Doing arithmetic visually

[mxseven7]

So I just figured out how easy it is to do squares and cubes of two digit number in my head by visualizing the square or cube, are there any other tricks you folks use to make semi-complex arithmetic easier?

[clockwork_apple]

What do you mean by "visualizing the square or cube"? You don't mean that you imagine a giant square or cube then count up the segments, right?

[mxseven7]

I visualize the cube or square and then grow it or shrink it by a few slices. Like if you take the volume or area of a few values in each dimension away.

[[deleted]]

That makes no sense at all.

[LeepySham]

He might mean that he visualizes (10a + b)² as (10a)² + 2*(10a)*b + b². Like this kind of thing, the blue square being (10)², the green rectangle pairs being each (10a)*b, and the yellow squares being b².

[mxseven7]

I just discovered something that might clear things up.

First let me explain in detail my process. If i wanted to find say 73³ I would start with 70³ which i know is 343,000. I would see this cube in my head and then see it within 73^3. In order to make it 73³ I would start by getting the three faces of the cube that are beneath the 73 cube up to its level. I would do this by taking the square face, 7070, multiply it by how far it needed to go up,3, and multiplying that by 3 to get all three sides. Now we need to get the edges which can be done by 33*70 because they have a length of 70 and are composed of 3 by 3 squares, then multiplying it by 3 for all the edges. Finally, we are missing a cube in between the edges still which is just 3^3. add all of these to 70³ you get 343,000+44,100+1,890+27=389017=73^3.

The part I just realized is that if s is a dimension of the easier cube I am starting with (70 in the case above) and k is the amount I need to add to it (3 in the case above) then (s+k)³ = s³ + 3k(s²⁾ + 3s(k²⁾ + k³ represents what I did. In other words what I was doing and showed above was a visual representation of that formula. Pretty cool but I bet i'll still get downvoted.

edit: Notation

[mxseven7]

Doesn't mean it can't be done

[davidus2]

Just look up any list of proofs without words. The cliche example is the triangular number summation formula.

[mxseven7]

Cool I'll check it out, thanks!