Product of Negative Real Numbers: A Geometric Proof

[Leo Authersh]

What is the Geometric Proof for the product of two negative real numbers being a positive real number?

 

[fresh_42]

What is the Geometric Proof for the product of two negative real numbers being a positive real number?

First define a negative number and a negative product geometrically please. Otherwise we wouldn't know what to discuss.

 

[Leo Authersh]

First define a negative number and a negative product geometrically please. Otherwise we wouldn't know what to discuss.

Hi fresh_42,

Excuse my failure to provide a well-illustrated extension of my question. I haven't learned Geometry that deals with Negative and Complex Numbers. And I find it hard to define Negative numbers and operators in terms of Euclidean Geometry (which, as per my understanding, is a positive subset of Algebra that can be visualized). But I still there would be an axiomatic proof for negative numbers in terms of other Geometric studies.

 

[symbolipoint]

What is the Geometric Proof for the product of two negative real numbers being a positive real number?

I'm not sure that the necessary Geometry needs to be sophisticated. Would a number line be enough for your purpose? One should be able to show both the addition of commonly-understood positive numbers, and also the addition of positive and negative numbers. The negative values on the number line are those which are toward the LEFT side of zero.

 

[Leo Authersh]

I'm not sure that the necessary Geometry needs to be sophisticated. Would a number line be enough for your purpose? One should be able to show both the addition of commonly-understood positive numbers, and also the addition of positive and negative numbers. The negative values on the number line are those which are toward the LEFT side of zero.

Hi symbolipoint,

As I now think about it, I can remember that the Euclidean Geometry is only applicable for positive algebraic operations and representations. It cannot represent negative numbers. It can only represent the subtraction of positive numbers.

Now, I have clarified my question in my reply for fresh_42. Wish you will provide me your understanding on that.

 

[hilbert2]

If you define complex number multiplication geometrically (moduli of the numbers are multiplied and phase angles are added), you can show in a kind of geometric way that the product of two negative numbers $0>a,b\in \mathbb{R} \subset \mathbb{C}$ is positive.

As the moduli of the complex numbers are positive, you can define their product as the area of a rectangle.

 

[jbriggs444]

As I now think about it, I can remember that the Euclidean Geometry is only applicable for positive algebraic operations and representations. It cannot represent negative numbers. It can only represent the subtraction of positive numbers.

If one considers the angle made at the intersection of two line segments, AB and BC on the Euclidean plane, would you accept that the angle can be characterized as being less then a straight angle (counter-clockwise rotation at the intersection) or greater than a straight angle (clockwise rotation at the intersection)?

 

[Leo Authersh]

If one considers the angle made at the intersection of two line segments, AB and BC on the Euclidean plane, would you accept that the angle can be characterized as being less then a straight angle (counter-clockwise rotation at the intersection) or greater than a straight angle (clockwise rotation at the intersection)?

This is a great idea. If we consider two lines A and B and if magnitude of A is considered as the positive value then the magnitude of B loses its arithmetic meaning with respect to the Geometrical Line A only at two angles 90° and 180°.

At 90°, the arithmetic value of B on A is zero.

At 180°, the arithmetic value of B on A is Geometrically incalculable since at this angle B changes its sign value (direction) which is irrepresentable in the Geometric World.

But at all other angles between 0° and 180°, B has arithmetic value on A. When the intercepting angle is acute, B is addend of A and when the intercepting angle is obtuse, B is the subtractend of A.

Even though, this representation is limited within 180° degrees, this provides a very good intuition on the nature of Geometrical representation of Arithmetics.

Thank you for this valuable suggestion.

 

[Leo Authersh]

If you define complex number multiplication geometrically (moduli of the numbers are multiplied and phase angles are added), you can show in a kind of geometric way that the product of two negative numbers $0>a,b\in \mathbb{R} \subset \mathbb{C}$ is positive.

As the moduli of the complex numbers are positive, you can define their product as the area of a rectangle.

This is the representation I've been looking for. Can you please provide me any reference article that explains this Complex Geometry?

 

[jbriggs444]

At 90°, the arithmetic value of B on A is zero.

That amounts to an assessment of the cosine of the angle between the two line segments.

What I had in mind was imparting a "handedness" to the Euclidean plane and thinking about the sine. At 90 degrees, the sine of the angle is either +1 or -1 depending on whether ∠ABC is counter-clockwise (+90 degrees = left turn) or clockwise (-90 degrees = right turn).

This in turn allows one to consider counter-clockwise polygons with positive area and clockwise polygons with negative area.

Edit: updated the text to match the sign convention I really had in mind -- which matches the picture produced by @fresh_42

 

[fresh_42]

How about this solution?

with orientation x first then y.