Rational Numbers
23.09.2013 15:05

A number that can be written as a fraction involving only integers is called a rational number or 'R". Examples of rational numbers (Read more) are ½, -4/3, 7/9, 102/5, -124, 5568 etc. There can be infinitely many rational numbers. It is possible to insert any number of rational (abbreviated as r’al) number (abbreviated as num) between any two rational numbers. In spite of that, the set of rational numbers is not a complete set. That means if they are plotted on a num line, there still would be gaps that need to be filled, but there are no rational numbers available to do so.  In other words, if we  have a continuous line that extends to infinity on both sides, there would be points on this line, for which there is no r’al num. The set of rational number is denoted by the alphabet Q.

Properties of rational numbers:
1. Property of closure: The set of rational numbers is closed under addition, subtraction and multiplication. That means, for any two or more rational numbers, their sum, differences as well as their products are also always rational numbers.
2. Commutative property: The process of adding or multiplying rational numbers is commutative in nature. That means for two r’al num, a and b, a + b is same as b + a and a * b is same as b * a.
3. Associative property (Read more): When multiplying or adding r’al nums, the associative property holds. Thus, for any three numbers a, b and c, a + (b+c) is the same as (a+b) + c and a * (b*c) is the same as (a*b) * c.
4. Additive identity property: This is the property by which we say that the when zero is added to any "R" num, then the value of that num does not change. Thus the additive identity of "R" nums is 0.
5. Multiplicative identity property: According to this property, the multiplicative identity of r’al nums is 1. Any "R" num multiplied by 1 yields the same "R" num, thereby not altering its value.
6. Additive inverse property: For any r’al number p/q, its additive inverse would be –p/q. Thus p/q + (-p/q) = 0.
7. Multiplicative inverse property: For any such num p/q, its multiplicative inverse would be q/p. This holds only if neither of p and q are equal to zero. The q/p is also called the reciprocal of p/q and vice versa. p/q * q/p = 1. Thus product of a "R" num and its reciprocal would always be 1.
8. Distributive property: For any three random r’al nums, a, b and c, the product of a * (b+c) is the same as the sum of the products a*b + a*c. Thus, a * (b+c) = a*b + a*c and conversely, a*b + a*c = a* (b+c). This is called the distributive property.

Note that r'al nums are not commutative under subtraction. That means a - b is not the same as b - a for any two r'al numbers a and b.

Examples of probability (Read more): This is a different topic altogether, thus will be discussed under a separate head.

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Integration Examples


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