Proving 13 + 2√6 is an Irrational Number with Proof by Contradiction

In summary, we are asked to prove or disprove the statement that 13 + 2√6 is an irrational number, given that √6 is irrational. To solve this, we assume that 13 + 2√6 is a rational number and use the definition of a rational number to manipulate the expression. However, this leads to a contradiction as we know that √6 is irrational. Therefore, the statement is proven to be true.
  • #1
Brooke73
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Homework Statement



Prove or disprove the statement:
13 + 2√6 is an irrational number
Given that √6 is irrational


Homework Equations



Rational number = p/q where p and q are integers

The Attempt at a Solution


Assume that 13 + 2√6 is a rational number
Rational number = p/q where p and q are integers
Let 13 = n where n is an integer
Let 2√6 = x
x + n = p/q
x = ( p/q) – n
x = (p – qn)/q
We have shown that x can be expressed as a ratio of two integers. This is a contradiction because we know that x is irrational. Therefore, the statement: 13 + 2√6 is an irrational number is proven true.
 
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  • #2
Correct idea, but x = 2√6, and you are given that √6 is irrational.
 

1. What is "Proof by Contradiction" in mathematics?

"Proof by Contradiction" is a method used in mathematics to prove a statement or theorem by assuming the opposite or negation of the statement and showing that it leads to a contradiction or absurdity. This proves that the original statement must be true.

2. How does "Proof by Contradiction" work?

In "Proof by Contradiction", we start by assuming the opposite of the statement we want to prove is true. We then use logical deductions and mathematical principles to arrive at a contradiction. This shows that our initial assumption must be false and the original statement is therefore true.

3. When is "Proof by Contradiction" used?

"Proof by Contradiction" is used to prove statements in mathematics that cannot be easily proven using direct or indirect methods. It is often used in proving theorems related to real numbers, sets, and inequalities.

4. What is the difference between "Proof by Contradiction" and "Proof by Contrapositive"?

"Proof by Contradiction" and "Proof by Contrapositive" are both indirect proof methods. However, "Proof by Contradiction" starts by assuming the opposite of the statement and leads to a contradiction, while "Proof by Contrapositive" starts with the negation of the conclusion and shows that it implies the negation of the hypothesis.

5. What are the limitations of "Proof by Contradiction"?

"Proof by Contradiction" cannot be used to prove all statements or theorems in mathematics. It can only be used when there is a logical contradiction that can be reached from the opposite of the statement. Additionally, it may not provide an intuitive understanding of why a statement is true, as it relies on the assumption of its opposite being false.

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