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Math18

Sum

Content

What is addition or sum?

The sum (also known as addition) is one of the four basic operations of the arithmetic of integers, decimals, fractions, real and complex numbers, and also in algebraic expressions or on structures associated with them, such as vector spaces.

Symbol or sign of the sum

The sign of the sum is represented by means of a “+” cross that is known as “plus” or “positive”.


Parts of the sum

When performing an addition operation, there are two parts or elements:

  • Addends: Correspond to the numbers to add.
  • Sum: The sum or total result
+
2 ← Addends 1 ← Addends / 3 ← Sum

Another way to represent the previous sum would be: 1 + 2 = 3 (1 is an addend, 2 is an addend and 3 is the Sum Total or Total).

Note: The word sum represents both the operation to be performed and the result obtained from it.


Sum properties

There are 5 basic properties of a sum:

  • Commutative: “The order of the addends does not alter the result or sum”. this means that 2 + 5 = 7 is the same as 5 + 2 = 7.
  • Uniformity: “The sum of several given numbers has a unique value”. When adding the same numbers the resulting total will not change, even if the things being added are different things, for example:
    5 shoes + 3 shoes = 8 shoes
    5 hats + 3 hats = 8 hats
    We see that in adding 5 and 3, regardless of the nature of the sets that the addends represent (shoes, hats, etc.), the total is always 8.
  • Associative: “In adding three or more numbers, the sum is the same regardless of the grouping of the addends”. This means that in adding several numbers, or addends, the order in which they are added does not change the total. For example: In adding 6 + 4 + 3, assuming that first I add 4 + 3 = 7 and then 7 + 6 = 13, the same result can be obtained by adding 6 + 3 = 9 and then 9 + 4 = 13.
  • Distributive: “Multiplying the sum of two or more addends by a number will result in the same total as multiplying each individual addend by the number, and then adding the products together”. This property indicates that it is possible to decompose the sum by adding two or more addends of lesser value, without altering the result, for example: If I want to add 23 + 45 it is possible to decompose into 23 = 20 + 3 and 45 = 40 +5 and then add 20 + 40 + 3 + 5 = 68.

Note: "0" is the only number that added to another does not alter it. (Identity Property).


How do we add?

To begin learning to add, it is advisable to use familiar objects and a simple math game. For example: Begin with a box containing 2 blue balls and introduce a green ball to the same box. This will result in a total of 3 balls in the box (2 blue balls + 1 green ball).

There are different learning methods for addition, among these methods are two of significance, one of which is used for small quantity numbers and the other method for large quantity numbers.

  • Addition arranged in a line: This is used in adding numbers of low quantities. As experience is gained, the ease of use of this method for larger numbers increases. It should be noted that, for a beginner, 3 + 5 mathematics it can be a bit confusing, but the purpose is to gradually increase the difficulty. This method assists with the development of mental calculation.

    Examples:

    3 + 5 = 8
    2 + 7 = 9
    5 + 2 = 7
    2 + 4 = 6

    Exercise:

    A) 4 + 3 = ?
    B) 5 + 4 = ?
    C) 3 + 2 = ?
    D) 3 + 4 = ?
  • Add in column: This is a method for the addition of large addends. The addends are placed, one below another in columns which correspond to the place value of each integer. The place value structure of each number must be considered; aligning ones with one units,tens with tens, hundreds with hundreds and so on. To identify the units of ones, tens and hundreds of the number 218, we must first start from right to left, this means that we have 8 ones, 1 ten and 2 hundreds, which is equivalent to 8 + 10 + 200 = 218.

    Examples:

    +
    24 15 / 39
    +
    52 44 / 96
    +
    346 243 / 589
    +
    563 326 / 889

    Exercise:

    A) +
    12 16 / ?
    B) +
    32 45 / ?
    C) +
    45 54 / ?
    D) +
    92 05 / ?

In some sums we will have the case of a carried number, and this can complicate the the addition of larger addends. It is recommended to have an order to facilitate adding the columns of sums and obtain the correct result. What is a carried number? Assuming we have 34 + 28 = 62, we must first add the ones units 8 + 4 = 12 so we have “2” ones units and “1” tens unit. The ten obtained represented by a “1” must be “carried” to the next position corresponding to the tens unit column 3 + 2 + 1 (carried) = 6 tens. Or we could interpret as 3 + 2 = 5 and then add the carried “1” 5 + 1 = 6.

Examples:

+
1 16 25 / 41
+
1 53 19 / 72
+
01 127 154 / 281
+
11 364 187 / 551

The red number represents the carried number.

Exercise:

A) +
12 18 / ?
B) +
26 45 / ?
C) +
128 118 / ?
D) +
258 258 / ?

Sums with decimals

When adding numbers with decimals, it is important to align the numbers according to the quantity they represent by aligning the decimal points, and then each number according to its place value by these quantities: the ones, tens, hundreds and thousands units are aligned to the left of the point; to the right of the point the tenths, hundredths and thousandths must line up.

Examples:

+
0 5.2 4.6 / 9.8
+
  1   6.8   4.3 / 11.1
+
01 43.7 53.7 / 97.4
+
111 1 364.67 187.38 / 552.05

The red number represents the led.

Exercise:

A) +
8.2 1.6 / ?
B) +
15.42 13.68 / ?
C) +
111.82    8.63 / ?
D) +
342.86 652.96 / ?

Addition Tricks

Here are some techniques that may be useful to facilitate the operations to be performed.

  • Select the largest number:

    It is always easier to start with the greater sum, for example: suppose 2 + 18, it would be faster if we first consider the number 18 and count up 2 numbers to get the result of 20.

  • Get a closed number:

    When obtaining a closed number we refer to tens (10,20,30,40,50,60,70,80,90), hundreds (100,200,300,400,500,600,700,800,900) and so on, for example: in adding 6 + 3 + 4, it would be advisable to add 6 + 4 = 10 and then 10 + 3 to get the total result 13.


Identify the amount it represents

Ones units (A), tens (B), hundreds (C) and thousand units (D) are aligned to the left of the point; to the right of the point the tenths (E), hundredths (F) and thousandths (G) are aligned.

+
 DCBA . EFG 1 3 5 4 . 4 5 6 6 8 7 2 . 5 6 3 / 8 2 2 7 . 0 1 9

We have as a result: 8 thousand units, 2 hundreds, 2 tens and 7 ones to the left of the point; to the right of the point we have: 0 tenths, 1 hundredth and 9 thousandths.


Checking the sum

The sum can be checked with its inverse mathematical operation, which is subtraction. To verify a sum, one addend is subtracted from the total and from this subtraction the other addend must be obtained, indicating that the sum was correct.

Examples:

+
12 15 / 27
-
27 15 / 12

Taking the total of the sum (27) and subtracting one addend (15) we will obtain the the other addend as a result (12).


Sum of fractions

In order to perform addition with fractions, the same denominator must be obtained. When fractions do not have the same denominator, it is possible to add them together by following a slightly different procedure.

Examples:

5 / 3
+
2 / 3
=
5 + 2 /
3
=
7 / 3