Prove the Square Root of 2 is irrational

In summary, Algebra 2 question asks what is a rational number and how does this relate to geometry. The answer is that a rational number is of form a/b where a and be are mutually prime. This means that if p is an odd integer, then it can be written 2n+ 1 where n is any integer. p2 is of the form 2m+1 (m= 2n2+2n) which is odd, so a must be even.
  • #1
njkid
22
0
This is Algebra 2 question...

I have to prove that the square root of 2 is irrational...

First we must assume that

sqrt (2) = a/b

I never took geometry and i don't know proofs...

Please help me.

Thank you.
 
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  • #2
a rational number is of form a/b where a and be are mutually prime. I will give you a hint: you must prove that and and b cannot possibly be mutually prime.

and what does this have to do with geometry?
 
  • #3
You're off to a good start. Let "a" and "b" be natural numbers.

[tex]\sqrt{2}=\frac{a}{b}\implies b\sqrt{2}=a[/tex]

Now, how could "a" be a natural number? Well, "b" would be some multiple of [itex]\sqrt{2}[/itex]. This in turn, would mean that "b" isn't a natural number.

Can you see where this is going? You need to prove this by contradiction.
 
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  • #4
"Now, how could "a" be a natural number? Well, "b" would be some multiple of [itex]\sqrt{2}[/itex]. This in turn, would mean that "b" isn't a natural number."
How does that follow? Saying b*(1/2) , for example, equals a natural number does not imply that b isn't a natural number! Of course 1/2 isn't an irrational number but the whole point here is to prove that [itex]\sqrt{2}[/itex] is irrational.

Better to note that if [itex]\frac{a}{b}= \sqrt{2}[/itex] then, squaring both sides, [itex]\frac{a^2}{b^2}= 2[/itex] so that a2= 2b2 showing that a2 is even.

Crucial point: the square of an odd integer is always odd:

If p is an odd integer, then it can be written 2n+ 1 where n is any integer.

p2= (2n+1)2= 4n2+ 4n+ 1= 2(2n2+2n)+1.

Since 2n2+ 2n is an integer, p2 is of the form 2m+1 (m= 2n2+2n) and so is odd.

Do you see why knowing that tells us that a must be even?
 
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  • #5
HallsofIvy said:
"Now, how could "a" be a natural number? Well, "b" would be some multiple of [itex]\sqrt{2}[/itex]. This in turn, would mean that "b" isn't a natural number."
How does that follow? Saying b*(1/2) , for example, equals a natural number does not imply that b isn't a natural number! Of course 1/2 isn't an irrational number but the whole point here is to prove that [itex]\sqrt{2}[/itex] is irrational.

Better to note that if [itex]\frac{a}{b}= \sqrt{2}[/itex] then, squaring both sides, [itex]\frac{a^2}{b^2}= 2[/itex] so that a2= 2b2 showing that a2 is even.

Crucial point: the square of an odd integer is always odd:

If p is an odd integer, then it can be written 2n+ 1 where n is any integer.

p2= (2n+1)2= 4n2+ 4n+ 1= 2(2n2+2n)+1.

Since 2n2+ 2n is an integer, p2 is of the form 2m+1 (m= 2n2+2n) and so is odd.

Do you see why knowing that tells us that a must be even?

Wow! You are so good at teaching! Thank you everybody!
 

What is the definition of an irrational number?

An irrational number is a number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction with a non-zero denominator.

What is the proof that the square root of 2 is irrational?

The proof that the square root of 2 is irrational is a classic mathematical proof known as a proof by contradiction. It begins by assuming that the square root of 2 is rational, and then uses algebraic manipulations to arrive at a contradiction, thus proving that the initial assumption was incorrect.

Why is the proof of the square root of 2 being irrational important?

The proof of the square root of 2 being irrational is important because it demonstrates that there are numbers that cannot be expressed as fractions, expanding our understanding of numbers beyond the realm of rational numbers.

Are there other methods to prove the irrationality of the square root of 2?

Yes, there are other methods to prove the irrationality of the square root of 2. One method is to use the unique prime factorization of integers and show that the prime factorization of the square of an integer cannot include a factor of 2. Another method is to use the concept of continued fractions.

What are some real-world applications of the proof of the square root of 2 being irrational?

The proof of the square root of 2 being irrational has many real-world applications, including in geometry, number theory, and cryptography. It also has implications in fields such as physics and engineering, where irrational numbers are used in calculations and measurements.

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