# Proving Statements for Field Math: F/b = (ab^-1) and Other Equations

• karnten07
In summary, the conversation discusses proving statements about fields and their elements using alternative values and inverse operations. The statements cover topics such as equality, addition, multiplication, and division within a field. The individual provides a solution for the first statement, using alternative values and inverse operations to show that a/b and a'/b' are equal when ab' = a'b.
karnten07

## Homework Statement

Let F be a field. For any a,b $$\in$$ F, b$$\neq$$0, we write a/b for ab^-1. Prove the following statements for any a, a' $$\in$$F and b, b' $$\in$$ F\{0}:

i.) a/b = a'/b' if and only if ab' =a'b
ii.) a/b +a'/b' = (ab'+a'b)/bb'
iii.) (a/b)(a'/b') = aa'/bb'
iv.) (a/b)/a'/b') = ab'/a'b (if in addition a'$$\neq$$0)

## The Attempt at a Solution

I'm struggling to understand how i am to prove these statements. What am i to take the dashes to mean, because they are often used to show inverses? So for the first one:

a/b=ab^-1 which = a^-1b = a'/b'?

My guess is a dash means "alternative value." For example, a = 1, and a' = 2.

EnumaElish said:
My guess is a dash means "alternative value." For example, a = 1, and a' = 2.

I also thought that, i will go with that and see what i come up with, thanks

EnumaElish said:
My guess is a dash means "alternative value." For example, a = 1, and a' = 2.

Does ab^-1 mean a.b^-1 or (a.b)^-1? I think it might be the former.

If it is, i get:

i) a.b^-1 = a'.b'^-1 when written out fully. So if ab' = a'b, then rearranged gives a= a'b/b' and a' = ab'/b. So inserting them into a.b^-1 = a'.b'^-1 we get:

a'b.b^-1/b' = ab.b^-1/b

and then we get indentity elements leaving a'/b' = a/b

Is this right?

## 1. What is the purpose of proving statements in field math?

The purpose of proving statements in field math is to provide a logical and rigorous justification for mathematical equations and concepts. This helps ensure the validity and accuracy of mathematical theories and enables mathematicians to build upon existing knowledge and make new discoveries.

## 2. What does the equation F/b = (ab^-1) represent?

This equation represents the division of two elements in a field, where F is the dividend, b is the divisor, and a is the multiplicative inverse of b. In other words, it shows the relationship between the quotient of two elements and their multiplicative inverses.

## 3. How do you prove an equation like F/b = (ab^-1)?

To prove an equation like F/b = (ab^-1), you would start by assuming that the equation is true and then use logical steps and previously established mathematical principles to derive a valid proof. This may involve using properties of fields, such as the commutative and associative properties, as well as the definition of multiplicative inverses.

## 4. Can you provide an example of proving an equation in field math?

Sure, let's say we want to prove that (a+b)^2 = a^2 + 2ab + b^2 in a field. We can start by expanding the left side using the distributive property: (a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b). Then, using the distributive property again, we get a^2 + ab + ba + b^2. Since this is a field, the commutative property applies and we can rewrite the second term as ab instead of ba. This gives us a^2 + 2ab + b^2, which is the same as the right side of the equation, proving that (a+b)^2 = a^2 + 2ab + b^2.

## 5. Why is it important to prove equations in field math?

Proving equations in field math is important because it provides a solid foundation for understanding and applying mathematical concepts. It also allows for the discovery of new mathematical principles and the development of more complex theories. Additionally, proofs can be used to verify the accuracy of calculations and to identify any errors or inconsistencies in mathematical reasoning.

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