Proofs concerning irrational numbers

Here are the key points to understand when studying the proof that √2 is irrational:

1. Definition of Rational Numbers: A rational number can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$.

2. Assumption: Start by assuming that √2 is rational, so it can be expressed as $\frac{p}{q}$, with $p$ and $q$ in their simplest form (no common factors).

3. Equation Setup: From the assumption, you derive the equation $\sqrt{2} = \frac{p}{q}$ and square both sides to get $2 = \frac{p^2}{q^2}$, leading to $p^2 = 2q^2$.

4. Even Numbers: Since $p^2 = 2q^2$, it follows that $p^2$ is even, which implies that $p$ must also be even (the square of an odd number is odd).

5. Substitution: If $p$ is even, it can be expressed as $p = 2k$ for some integer $k$. Substitute back into the equation to get $(2k)^2 = 2q^2$, simplifying to $4k^2 = 2q^2$ or $q^2 = 2k^2$.

6. Contradiction: The conclusion $q^2 = 2k^2$ indicates that $q^2$ is also even, hence $q$ must also be even. This means both $p$ and $q$ are even, contradicting the assumption that $\frac{p}{q}$ is in simplest form.

7. Conclusion: Since the assumption that √2 is rational leads to a contradiction, it follows that √2 must be irrational.

These points outline the classic proof by contradiction used to demonstrate the irrationality of √2.

Here are the key points for studying the proof that the square roots of prime numbers are irrational:

1. Definition of Irrational Numbers: Understand that an irrational number cannot be expressed as a fraction of two integers.

2. Proof by Contradiction: Familiarize yourself with the method of proof by contradiction, which involves assuming the opposite of what you want to prove and showing that this leads to a contradiction.

3. Assumption: Start by assuming that the square root of a prime number $p$ is rational, meaning it can be expressed as $\frac{a}{b}$, where $a$ and $b$ are integers with no common factors (in simplest form).

4. Squaring Both Sides: Square both sides of the equation to derive $p = \frac{a^2}{b^2}$ or $a^2 = pb^2$.

5. Divisibility: Analyze the implications of $a^2 = pb^2$. It shows that $a^2$ is divisible by $p$, and thus $a$ must also be divisible by $p$ (because if a prime divides a square, it divides the base).

6. Substituting Back: Let $a = pk$ for some integer $k$, and substitute back into the equation to show that $b$ must also be divisible by $p$.

7. Contradiction: This leads to a contradiction because it implies that both $a$ and $b$ have $p$ as a common factor, which contradicts the original assumption of being in simplest form.

8. Conclusion: Thus, the assumption that $\sqrt{p}$ is rational leads to a contradiction, proving that $\sqrt{p}$ is irrational.

By focusing on these points, you can effectively grasp the essentials of the proof that the square roots of prime numbers are irrational.

When studying the proof that there is an irrational number between any two rational numbers, focus on these key points:

1. Understanding Rational Numbers: Rational numbers are numbers that can be expressed as the quotient of two integers (e.g., $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$).

2. Irrational Numbers: These are numbers that cannot be expressed as a fraction of two integers, such as $\sqrt{2}$ or $\pi$.

3. Choosing Rational Numbers: Let two rational numbers $r_1$ and $r_2$ such that $r_1 < r_2$.

4. Constructing the Interval: The goal is to find an irrational number $x$ such that $r_1 < x < r_2$.

5. Using an Irrational Offset: A common approach is to consider an irrational number (like $\sqrt{2}$) and adjust it so that it lies between $r_1$ and $r_2$. For example, using $x = r_1 + (r_2 - r_1) \cdot \frac{\sqrt{2}}{2}$.

6. Verifying the Boundedness: Show that the constructed number $x$ remains between $r_1$ and $r_2$.

7. Generalization: This method can be generalized for any pair of rational numbers, illustrating that there are infinitely many irrational numbers between any two rationals.

Understanding these aspects provides a solid foundation for comprehending the nature of rational and irrational numbers and the idea that the set of irrational numbers is dense within the rational numbers.