To prove that 5 is irrational, we first need to recall the definition of an irrational number. A number is considered irrational if it cannot be expressed as a fraction of two integers, i.e., it cannot be written in the form p/q, where p and q are integers and q ≠ 0.
Next, we need to check if 5 can be expressed in this way. In fact, 5 can be written as 5/1, which shows that it is a rational number, not an irrational one. Thus, we have established that:
- 5 is a rational number, not irrational.
Now, moving on to the expression 2√5. Since we have established that 5 is rational, we can analyze √5. The square root of a rational number (like 5) or a whole number is irrational when the number is not a perfect square. Specifically, √5 is an irrational number because it cannot be expressed as a fraction of integers.
To show that 2√5 is also irrational, we can use the property that the product of a rational number and an irrational number is always irrational. Here, 2 is rational, as it can be expressed as 2/1, and √5 is irrational. Therefore, the product 2√5 must also be irrational.
In conclusion, while 5 is rational and not irrational, the statement 2√5 is indeed irrational.