The number 5 itself is rational. However, proving that the square root of 5 (√5) is irrational is a common exercise in number theory. Here's how to demonstrate that √5 cannot be expressed as a fraction of two integers:
Proof by Contradiction
This is a standard method for proving irrationality.
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Assume the opposite: Suppose, for the sake of contradiction, that √5 is rational. This means we can write it as a fraction a/b, where a and b are integers and b ≠ 0. We can also assume that the fraction a/b is in its simplest form (i.e., a and b have no common factors other than 1).
√5 = a/b
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Manipulate the equation: Square both sides of the equation:
5 = a²/b²
Multiply both sides by b²:
5b² = a²
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Deduce a divisibility relationship: The equation 5b² = a² tells us that a² is a multiple of 5. Therefore, a² is divisible by 5.
- A key point: If a² is divisible by a prime number (like 5), then a itself must also be divisible by that prime number. (This can be proven separately, but we'll accept it as true for this proof.)
Therefore, a is divisible by 5. This means we can write a as 5k, where k is some integer.
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Substitute and repeat: Substitute a = 5k back into the equation 5b² = a²:
5b² = (5k)²
5b² = 25k²
Divide both sides by 5:
b² = 5k²
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Deduce another divisibility relationship: Now we see that b² is a multiple of 5, meaning b² is divisible by 5. Following the same logic as before, this means b is also divisible by 5.
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Reach a contradiction: We have shown that both a and b are divisible by 5. However, we initially assumed that a/b was in its simplest form, meaning a and b have no common factors other than 1. The fact that they are both divisible by 5 contradicts this assumption.
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Conclude the proof: Since our initial assumption led to a contradiction, our assumption must be false. Therefore, √5 cannot be expressed as a fraction of two integers, which means √5 is irrational.
Why This Works
The core of this proof hinges on the properties of prime numbers and divisibility. The fact that if a prime divides a square, it must divide the original number is essential. The contradiction arises because assuming √5 is rational leads to the conclusion that its fractional representation can never be in simplest form, no matter how much we try to reduce it.
