Prove that 5 is irrational and hence show that 3 + 2 5 is irrational.
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Step-by-Step Solution
Step 1: Assume 5 is rational
To prove that 5 is irrational, we use the method of contradiction. We start by assuming the opposite, that 5 is a rational number. A rational number can always 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., they are coprime).
Step 2: Square both sides and rearrange
If 5=ba, then squaring both sides gives 5=b2a2. Multiplying both sides by b2 gives us the equation 5b2=a2. This equation implies that a2 is a multiple of 5, which means that a itself must be a multiple of 5.
Step 3: Substitute a=5c and find a contradiction
Since a is a multiple of 5, we can write a=5c for some integer c. Substituting this into the equation 5b2=a2, we get 5b2=(5c)2, which simplifies to 5b2=25c2. Dividing both sides by 5 gives b2=5c2. This implies that b2 is a multiple of 5, and therefore b must also be a multiple of 5.
Step 4: Conclude 5 is irrational
We have shown that both a and b are multiples of 5. This means that a and b have a common factor of 5. However, our initial assumption was that a and b are coprime (have no common factors other than 1). This is a contradiction. Therefore, our initial assumption that 5 is rational must be false. Hence, 5 is irrational.
Step 5: Assume 3+25 is rational
Now we need to prove that 3+25 is irrational. Again, we use the method of contradiction. Let's assume that 3+25 is a rational number. This means it can be written as qp, where p and q are integers, q=0, and p and q are coprime.
Step 6: Isolate 5
From the equation 3+25=qp, we can isolate 5. First, subtract 3 from both sides: 25=qp−3. Combine the terms on the right side: 25=qp−3q. Finally, divide by 2: 5=2qp−3q.
Step 7: Conclude 3+25 is irrational
Since p, q, 2, and 3 are all integers, the expression 2qp−3q must be a rational number. This means that if 3+25 were rational, then 5 would also be rational. However, we have already proven that 5 is irrational. This is a contradiction. Therefore, our assumption that 3+25 is rational must be false. Hence, 3+25 is irrational.