Step by Step Proof That Root 2 Cannot Be Rational Number
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 Euclid Proof That Square Root of 2 Is Irrational
1. Why is √2 irrational?
The square root of 2 is irrational because it cannot be written as a ratio of two integers in the form a/b. Euclid’s proof uses contradiction:
- Assume √2 = a/b in lowest terms.
- Squaring gives 2 = a²/b², so a² = 2b².
- This implies a² is even, so a is even.
- If a is even, then b must also be even.
- This contradicts the assumption that a/b is in lowest terms.
2. What is Euclid’s proof that √2 is irrational?
Euclid’s proof shows that √2 is irrational using proof by contradiction. The steps are:
- Assume √2 = a/b, where a and b have no common factors.
- Square both sides: 2 = a²/b².
- Rearrange: a² = 2b².
- This means a² is even, so a is even.
- Substitute back to show b is also even.
- This contradicts the lowest-term assumption.
3. What does it mean for √2 to be irrational?
Saying √2 is irrational means it cannot be expressed as a fraction of two integers and its decimal expansion never terminates or repeats. In decimal form, √2 ≈ 1.414213..., and the digits continue infinitely without a repeating pattern. Therefore, it is not a rational number.
4. How do you prove √2 is irrational using contradiction?
To prove √2 is irrational using contradiction, assume the opposite and reach an impossibility.
- Assume √2 = a/b in simplest form.
- Square both sides to get a² = 2b².
- Conclude a is even, so write a = 2k.
- Substitute to show b is also even.
- This contradicts the assumption that the fraction is simplified.
5. Is √2 a rational or irrational number?
The number √2 is irrational because it cannot be written as a ratio of integers and has a non-terminating, non-repeating decimal expansion. Unlike rational numbers such as 1/2 = 0.5, the decimal of √2 ≈ 1.414213... continues infinitely without repetition.
6. What is the decimal value of √2?
The decimal value of √2 ≈ 1.414213562..., and it continues infinitely without repeating. This non-terminating, non-repeating decimal expansion confirms that √2 is an irrational number.
7. Why does a² = 2b² imply that a is even in Euclid’s proof?
The equation a² = 2b² implies a is even because a² is divisible by 2. If a square number is even, then the original number must also be even. Since a² equals 2b², it is divisible by 2, so a must be even.
8. Can √2 be written as a fraction?
No, √2 cannot be written as a fraction because it is an irrational number. Any attempt to express it as a/b in simplest form leads to a contradiction where both a and b must be even, which violates the definition of a simplified fraction.
9. What type of proof did Euclid use to show √2 is irrational?
Euclid used a proof by contradiction to show that √2 is irrational. This method assumes the opposite statement (that √2 is rational) and logically derives a contradiction, proving the original assumption false.
10. Why is the proof that √2 is irrational important in mathematics?
The proof that √2 is irrational is important because it was one of the first discoveries of irrational numbers and showed that not all numbers are rational. It introduced rigorous logical reasoning, especially proof by contradiction, which remains fundamental in number theory and mathematics.
