Prove that the square root of 2 is irrational.
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Step-by-Step Solution
Step 1: Assume 2 is rational
To prove that 2 is irrational, we will use the method of proof by contradiction. We start by assuming the opposite: that 2 is a rational number. By definition, a rational number can be expressed as a fraction ba, where a and b are integers, b=0, and a and b have no common factors other than 1 (i.e., the fraction is in its simplest form).
Step 2: Square both sides
Next, we square both sides of the equation 2=ba to eliminate the square root. This gives us 2=b2a2. Rearranging the terms, we get a2=2b2. This equation tells us that a2 is an even number, because it is equal to 2 times another integer (b2).
Step 3: Deduce a is even
If a2 is an even number, then a itself must also be an even number. This is because the square of an odd number is always odd ((2n+1)2=4n2+4n+1=2(2n2+2n)+1), and the square of an even number is always even ((2n)2=4n2=2(2n2)). Therefore, we can write a as 2k for some integer k.
Step 4: Substitute a into the equation
Now we substitute a=2k back into our equation a2=2b2. This gives us (2k)2=2b2, which simplifies to 4k2=2b2. Dividing both sides by 2, we get 2k2=b2. This equation shows that b2 is an even number, because it is equal to 2 times another integer (k2).
Step 5: Deduce b is even and find contradiction
Similar to the previous step, if b2 is an even number, then b itself must also be an even number. So, we can write b as 2m for some integer m. This means that both a and b are even numbers, which implies they both have a common factor of 2. However, this contradicts our initial assumption that a and b have no common factors other than 1. Since our initial assumption leads to a contradiction, it must be false.
Step 6: Conclusion
Since our initial assumption that 2 is rational led to a contradiction, we must conclude that the assumption is false. Therefore, 2 cannot be expressed as a fraction of two integers and is, by definition, an irrational number.