Addition, subtraction, multiplication and division are the main basic mathematical operations that can be performed and the first ones you learn. Addition is the simplest, most basic mathematical operation of adding two or more quantities together and subtraction is the inverse operation of addition. In other words, it is the operation of removing quantities from other quantities. The properties of commutativity and associativity apply for addition, but not for subtraction. Multiplication replaces addition of large number of numerals. What would be written as 5 + 5 + 5 + 5 + 5 + 5 during addition, can be written as 5 * 6 using multiplication, where 6 is the number of times number 5 is being added. The properties of commutativity, associativity and distributivity apply for multiplication, but not for division, which is the inverse operation of multiplication. Division is an operation with which you can determine how many times the number or quantity you are dividing is larger than the number or quantity you are dividing by. The resulting number may or may not be an integer.
Properties of commutativity, associativity and distributivity for basic mathematical operations
Commutativity:
a + b = b + a
Associativity:
(a + b) + c = a + (b + c)
Distributivity:
a * (b + c) = a * b + a * c
Basic mathematical operations and integers
Addition is pretty straightforward when performed using positive integers. The result of adding two or more positive integers is always a positive integer. Adding two or more negative integers is also relatively simple because the result is always a negative integer. Things get a bit more complicated when adding positive and negative integers. In that case, addition performs as subtraction of two positive integers in which the result may be either a positive or a negative integer.
Subtraction changes the sign of a number. Subtracting a negative number from a negative number changes the sign of the number that is being subtracted and the operation performs as the subtraction of two positive integers. If a positive integer is being subtracted from a negative integer, the result will always be a negative integer with a value equal to the sum of the two integers. When subtracting a negative integer from a positive integer, you will always get a positive integer as a result.
When it comes to the multiplication of integers, the important thing to remember is that multiplying two or more positive integers or an even number (2, 4, 6, etc…) of negative integers always yields a positive integer as a result. Multiplication of a positive and a negative integer or an uneven number (1, 3, 5, etc…) of negative integers always results in a negative integer. The same rules apply with division.
Those were the basic mathematical operations and the ways they are applied to integers. These mathematical operations are universal and can also be applied on other kinds of numbers.
Basic mathematical operations and fractions
Fractions are rational numbers that are not integers and that can be represented as a ratio of two integers. Rules that apply when performing the basic mathematical operations on fractions are a bit different than when the same mathematical operations are applied on integers.
In order to perform addition on two or more fractions, they have to have a common denominator. If they already do, than simply adding the numerators is all that is necessary. If not, the first thing that needs to be done is to find the lowest common multiple of the denominators in question. Then multiply the whole (!) fraction with the number the denominator needs to be multiplied with in order to get to the value of the lowest common multiple. After that, simply add the numerators to get to the final result. The same procedure applies for subtraction.In order to add a fraction to an integer, you have to turn that integer into a fraction and then perform the procedure for adding two fractions. The easiest way to do that is to consider the integer as a fraction with “1″ being the value of the denominator. Then just multiply that whole fraction with the denominator of the fraction you wanted to be addedt to the integer.
Multiplication of two or more fractions is fairly simple. You have to multiply the numerators with other numerators and the denominators with other denominators. The numerator of the resulting fraction will be the product of all the numerators and the denominator will be the product of all the denominators.When multiplying a fraction and an integer, you just have to multiply that integer with the nominator of the fraction in question.
Division of fraction by another fraction can be performed as multiplication of the first fraction with the reciprocal of the other fraction. Dividing a fraction by an integer can be accomplished in two ways. You can either divide the numerator of that fraction by the integer or multiply the denominator of said fraction with that integer.