Step-by-Step Proof: Euclid’s Method for Irrational Numbers
Before proving \[\sqrt{2}\] as irrational, first, let us understand what an Irrational number.
What is an Irrational Number?
For any integers, an irrational number is a number that can not be represented as a fraction, and irrational numbers have decimal expansions that do not terminate.
The best example of an irrational number is Pi (𝝅) which is has a non-terminating number 3.14159265359.
Here we have to prove the irrationality of \[\sqrt{2}\]. This proof is a classic example of Proof by Contradiction.
In proof by contradiction, at the start of the proof, the opposite is believed to be valid. The assumption is shown not to be valid after rational reasoning at every stage. This is also known as indirect proof and proof by assuming the opposite.
Euclid developed this proof by contradiction and applied for \[\sqrt{2}\] to prove as an irrational number
Euclid Square Root 2 Irrational Proof
According to proof by contradiction given by Euclid, the first step of the proof, we will assume the opposite is true. In the same way here will we assume that \[\sqrt{2}\] is equal to some rational number a/b.
\[\sqrt{2}\] = \[\frac{a}{b}\]……………………(1)
Now, we will square on both sides of equation (1),
(\[\sqrt{2}\])\[^{2}\] = (\[\frac{a}{b}\])\[^{2}\]…………………(2)
We will simplify and rewrite the equation (2) as
2b2 = a2……………………….(3)
If we observe, here the value of a2 will be positive because b2 is multiplied by an even number ‘2’. Since a2 is positive, we can conclude that a is also positive. Since ‘a’ is positive, we can write a=2c where c is any whole number. Since ‘a’ is even number 2 multiplied by any whole number will satisfy the definition of even number.
Now let substitute a=2c in equation (3),
2b2 = (2c)2
2b2 = 4c2………………..(4)
Now divide by 2 into both sides of equation (4), we get
b2 = 2c2……………………(5)
Here b2 is multiplied by 2 and c2 which satisfies the definition of even number. Therefore, b2 is also an even number which concludes that ‘b’ is an even number.
So, we have proved ‘a’ and ‘b’ even numbers.
In the next step, let us assume that b=2d in the same of assuming a=2c which satisfies the even number definition.
Now let us substitute a=2c and b=2d in equation (1) where we have assumed
\[\sqrt{2}\] = \[\frac{a}{b}\]
\[\sqrt{2}\] = \[\frac{2c}{2d}\]
\[\sqrt{2}\] = \[\frac{c}{d}\] ………………(6)
Now we have obtained c/d which is in simpler form compared to p/q. Also from equations (1) and (6)
a/b = c/d ……………….(7)
Here we can further simplify c/d into say e/f by carrying out the same process. Again e/f will be put through the same process and we obtain g/h is simpler.
Rational number definition states that “a number cannot be simplified indefinitely it has to terminate at some point.
So, the basic assumption that \[\sqrt{2}\] is a rational number will fail here. So the answer contradicts the basic assumption that \[\sqrt{2}\] as a rational number is unreasonable.
So, we can conclude that the contradiction has been reached that \[\sqrt{2}\] is not a rational number.
Hence we have proved that \[\sqrt{2}\] is irrational.
FAQs on Why Is the Square Root of 2 Irrational? Euclid’s Explanation
1. What is an irrational number in the context of the CBSE Class 10 syllabus?
An irrational number is a type of real number that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. For the purpose of the CBSE Class 10 curriculum, the key characteristic is that their decimal expansion is non-terminating and non-repeating. A classic example studied is the square root of 2 (√2).
2. What is the main method used to prove that the square root of 2 is irrational?
The primary method used to prove that √2 is irrational is the 'proof by contradiction'. This logical technique begins by assuming the opposite of what we want to prove. We start by assuming that √2 is a rational number. Through a series of logical steps, this assumption leads to a contradiction, thereby proving that the initial assumption must have been false and that √2 is indeed irrational.
3. In the proof that √2 is irrational, why is it essential to assume it equals a fraction p/q in its simplest form?
Assuming that √2 = p/q, where p and q are coprime integers (meaning their only common factor is 1), is the core of the contradiction. The proof proceeds to show that if √2 = p/q, then both 'p' and 'q' must be even. If both are even, they share a common factor of 2. This directly contradicts our crucial starting assumption that p/q was in its simplest form, thus invalidating the premise that √2 is rational.
4. What is the role of the Fundamental Theorem of Arithmetic in proving the irrationality of √2?
The Fundamental Theorem of Arithmetic is vital. It states that every composite number can be uniquely expressed as a product of primes. When the proof shows that p² is an even number (p² = 2q²), this theorem allows us to conclude that 'p' itself must be an even number. This is because for p² to have a prime factor of 2, 'p' must also have a prime factor of 2. This step is essential for reaching the final contradiction.
5. Why can't a calculator be used to definitively prove that √2 is irrational?
A calculator only displays a finite number of decimal places (e.g., 1.41421356...). While this display doesn't show a repeating pattern, it doesn't prove one doesn't exist. The pattern could theoretically begin after a trillion digits, which is beyond the calculator's capacity. A formal mathematical proof is necessary to show with certainty that the decimal expansion is infinite and non-repeating, which is the absolute definition of an irrational number.
6. Who is generally credited with first proving that the square root of 2 is irrational?
The discovery that √2 is irrational is widely attributed to Hippasus of Metapontum, who was a member of the ancient Greek school of Pythagoreans. This was a groundbreaking and disruptive discovery because the Pythagorean worldview was based on the belief that all phenomena could be described by whole numbers and their ratios.
7. Can the 'proof by contradiction' method for √2 be used to prove other numbers are irrational?
Yes, this powerful method is not limited to just √2. The same logical framework can be adapted to prove the irrationality of the square roots of any prime number. For example, as per the CBSE syllabus, students can apply this same technique to prove that √3, √5, and √7 are also irrational numbers by assuming they are rational and working towards a similar logical contradiction.
