# Multiplicative Inverse Proof Problem

• MHB
• A.Magnus
In summary, the conversation discusses a proof problem involving ordered fields and the multiplicative and additive properties. The steps to solve the problem involve using the property of multiplicative inverse and unity, as well as the associative and commutative properties. Revisions were made to the initial steps to ensure accuracy.
A.Magnus
I am working on a proof problem on ordered field from a textbook, which lists additive and multiplicative properties similar to the ones here:

If $x \ne 0$, show that $\frac{1}{\frac{1}{x}} = x.$

The followings are what I was able to come out -- I just wanted to make sure that they are acceptable:

(a) By the multiplicative inverse property, there must exists $\frac{1}{x}$ such that $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = 1$.

(b) By the same property, there must exists $x$ such that $\frac{1}{x} \cdot x = 1$.

(c) By equating the (a) and (b) above, we have $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = \frac{1}{x} \cdot x$.

(d) By multiplying both sides with $x$, then we have $x \cdot \frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = x \cdot \frac{1}{x} \cdot x$, showing that $\frac{1}{\frac{1}{x}} = x$, as desired.

Thank you for your time and gracious helps. ~MA

MaryAnn said:
I am working on a proof problem on ordered field from a textbook, which lists additive and multiplicative properties similar to the ones here:

The followings are what I was able to come out -- I just wanted to make sure that they are acceptable:

(a) By the multiplicative inverse property, there must exists $\frac{1}{x}$ such that $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = 1$.

Hey again MaryAnn,

This is not quite correct.
Instead we have that given $x\ne 0$ by the multiplicative inverse property, there must exist an $\frac{1}{x}$ such that $\frac{1}{x} \cdot x = 1$.

Now we know that $\frac{1}{x}$, but we do not know if it is equal to $0$ or not.
Therefore we cannot deduce yet either that its multiplicative inverse exists, can we?
It's only as soon as we can conclude that $\frac{1}{x}\ne 0$, that we can conclude by the multiplicate inverse property that $\frac{1}{\frac{1}{x}}$ must exist, and that $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = 1$.

MaryAnn said:
(b) By the same property, there must exists $x$ such that $\frac{1}{x} \cdot x = 1$.

This is the wrong way around. We conclude from the existence of $x\ne 0$ that $\frac 1x$ exists.

MaryAnn said:
(c) By equating the (a) and (b) above, we have $\frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = \frac{1}{x} \cdot x$.

This is a correct deduction.

MaryAnn said:
(d) By multiplying both sides with $x$, then we have $x \cdot \frac{1}{x} \cdot \frac{1}{\frac{1}{x}} = x \cdot \frac{1}{x} \cdot x$, showing that $\frac{1}{\frac{1}{x}} = x$, as desired.

This is true, but you are skipping a couple of steps, and you are not mentioning which properties you are using.
As it is, you are using the property of associativity (twice), the property of the multiplicative inverse (twice), and the property of unity (twice).

MaryAnn said:
Thank you for your time and gracious helps. ~MA

Thank you for your response! ~MA

How about these revised steps - hope these come out right:

(a) Since $x \neq 0$, there must exist $\frac{1}{x}$ such that $x \cdot \frac{1}{x} = 1$. (Multiplicative Inverse Property)

(b) By Commutative Property, we have $\frac{1}{x} \cdot x = 1$.

(c) Multiplying both sides of the equation by $\frac{1}{\frac{1}{x}}$, we have $\frac{1}{\frac{1}{x}} \cdot \frac{1}{x} \cdot x = 1 \cdot \frac{1}{\frac{1}{x}}$.

(d) By Property of Unity, in the right hand side we have $\frac{1}{\frac{1}{x}} \cdot \frac{1}{x} \cdot x = \frac{1}{\frac{1}{x}}$.

(e) By Associative Property, we then have $(\frac{1}{\frac{1}{x}} \cdot \frac{1}{x}) \cdot x = \frac{1}{\frac{1}{x}}$.

(f) By Property of Unity, we finally have $1 \cdot x = \frac{1}{\frac{1}{x}}$, and by the same property again $x = \frac{1}{\frac{1}{x}}$, as desired.

Thank you again for all your helps. ~MA

MaryAnn said:
How about these revised steps - hope these come out right:

(a) Since $x \neq 0$, there must exist $\frac{1}{x}$ such that $x \cdot \frac{1}{x} = 1$. (Multiplicative Inverse Property)

(b) By Commutative Property, we have $\frac{1}{x} \cdot x = 1$.

(c) Multiplying both sides of the equation by $\frac{1}{\frac{1}{x}}$, we have $\frac{1}{\frac{1}{x}} \cdot \frac{1}{x} \cdot x = 1 \cdot \frac{1}{\frac{1}{x}}$.

Careful, we should multiply on the same side, since we should not assume commutativity of multiplication.
And we should not assume associativity by leaving out the parentheses.
So it should be: $\frac{1}{\frac{1}{x}} \cdot \left(\frac{1}{x} \cdot x\right) = \frac{1}{\frac{1}{x}} \cdot 1$

MaryAnn said:
(d) By Property of Unity, in the right hand side we have $\frac{1}{\frac{1}{x}} \cdot \frac{1}{x} \cdot x = \frac{1}{\frac{1}{x}}$.

(e) By Associative Property, we then have $(\frac{1}{\frac{1}{x}} \cdot \frac{1}{x}) \cdot x = \frac{1}{\frac{1}{x}}$.

(f) By Property of Unity, we finally have $1 \cdot x = \frac{1}{\frac{1}{x}}$, and by the same property again $x = \frac{1}{\frac{1}{x}}$, as desired.

That's the property of multiplicative inverse instead of unity.
The second part is using the property of unity though.

## 1. What is a multiplicative inverse?

A multiplicative inverse, also known as a reciprocal, is a number that when multiplied by a given number, results in a product of 1. For example, the multiplicative inverse of 5 is 1/5, because 5 x 1/5 = 1.

## 2. What is the Multiplicative Inverse Proof Problem?

The Multiplicative Inverse Proof Problem is a mathematical problem that involves proving the existence of a multiplicative inverse for a given number. This is typically done by showing that the product of the number and its inverse is equal to 1.

## 3. How do you solve the Multiplicative Inverse Proof Problem?

The Multiplicative Inverse Proof Problem can be solved by using algebraic manipulation and properties of multiplication. For example, if the given number is a fraction, you can multiply it by its reciprocal to show that the product is equal to 1.

## 4. Why is the Multiplicative Inverse Proof Problem important?

The Multiplicative Inverse Proof Problem is important because it is a fundamental concept in mathematics and is used in various fields such as engineering, physics, and economics. It also helps to understand the concept of fractions and how they relate to each other.

## 5. Are there any real-world applications of the Multiplicative Inverse Proof Problem?

Yes, the Multiplicative Inverse Proof Problem has various real-world applications. For example, it is used in calculating interest rates in finance, finding equivalent resistance in electrical circuits, and determining concentration and dilution in chemistry.

Replies
11
Views
2K
Replies
6
Views
999
Replies
1
Views
1K
Replies
3
Views
2K
Replies
10
Views
1K
Replies
16
Views
1K
Replies
4
Views
2K
Replies
17
Views
2K
Replies
14
Views
3K
Replies
3
Views
2K