What are the basic rules of summation (axioms) in mathematics (algebra), and how do we apply the algebraic rules? In this article, we will explain the basic calculation rules. First, we will look at the commutative, associative, and distributive laws of summation. You will also learn about the neutral and the inverse element of counting the sum of numbers. You will need this knowledge in general mathematics, business mathematics, statistics, and everyday life.
Introduction to Basic Rules of Summation
The axioms (basic rules) of summation are mathematical arguments of logical algebra. Every day we are confronted with mathematical problems where we have to apply the addition rules (axioms). For example, when counting objects in everyday scenarios in the supermarket and all areas of life. Suppose you have observed a shelf in a supermarket and there are similar objects on the racks. You then decide to count the items on each level and finally add up all the items on all levels of the counters. Assume there are only two levels of shelving, A and B.
Now consider that there are only two levels of shelves, A and B. Level A has $a=50$ like items, and level B has $b=24$ like items.
Now, how do we get the sum $S=a+b=50+24=74$ of the homogeneous objects on the two shelf levels? We will use the summation rules to show that the result obeys the basic rules.
Commutative Law of Summation
The commutative law of summation means that the order of addition does not influence the result.
Let us assume that the store manager changed the shelving system into a three-tier system. He then fills the shelf level C with $c=26$ homogeneous items. The commutative law of addition will also be satisfied:
S=a+b+c \\=a+c+b \\=c+a+b \\= c+b+a \\ =b+c+a\\=100
Associative Law of Summation
The associative law of addition means that spontaneous sequential summation by adding terms does not change the results.
S=a+b+c \\ =(a+b)+c \\ =a+(b+c) \\ (a+c)+b \\ = 100
Distributive Law of Summation
How would you sum up the total number of items if you saw three rows of shelves $n=3$ with the same number of like items on each level as on the first shelf? In this case, you would use the distributive law of addition. The distributive law of addition proves how to factor a constant within a summation and open the brackets (or multiply out) in mathematics.
S=n(a+b+c) \\ =an+bn+cn \\=3\cdot 100=300
Neutral Element of the Summation
The neutral element of summation is the number $0$, which does not change the sum of terms in mathematics.
S=a+b+c+0 \\ =a+b+c=100
In our supermarket shelves example, we can treat each space on the shelf as a zero (the neutral element of summation). We can say that the neutral component of addition is empty.
The inverse Element of Summation
The inverse element of summation is the negative term $(-)$ of the positive expression $(+)$ so that their sum is equal to the neutral element of summation (zero).
a+(-a)=0 \\b+(-b)=0 \\ c+(-c)=0