# How to prove rational number is a commutative field

1. Sep 10, 2011

### colstat

I was wondering how to prove rational number is a commutative field.

Personally, I didn't think the word "commutative" is necessary, how about others?

Do I simply prove it is commutative under multiplication?

2. Sep 10, 2011

### micromass

Staff Emeritus
You'll need to check the field axioms. See http://en.wikipedia.org/wiki/Field_(mathematics)

That is, you need to check

• Associativity of addition and multiplication
• Commutativity of addition and multiplication
• Existence of the additive and multiplicative identities
• Existence of the additive and multiplicative inverses for each nonzero element
• Distributivity

So;, which one is troubling you??

3. Sep 10, 2011

### colstat

Hey, micromass. How do I check field axioms?
Do I say a and b belongs to rational numbers and go through the axiom?

4. Sep 10, 2011

### micromass

Staff Emeritus
OK, let me do an example. Let me check commutativity of addition. Take two fraction a/b and c/d. We must prove that

$$\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}$$

Indeed:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

and

$$\frac{c}{d}+\frac{a}{b}=\frac{bc+ad}{db}$$

because addition and multiplications in the integers is commutative, we got that the two right-hand sides above are equal. Thus the left-hand-sides are also equal. Thus addition is commutative.

Can you prove all the other ones?

5. Sep 10, 2011

### colstat

Thanks so much, I think I got it, that was very helpful! :)
The text passage is a discussion thread where a user asks for guidance on proving that rational numbers form a commutative field, and another user provides a detailed response with an example of how to check one of the field axioms.
no_question

the passage does not contain a mathematical question in the form of a problem to solve or a calculation to perform. instead, it is a request for guidance on a proof, which is more of a conceptual question. there is no specific mathematical question with an answer provided in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# How to prove rational number is a commutative field

1. Sep 10, 2011

### colstat

I was wondering how to prove rational number is a commutative field.

Personally, I didn't think the word "commutative" is necessary, how about others?

Do I simply prove it is commutative under multiplication?

2. Sep 10, 2011

### micromass

Staff Emeritus
You'll need to check the field axioms. See http://en.wikipedia.org/wiki/Field_(mathematics)

That is, you need to check

• Associativity of addition and multiplication
• Commutativity of addition and multiplication
• Existence of the additive and multiplicative identities
• Existence of the additive and multiplicative inverses for each nonzero element
• Distributivity

So;, which one is troubling you??

3. Sep 10, 2011

### colstat

Hey, micromass. How do I check field axioms?
Do I say a and b belongs to rational numbers and go through the axiom?

4. Sep 10, 2011

### micromass

Staff Emeritus
OK, let me do an example. Let me check commutativity of addition. Take two fraction a/b and c/d. We must prove that

$$\frac{a}{b}+\frac{c}{d}=\frac{c}{d}+\frac{a}{b}$$

Indeed:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

and

$$\frac{c}{d}+\frac{a}{b}=\frac{bc+ad}{db}$$

because addition and multiplications in the integers is commutative, we got that the two right-hand sides above are equal. Thus the left-hand-sides are also equal. Thus addition is commutative.

Can you prove all the other ones?

5. Sep 10, 2011

### colstat

Thanks so much, I think I got it, that was very helpful! :)
user
The text passage is a discussion thread where a user asks for guidance on proving that rational numbers form a commutative field, and another user provides a detailed response with an example of how to check one of the field axioms.

TYPE: NO_QUESTION

The passage does not contain a mathematical question in the form of a problem to solve or a calculation to perform. Instead, it is a request for guidance on a proof, which is more of a conceptual question. There is no specific mathematical question with an answer provided in the text.