Where I left off in my last post was young children making lovely symmetrically reflective arrangements.
Where I am taking you from there is how to understand, through the lens of symmetry, why multiplying two negative numbers together makes a positive.
Not being able to visualize this has been a chronic thorn in my side.
Then I saw a post by David Wees
He didn’t mention symmetry by name but he did talk about reflection, so, yeah, symmetry.
Above you see reflective symmetry. created by a couple of young children. (Please think of the shades of greens as just green) They reflect across an imaginary line.
Now think of a number line, one which includes negative numbers. It’s the same idea as visualized above. except now it’s the numbers that are reflected. The negative just indicates which side of zero the numbers are situated.
Think of the numbers as defining the distance they are away from zero. For instance, the 4 is four spaces away from zero. The negative four is also four spaces from zero, but the negative sign indicates its position is a reflection of the placement of the positive four.
(Just to let you know, this is not something I am making up on the fly. Mathematicians often look at numbers as their distance away from zero, sometimes referring to this as the number’s magnitude, or its absolute value. This is not something you need to sort out, I just want you to know I’m not making this stuff up.)
In the photograph above -Reflected Shapes- you can see that both deep purple shapes are four units away from either side of the center, right? We could say that the deep purple shape on the right is at the positive four spot and the deep purple shape on left is at the negative four spot.
So what does this have to do with multiplication?
When multiplying, what the negative sign does is that it REFLECTS the quantity across the zero.
Here’s three times two. There’s no negatives. Three times two means two groups of three, so, six.
Now, what happens, visually, when just one of the numbers in our problem is wearing a negative sign?
In this equation, negative three times two, I’m first thinking as three times two because that’s the distance from zero that this equation is expressing. So, again, I get six BUT NOW, because there is a negative sign, that’s my signal to reflect the product (6) over across the zero line. Multiplying by a negative does this: it reflects across the zero line.
Now, what about if there are two negative signs in the multiplication problem?
The same thinking applies. First, multiply the numbers three times two. Because the three is negative, reflect the product over the zero line. Now, because the two is also negative, reflect across the zero line again, which lands you back into positive territory.
To see me reflecting these quantities via youtube, click below:
I shouldn’t gush over my own posts, but this particular one, is one I am so excited by because it’s something I’ve failed to visualize for such a long time.
Many thanks to David Wees.
This is the section from his post that turned the light on for me: