# Negative times negative is positive?

• Bipolarity
In summary, the conversation is about proving that the multiplication of two negatives yields a positive. The guest is looking for a rigorous proof and the host asks about the axioms they are accepting. They then use axiom II.c to prove the statement and discuss the same statement for a product of two positive numbers in an ordered field. The guest then asks if this is only true for ordered fields and the host gives an example of a number system that is not ordered.
Bipolarity
Probably the stupidest question I have ever asked, but is it possible to prove that the multiplication of two negatives yields a positive? Go easy on me I've asked better questions :D

BiP

How rigorous a proof are you looking for?

As rigorous as rigorous gets :D

Bipolarity said:
As rigorous as rigorous gets :D

What axioms are you accepting??

These are the usual axioms of real numbers system:

http://www.gap-system.org/~john/analysis/Lectures/L5.html

Specifically, look at the 2nd axiom of order.

EDIT:
Then, prove the following:
$$a < 0 \Rightarrow -a > 0$$
By axiom II.c
$$0 > a \Rightarrow 0 + (-a) > a + (-a) \Leftrightarrow -a > 0$$
Q.E.D.

Then, look at the following:
$$a, b < 0 \Rightarrow a \cdot b = (-a) \cdot (-b)$$
Then you have a product of two positive numbers, which by the quoted axiom is positive.

Ahh, so this is only true for ordered fields then? Is there any number system where the field is not ordered (i.e. does not satisfy axiom II) ?

BiP

Bipolarity said:
Ahh, so this is only true for ordered fields then? Is there any number system where the field is not ordered (i.e. does not satisfy axiom II) ?

BiP

Sure, $\mathbb{Z}_2$ is not an ordered field.

If you don't know what it is: it's just the set {0,1} with

0+0=1+1=0, 1+0=0+1=1
0*0=0*1=1*0=0, 1*1=1

## 1. What is the rule for multiplying two negative numbers?

The rule for multiplying two negative numbers is that the product is always positive.

## 2. How do you explain the concept of negative times negative equals positive?

Negative times negative equals positive because when you multiply two negative numbers, you are essentially combining two sets of opposite values. This results in a positive value because the negatives cancel each other out.

## 3. Can you give an example to illustrate why negative times negative is positive?

For example, if we have -2 and -3, when we multiply them, we get a positive 6. This is because -2 represents a debt of 2, and -3 represents a debt of 3. When we combine these debts, we get a positive value of 6.

## 4. What is the difference between multiplying two negative numbers and adding two negative numbers?

The main difference between multiplying and adding two negative numbers is that the product of two negative numbers is always positive, while the sum of two negative numbers is always negative. This is because when multiplying, we are combining two sets of opposite values, while when adding, we are simply combining two negative values.

## 5. Is there any exception to the rule of negative times negative equals positive?

No, there is no exception to this rule. Negative times negative will always result in a positive value, regardless of the numbers being multiplied.

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