Help (1 Viewer)

[mmmmmmmmaaaaaaa]

I need help for part b for the two questions below. I have completed part a for both

 

[jimmysmith560]

Regarding Question 6, you have proven that is irrational.

Assume, for a contradiction, that this is not the case, and for some and . Since , we see that . Thus,

​

Since the number on the left-hand side is divisible by 5, while the number on the right-hand side is not divisible by 5, we reach a contradiction.

Now, since is irrational, it follows that:

is irrational.
​

I hope this helps!

 

[Paradoxica]

The first part of 7 is a standard exercise which can easily be done using contradiction (assuming you have seen the algebraic bash proof that √2 is irrational)

The second part: If x is irrational, then 2x is irrational (is what we want to show here, for a specific value of x)

Taking the contrapositive: If 2x is rational, then x is rational. This can be proven directly by writing 2x as a quotient of two integers.

In classical logic, the contrapositive of a statement is equivalent to the original statement.

Hence, the original statement is true, and 2√11 = √44 is irrational.

 

[mmmmmmmmaaaaaaa]

The first part of 7 is a standard exercise which can easily be done using contradiction (assuming you have seen the algebraic bash proof that √2 is irrational)

The second part: If x is irrational, then 2x is irrational (is what we want to show here, for a specific value of x)

Taking the contrapositive: If 2x is rational, then x is rational. This can be proven directly by writing 2x as a quotient of two integers.

In classical logic, the contrapositive of a statement is equivalent to the original statement.

Hence, the original statement is true, and 2√11 = √44 is irrational.

Regarding Question 6, you have proven that is irrational.

Assume, for a contradiction, that this is not the case, and for some and . Since , we see that . Thus,

​

Since the number on the left-hand side is divisible by 5, while the number on the right-hand side is not divisible by 5, we reach a contradiction.

Now, since is irrational, it follows that:

is irrational.
​

I hope this helps!

Thank you for the help. Much appreciated