# Proof involving quantifiers and rationality/irrationality

• Oxygenate
In summary, the problem is to find two irrational numbers that will make the sum of them equal a rational number. It is proved that for every rational number z, there exist irrational numbers x and y such that x + y = z.
Oxygenate
I have no idea how to begin this proof.

## Homework Statement

Prove that for every rational number z, there exist irrational numbers x and y such that x + y = z.

## The Attempt at a Solution

I can't think of even a way to start this proof...it's just quite obvious that the sum of two irrational numbers will equal a rational number, somehow... Please give me some pointers to start this proof with.

"it's just quite obvious that the sum of two irrational numbers will equal a rational number" that, which is the statement converse of what you are asked, is not obvious and not true in general.

For the problem, you do know of a rational number A and an irrational x one that is less than it. Can you prove that (A - x) is also irrational?

Then you can do it for any pair of rationals.

Okay, so suppose A is irrational number. Then let x = 1/2z – a. And let y = 1/2z + a. Then x + y = z, and x and y are irrational.

Oxygenate said:
Okay, so suppose A is irrational number. Then let x = 1/2z – a. And let y = 1/2z + a. Then x + y = z, and x and y are irrational.

Yes, understand you mean x = z/2 – a. And let y = z/2 + a

But you have to prove explicitly that if z is irrational, so is z/2 + a etc.

Okay, so:

Let y = z - x. Then suppose y is rational, then m/n - x = a/b, in which case x = m/n - a/b = (mb - na) / (nb), which is a rational number. But this is a contradiction to the fact that x is an irrational number. Thus the statement (for every rational number z, there exist irrational numbers x and y such that x + y = z) is true.

I essentially just took your idea and developed it. Thanks for your help. :)

## 1. What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as a ratio and have an infinite number of non-repeating decimal places.

## 2. How do you prove that a number is rational?

A number can be proven rational if it can be expressed as a fraction in lowest terms, or if it has a repeating decimal pattern.

## 3. What is the contrapositive of a statement involving quantifiers?

The contrapositive of a statement involving quantifiers is the statement where the order of the quantifiers is reversed and the negation of the original statement is taken.

## 4. How can you use proof by contradiction to show a number is irrational?

To use proof by contradiction, we assume that the number is rational and then show that this leads to a contradiction. This contradiction proves that the number cannot be rational, meaning it must be irrational.

## 5. Can all irrational numbers be represented using mathematical symbols and operations?

No, some irrational numbers, such as pi and e, cannot be represented exactly using mathematical symbols and operations. They can only be approximated using decimal representations or other mathematical methods.

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