Sum of decimal numbers

Before advancing on the definition and correct way in which the Sum of decimal numbers must be solved, perhaps the best thing to do is to take into consideration some concepts that will allow this mathematical operation to be understood in its precise context.


Fundamental definitions

In this sense, it may also be necessary to delimit this revision to two specific notions: the first of them, the concept of Decimal Numbers itself, since this will allow us to know the nature of the numbers involved in this operation. In the second place, it will be equally pertinent to look at the very definition of Sum, since it is the operation carried out with this type of numbers. Next, each one of these concepts.

Decimal numbers

In this way, we will begin by saying that Mathematics has defined Decimal Numbers as those types of numerical elements, used to represent both rational Numbers (written expression of the quotient between an integer and a natural number, which can be annotated as a decimal where incomplete units are limited or repeated) and irrational Numbers (numbers that cannot be expressed as a fraction, because their incomplete units extend to infinity, without any series of numbers being repeated in them).

For the same reason, the mathematical discipline indicates that the Decimal Numbers will be conformed by two parts: the first of them, known as Units, will always be constituted by an integer, which can be positive, negative or even the same zero (0); as for the second part of this type of numbers, Mathematics recognizes them as incomplete Units -or also decimal part-, being conformed by a number that is always smaller than the unit, and that in the Numerical Line can be located between 0 and 1.

Both parts of this type of number will be separated – and at the same time joined – by a comma. To the left of this symbol, the Units will always be annotated -and without exception-, that is to say, the whole part of the Decimal Number. On the right side of the comma, the incomplete Units, or the decimal part of this type of numbers, should be disposed. In some mathematical currents the use of the comma is not chosen, but the point is preferred.

Sum

In another order of ideas, it will also be necessary to review the concept of Sum, which is understood as a mathematical operation, where the main objective is to combine the different values of its factors -which will be recognized as summands- in order to obtain a combined value, which is generally called total. This operation responds to several mathematical properties such as the commutative Property, the associative Property, the distributive Property with respect to multiplication, the Property of the neutral element, among others.

Sum of decimal numbers

Once each of these definitions has been reviewed, it may certainly be much simpler to approach an explanation of the Sum of Decimal Numbers, which can basically be seen as the mathematical operation, whose main objective is to combine the values of decimal numbers -which act as summands- in order to obtain a total from them.

However, considering that decimal Numbers are formed by the Units, the comma and the incomplete Units, and at the same time each one of these parts are constituted by other elements, perhaps the best thing before advancing in the form in which the sum of decimals must be made, is to take into account the way each one of the parts of these numbers is constructed, as can be seen below:

Elements that make up Decimal Numbers

As its definition states, this type of number will be made up of two parts: an integer and a decimal, in which each of the different elements that make it up, being inscribed within the decimal numbering system, will have a positional value. In this way, the following structure can be found in each of them:

  • Elements of the Units: first, the whole part of the Decimal Numbers will be extended, to the left and to the left of the comma. In it, the elements will be counted equally in that direction, finding respectively and in order the following: units, tens, hundreds, units of a thousand, tens of a thousand, hundreds of a thousand:
  • Elements of incomplete Units: on the other hand, the decimal part of this type of numbers will extend to the right, and to the right of the comma. There is also a positional value in them, while each one of the elements will represent a specific quantity (always a multiple of 10) of how many parts the unit is divided into. In this part of the decimal number, the tenths (10 tenths = 1), the hundredths (100 hundredths = 1), the thousandths (1000 thousandths = 1) and the ten thousandths (10000 = 1) will be found in order:

As for the procedure to be performed to execute the sum of two or more decimal numbers, mathematics indicates a series of steps or rules to consider, which can be explained as follows:

To begin the sum of two or more decimal numbers, they must be arranged in columns, one on top of the other, taking care that each element -both of the Units and of the incomplete Units- of a decimal number, positively coincides the elements of the other decimal number with which it is summed. For example:

If you have two decimal numbers, and you want to have them for the sum, they should be noted in the form of a column, making each element assume and coincide as to its position:

Assuming the numbers to be added are 234,321 and 345,983. These should be arranged according to the positional structure of the decimal numbers:

2.- Arranged in this way, each one of the numbers will be added with its equal, positionally speaking. To do so, both will be taken as integers. However, the sum should be started from right to left, noting only one number of the subtotal. If this corresponds to a figure of two numbers, the last figure would be noted in the total, while the first would be added to the column immediately to the left. For example:

Finally, Mathematics also points out a last consideration regarding the Sum of decimals: if for any reason one of the summands does not have the totality of numbers or elements, which has the decimal number with which it holds the sum -whether in Units or in incomplete units-, it is assumed that this is equal to zero, and the sum is made, without even noting it down. That is to say, the Property of the neutral element is assumed. For example:

4.- Finally, although it is not usual to perform a decimal number addition operation, it is also possible to perform a procedure, aimed at checking whether the operation has really led to solve correctly the combination between the two decimal numbers that have exerted as summands.

In this way, Mathematics points out that the correct way to check an addition of decimal numbers, and in fact an addition in general, will be resorting to its inverse operation: that is, the Subtraction. In this order of ideas, whenever you want to check if the sum of decimal numbers has been done correctly you must submit to an operation of subtraction the total obtained and some of the numbers that have served as additions, the result must then be the other addition, to be considered correct.

Examples of sum of decimal numbers

However, perhaps the best way to complete an explanation of the sum of decimal numbers is through the presentation of a concrete example, which allows us to see the correct way in which this type of operations must be resolved. Here is an example of the sum of decimal numbers.

Assuming you have the numbers 23,98 and 234,91, you should add them together:

  • The first thing to do is to arrange each one of these numbers in the form of a column, making each one of its elements coincide positionally with its equal, and regardless of whether any of the numbers is missing or not. Since the Sum is a commutative operation (where the order of the factors does not alter the product) it is chosen for aesthetic reasons to place the largest number on the one with the fewest digits:

  • Each of the positional columns will be summed, remembering that if any summing had an element that does not find an echo in the other summing, the property of the neutral Element is assumed, and this number is added with an imaginary zero, noting the result in the total. If, on the other hand, the sum is made, and the subtotal has two digits, only one would be noted in the total, while the first digit would be added with the column immediately to the left:

  • Once this procedure has been carried out, the result is assumed to be the final solution of the proposed sum. To confirm if it has been done correctly, it should first verify that both units and incomplete units have been added according to the position of each of its elements. Likewise, the total can be subtracted with one of the summands, and this operation must give exactly the other sum that has participated in the operation. For example:

  • Once this has been done, and the other sum is obtained as a result, the sum of decimal numbers is considered checked.

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Sum of decimal numbers
Source: Education  
October 29, 2019


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