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Home » Math Homework Help » Algebra Homework Help » Rational Numbers
Rational Numbers
We have observed that in the set of natural numbers N, there was no additive identity so it was extended to a set W, the set of whole numbers which we defined as W = {0, 1, 2, 3, …. } and then we thought of the operation of subtractor, and found that the subtraction was not unrestrictedly possible in the system of whole and, therefore, W was extended to the set of integers I. We now draw our attention towards the operation of division and observe that 15/3 = 5 has a meaning in I whereas 15/4 is an undefined term in I. So to I we now attach new elements so as to obtain another family, to be denoted by Q, and call it the set of rational numbers.

The set Q of rational numbers

The set

Q = {p/q | p, q I and q ≠ 0}

is called the set of rational number, and we ask the readers to note that the divisions by 0 is not permissible and I ⊂ Q.

The fundamental compositions defined in Q are also addition and multiplication. We shall now state the properties of addition and multiplication and order relation in Q.

For convenience, by m in Q we shall understand an element of the form p/q; p, q I, q ≠ 0 etc.

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