But at the same time, you can find that the integer satisfies the equation . Similarly, if you want to find a number satisfying the equation then you can find the answer in the set of rational numbers as , but not in the set of integers.
Suppose you want to find a number satisfying the equation . . You can see that there is no rational number that satisfies this equation. So, we need a new set of numbers, namely, the set of irrational numbers on which we can find the solution to the equation as . In high school mathematics or in Mathematics I course, you have learnt that the set of rational numbers and the set of irrational numbers together constitute the set of real numbers which is denoted by . Given any two real numbers and , we can say that or or . Out of these options exactly one is true. This property is called the Law of Trichotomy . Note that the law of trichotomy holds good in ,, and the set of irrational numbers also. The following question may arise in your mind: Is there any equation for which there is no solution even in the set of real numbers? and the answer is `yes'. Consider the simple quadratic equation . Is there any real number satisfying this equation? One can conclude that there is no real number satisfying the equation . So, the real number system is also inadequate. In other words, we can say that there is some deficiency in the set of real numbers and it is to be expanded further. Such an expanded set is called the set of complex numbers and is denoted by . It can be shown that the equation where and has always a solution in the set of complex numbers.