an irrational number greater than one but less that two ?
PI/2
To find an irrational number between 1 and 2, we can start by understanding what makes a number irrational. An irrational number is any real number that cannot be expressed as a fraction or ratio of two integers.
One example of such an irrational number is the square root of 2 (√2). It is known that √2 is greater than 1 but less than 2. However, let me explain how to prove that √2 is irrational.
Proof that √2 is irrational:
1. Assume √2 is rational, which means it can be expressed as a fraction in the form of p/q, where p and q are integers with no common factors.
2. Square both sides of the equation: (√2)^2 = (p/q)^2
This simplifies to 2 = p^2/q^2.
3. Rearrange the equation to isolate p^2: p^2 = 2q^2.
4. Now, we can conclude that p^2 must be an even number because it is equal to 2 multiplied by q^2.
5. Consider the prime factorization of numbers: any integer can be expressed as a unique product of prime numbers.
Since p^2 is even, p must also be even (since the square of an odd number is odd).
6. If p is even, we can write it as p = 2k, where k is another integer.
7. Substituting p = 2k into the equation p^2 = 2q^2 gives (2k)^2 = 2q^2, which simplifies to 4k^2 = 2q^2.
8. Divide both sides by 2: 2k^2 = q^2.
9. From this equation, we can conclude that q^2 is even, so q must also be even.
10. If both p and q are even, they have a common factor of 2, contradicting our initial assumption that p/q is in its simplest form.
11. Therefore, our assumption that √2 is rational is false, proving that √2 is irrational.
Hence, √2 is an irrational number greater than 1 but less than 2.