the sum were positive, then it is assumed that the Sum operation really implies an addition. Likewise, as for its geometric interpretation in the Numerical Line, since the sum is constituted by two or more positive integers, the representation of the operation must go to the right, because it occurs between positive numbers, as well as because a positive number is obtained as total.
When performing the interpretation procedure, the first summand should be located on the Number Line. After this, as many digits will be counted as the value of the second sum involved, digits that will imply a shift to the right, that is to say, in a positive sense. When the point is found, it is marked on the Numerical Line, and with an arrow, the displacement that has occurred in this graph is represented.
However, the best way to explain how the geometric interpretation of the sum of two positive integers should be made may be through a specific example, such as the one below:
Suppose that a sum has been given between 1 and 4 (1+4= 5)
The result of this operation will be +5. When interpreting this operation geometrically, you must then mark on the number line 1, count 4 spaces to the right, as both are summands of positive integers. This displacement must originate a point on the +5, which is the total of the sum:
If integers of different signs are added together
On the other hand, it can also happen that the whole numbers in the sum have different signs. Consequently, according to the mathematical discipline, these numbers must be subtracted, and the result must have the sign of the greater addition. Therefore, it is the sign of the total obtained between these integers that will reveal if the displacement, from the starting point given by the first sum, must be given to the right, if the total is positive, or to the left, if the total is negative.
Here is a concrete example of how the geometrical interpretation of the sum of two integers of different signs should be made.
Assuming that the sum is between -3 and +6: -3 +6 = 3
Then place 6 on the number line, and then make a move to 3, which will involve a movement to the left, but within the segment of the number line intended for positive numbers:
However, it can also be the case that the integer of negative sign is the one established as the greater sumand, therefore the total obtained will be a negative number, which will imply a shift to the left.
However, the point of origin must also be taken into consideration, since if the first summing is a positive number, even if the displacement is to the left, and ends in the segment of the Number Line destined for negative numbers, then this movement will actually begin on the right side of the Number Line. The following is a concrete example of this type of case:
If one had to interpret geometrically the sum between 5 + (-9)= whose result of the operation is -4, this operation must be represented in the following way: Start from +5, on the right side of the Number Line, and move 9 spaces to the left, which should end in -4:
If negative integers are added
Finally, it can also happen that the whole numbers involved in the sum are both negative. In this case, because it has the same sign, the addition will simply be resolved, and the number obtained will be assigned the negative sign. At the moment of making its respective geometric interpretation in the Numerical Line, this must then be done in the segment arranged for the negative numbers. The starting point will be marked, which will be given by the first summing, while the displacement will be made to the left, ending in the point assigned to the total that has been obtained. An example of this will be the following:
If an operation of sum between (-5) + (-4) should be carried out =
The operation will then be resolved:
(-5) + (-4) =
-5 -4 = -9
And then we will proceed to interpret geometrically this result in the Numerical Line. To do this, note the starting point -5, and mark a displacement to the left of 4 places, which should result in -9:
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October 31, 2019