geometrical/physical interpretation of multiplication of real numbers (including negative)

Question

In calculus we see that the derivative has a physical interpretation as speed, and a geometric interpretation as slope, and that they are helpful when thinking intuitively about that concept. But this is just an example to motivate the question I want to make about a more elementary topic:

It is well know that multiplication of two positive real numbers can be interpreted as the area of a rectangle (and that it provides intuitive insight about the commutativity of multiplication). The problem is, what about negative numbers? The rectangle interpretation doesn't help in that case. Does it? So I ask:

Are there geometrical or physical interpretations of multiplication of two real numbers (which include negative numbers)?

Thank you.

Answer 1

There are a couple of different ways to express multiplication using geometry.

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One somewhat physical interpretation of multiplication involves stretching/compressing/reflecting the number line.

For example, multiplication by $3$ corresponds to stretching the numberline by a factor of $3$, where the stretching operation is centered on the point $0$; for example, the closed interval $[0,2]$ gets stretched out to to the interval $[0,6]$, corresponding to $3 \cdot 2 = 6$. In general: for each real number $r>0$, the closed interval $[0,r]$ on the number line gets stretched out to $[0,3r]$; and for each real number $r<0$ the closed interval $[r,0]$ gets stretched out to $[3r,0]$.

For another example, multiplication by $1/2$ corresponds to compressing the numberline by a factor of $2$. For example, the closed interval $[0,10]$ gets compressed to $[0,5]$, corresponding to $\frac{1}{2} \cdot 10 = 5$. In general, for each $r>0$ the closed interval $[0,r]$ gets compressed to $[0,r/2]$, and or each $r<0$ the closed interval $[r,0]$ gets compressed to $[r/2,0]$.

In general, multiplication by a positive number $s>0$ has the effect of stretching/compressing each closed interval of the form $[0,r]$ to $[0,sr]$, and similarly for $[r,0]$.

In order to multiply by a negative number, in addition to stretching/compression one must also reflect the line across the point $0$. For example multiplication by $-3$ corresponds to a composition of two operations, the first of which is stretching by a factor of $+3$, and the second of which is reflecting across $0$, so for example $[0,2]$ is first stretched to $[0,6]$ and then reflected to $[-6,0]$, corresponding to $(-3) \cdot 2 = -6$.

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There is also a more classical interpretation of multiplication using planar geometry; this interpretation first arises in Euclid, although nowadays it is easier to describe with the aid of Cartesian coordinates.

Instead of just a number line, let's consider a Cartesian coordinate plane, with $x,y$ coordinates. I'll give just a positive number example first, and then a negative number example.

To model multiplication by $3$, we consider two horizontal lines, namely $y=1$ and $y=3$. The line $L$ that passes through $(0,0)$ and intersects the line $y=1$ in the point $(2,1)$ then continues on to intersect the line $y=3$ in the point $(6,3)$, corresponding to the formula $3 \cdot 2 = 6$. In general, the line $L$ that passes through $(0,0)$ and intersects the line $y=1$ in the point $(r,1)$ then continues on to intersect the line $y=3$ in the point $(3r,3)$.

Multiplication by $-3$ is similarly modelled using the horizontal lines $y=1$ and $y=-3$: letting $L$ be the line that passes through $(0,0)$ and intersects the line $y=1$ in the point $(r,1)$, this line $L$ intersects $y=-3$ in the point $(-3r,-3)$.

Answer 2

I believe the geometrical interpretation still work, if you take the signed area of the rectangle. So, let's define an "oriented" rectangle as a rectangle with a particular orientation of its border (clockwise or anticlockwise), and define the area of the "oriented" rectangle to be:

• The same (positive, "as usual") value, if the border is oriented in anticlockwise direction.
• The negative of the "usual" value, if the border is oriented in the clockwise direction.

Now, place the rectangle in the co-ordinate system so that its consecutive corners are at $(0,0), (x,0), (x,y), (0,y)$ and use that sequence to define the orientation. As it happens, the area of this "oriented" rectangle, with that orientation, is $x\cdot y$, with the sign of the latter taken care of as well!

Answer 3

It is important to understand how the "Ancient civilizations" were defining the product of two positive quantities $a$ and $b$. @Lee Mosher has given such a definition.

Another one is through the area of a rectangle with sides $a$ and $b$. Here is for example a quotation from Van Der Waerden, the famous mathematician, well known for his "Moderne Algebra" (Springer 1930), but less known as an excellent historian of mathematics. Here is what he says page 71 of his "Geometry and Algebra in Ancient Civilizations" (Springer 1983);

[Omar Khayyam ; 11th century in Persia] writes: Every time we shall say in this book "a number is equal to a rectangle", we shall understand by the "number" a rectangle of which one side is unity, and the other a line equal in measure to the given number, in such a way that each of the parts by which it is measured is equal to the side which we have taken as unity. The same trick, namely in the introduction of a fixed unit of length e, was used by Rene Descartes [17th century] in his "Géométrie" for defining the product of two line segments $a$ and $b$ as a line segment $c$. He defined: $$ab=c \ \ \text{means} \ \ e/a=b/c.$$

(end of citation)

Remarks:

1. This concern (taking one of the lengths as a unit length) is a dimensional concern : this way allows to make sense of polynomial expressions like $ab + d$ removing the dimensional "nonsense" of adding (with our terms) square meters to meters...

2. In the case of the product of a negative number by a positive number (that wasn't widely understood - not to say accepted, at the time of Descartes), the intuition that was progressively developed was through a subtraction of areas.