# Sum of rational and irrational is irrational

• bigchaka
In summary: The proof uses a contradiction and a lemma. The contradiction is that if a+q is rational and q is rational, then aq is rational. The lemma is that if a and b are rational then ba is rational. So if a+q is rational, then aq is rational because ba is rational.
bigchaka
Summary:: i get a proof that sum of rational and irrational is rational
which is wrong(obviously)

let a be irrational and q is rational. prove that a+q is irrational.
i already searched in the web for the correct proof but i can't seem to understand why my proof is false.
my proof:

as you can see the proof i get is that the sum is rational.
can someone explain why thi

I don't understand why you talk about ##a+q## and write about ##a\cdot q##.
Anyway. What you did is: If ##a+q=q'## then ##a\in \mathbb{Q}##. This is correct. You wrote that ##a## is irrational, but you didn't use it. Quite the opposite is true: you started with an equation where ##a## is already rational.

This is the way to prove it anyway, both additive and multiplicative:

Given ##a\notin \mathbb{Q}## and ##q\in \mathbb{Q}##.
Assume ##a+q \in \mathbb{Q}##. Then ##a+q=q'\in \mathbb{Q}## i.e. ##a=q'-q \in \mathbb{Q}## since ##\mathbb{Q}## is closed under addition. But this contradicts ##a\notin \mathbb{Q}##, hence the assumption ##a+q \in \mathbb{Q}## was wrong, i.e. ##a+q \notin \mathbb{Q}##.

Multiplication goes the same way, except that you have to deal with ##q=0## as special case where the statement is wrong. The proof needs inverse elements and ##0## has no multiplicative inverse.

Last edited:
bigchaka and etotheipi
fresh_42 said:
I don't understand why you talk about a+q and write about a⋅q.
Anyway. What you did is: If a+q=q′ then a∈Q. This is correct. You wrote that a is irrational, but you didn't use it. Quite the opposite is true: you started with an equation where a is already rational.

This is the way to prove it anyway, both additive and multiplicative:

Given a∉Q and q∈Q.
Assume a+q∈Q. Then a+q=q′∈Q i.e. a=q′−q∈Q since Q is closed under addition. But this contradicts a∉Q, hence the assumption a+q∈Q was wrong, i.e. a+q∉Q.

Multiplication is the same way.
sorry my bad. it was mixed inside my head i did proofs all the day..
you are right.

You have shown "if a+q is rational and q is rational then aq is rational". Which is correct, but a+q and q both being rational implies a is rational.

bigchaka
mfb said:
You have shown "if a+q is rational and q is rational then aq is rational". Which is correct, but a+q and q both being rational implies a is rational.
I got it, thanks!

It's really hard to follow the scribble, but in general, I would attack problems of this form with an indirect proof.

... this is a case of "two out of three ain't possible".

A simple proof follows immediately from the sum of two rationals being rational.

## 1. What is the definition of a rational number?

A rational number is a number that can be expressed as a ratio of two integers, meaning it can be written as a fraction in the form a/b where a and b are integers and b is not equal to 0.

## 2. What is the definition of an irrational number?

An irrational number is a number that cannot be expressed as a ratio of two integers. It is a non-repeating, non-terminating decimal and cannot be written as a fraction.

## 3. Why is the sum of a rational and an irrational number always irrational?

This is because when we add a rational and an irrational number, the result is a non-repeating, non-terminating decimal. Since the irrational number cannot be expressed as a fraction, it cannot be simplified or combined with the rational number to create a rational number. Therefore, the sum will always be irrational.

## 4. Can the sum of two irrational numbers be rational?

Yes, it is possible for the sum of two irrational numbers to be rational. For example, the sum of √2 and -√2 is 0, which is a rational number. However, this is only possible when the irrational numbers have opposite signs and equal magnitudes.

## 5. Are there any exceptions to the rule that the sum of a rational and an irrational number is irrational?

No, there are no exceptions to this rule. The sum of a rational and an irrational number will always be irrational, regardless of the specific numbers used. This is because the definition of an irrational number is that it cannot be expressed as a ratio of two integers, and adding a rational number to it will not change this fact.

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