sum of rational number and irrational number is
The sum of a rational number and an irrational number is an irrational number
The sum of a rational number and an irrational number is an irrational number.
Let’s see a proof for this:
Suppose we have a rational number, represented by the fraction a/b, where a and b are integers and b is not equal to zero. We also have an irrational number, denoted by c.
The sum of these two numbers can be expressed as (a/b) + c.
Now, let’s consider two cases:
Case 1: When (a/b) + c is rational
If (a/b) + c is rational, then it can be represented as a fraction d/e, where d and e are integers and e is not equal to zero.
In this case, we have (a/b) + c = d/e.
Rearranging the equation, we get c = (d/e) – (a/b).
We know that the difference between two rational numbers is also a rational number. So (d/e) – (a/b) is rational.
This means that if (a/b) + c is rational, then c must also be rational. However, we assumed that c is an irrational number. Therefore, this case is not possible.
Case 2: When (a/b) + c is irrational
Since case 1 is not possible, we conclude that when we add a rational number and an irrational number, the sum is always irrational.
Therefore, the sum of a rational number and an irrational number is always an irrational number.
Note: This property is unique to irrational numbers. When you add two rational numbers, the sum can be either a rational or an irrational number.
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