Rethinking Arithmetic: A Visual Guide

Arithmetic gives us tools to smoosh, slide and stretch numbers. These transformations are handy: sometimes there’s things in the real world we want to smoosh, slide and stretch in the same way.

This is a companion discussion topic for the original entry at

The vector addition example for simple addition isn’t the best, IMHO, because to really add vectors like that you have to use squares and square roots. It would be probably better to just add scalar quantities.

Moreover, since you use blocks you can’t just add vectors because to walk blocks you use Manhattan distance and not cartesian distance, so 3 + 4 blocks is actually 7, and not 5.

Hi eliben, thanks for the comment. Yeah, the goal of the vector example wasn’t the distance metric, more the idea that you need to keep track of the parts that went in.

When adding apples, you have “3 + 4 = 7” and don’t really care that it was 3 and 4 before (2 + 5 gives the same result). With vectors, you get (3,4) and need to track each dimension separately (2 and 5 are very different). I’ll see if I can make this more clear.

Since this is a visual guide, I have a comment on the illustrations.

In the negative/inverse diagrams, the labels (x-2 and x1/2) are rather confusing. I originally interpreted “x-2” as “some quantity, X, minus two” rather than “multiply by negative two.” This is partly due to the spacing between characters, but it’s mostly caused by the ambiguity in the meaning of “x”, particularly when combined with “-”. The “scale by -3” diagram also has this problem, but I understood the correct meaning much faster because I had already figured out the previous diagram.

Hi Barb, thanks for the comment! Finding and fixing parts that aren’t clear is one of the great things about a blog.

I redrew the diagrams to use terms like “3x”, which I think is a more natural way to say “3 times” (on camera zooms, etc.). Hope this helps.

I’m a programmer - well, actually more of a ‘scripter’. I don’t recall how I originally ran across your site, but I’ve really been enjoying each post. I don’t deal with math on a level most others viewing this blog probably do - but I really enjoy the thought-provoking nature of the content you provide. Keep it up!

I really like the diagrams you use to illustrate your points, what are you using to generate these? I’d love to know so that I may do similar things in my own posts.


good post!!! I think what I got out of it the most was the idea of ‘context’. Having high school kids, it is easy to fall into the ‘plug and chug’ syndrome. thanks again “Professor Azad”

@Jed: Thanks for the note, glad you’ve found the site helpful. I try to do about 1 brain-bending post a week but hope the depth balances the frequency. I’ll keep cranking :).

I use PowerPoint 2007 to make the diagrams. The world needs more visuals, I’m happy to hear you’ll be adding to them!

@T: Thanks Mr. Rose! Yes, I find context is one of those “assumptions” that’s made but it’s nice to state explicitly.

I often wonder whether it’s better to teach plug-and-chug first to get the basics, then layer in understanding, and revisit plug-and-chug. Often times though, math teachers don’t get past the plug-and-chug step and go onto the next topic :). Glad you liked it.

I like your visualization of the Random Numbers example, although I think my brain would prefer to slide first and scale afterwards, because it’s a lot easier for me to visualize moving to the start point first and then stretching towards the end point. (Also, it’s easier to carry a small 1 rather than a big, scaled 5 that far! Hehe.)

Hi Travholt, thanks for the comment! Yes, sliding then stretching is possible but has a few subtleties:

Step 1: Slide over 5, so you have r + 5 [the random number range is 5-6].

Step 2: Now we want to stretch our current endpoint (6) to the max of 10. Basically, we want to tack on another “random” range 0-4 to the end. 0-4 is really 4r.

Step 3: Add it together: (r + 5) + 4r. This is the same as 5r + 5, just stated differently.

This gives the same result, but is a different way of thinking about it.

The problem with raw scaling is that it moves the left side as well as the right. So it works best when your left side is at zero.

That’s a great point though, I may extend the example above to include this also.

I like the way you use graphics to explain concepts. Very good post Kalid.
Keep them coming!

Thanks – I’ll keep cranking them out :).

Hm, I think I’m not communicating my point very well. Let me try again. The only thing I was trying to say was how my mind would attack this problem.

“I need to pick a random year between 5 and 10. So I’ll start with 5 and add a random number.”

5 + r

“My random number generator produces a number between 0 and 1, and my desired interval is 10 - 5 = 5, so I’ll multiply the random number by 5.”

5 + r * 5

So I end up with the same formula as you did (and not (r+5)+4r), but with the addends reversed, because this fits better with how my brain attacks the problem.

[…] Using visual diagrams, not just text, to understand the idea. […]

Nice Job Khalid. Started to read about Permutations - Ended reading for more than 3 hours. All articles are nicely put.
BTW what program do you use for drawing your images?

@Travholt: Thanks for the clarification, I think I overcomplicated it! (That’s 3 different ways to think about it). Sliding to 5 and then “tacking on” a random number from 0-5 works also. It’s fun seeing how different people approach the same problem.

@Ashok: Thanks, glad you found it useful! I use PowerPoint 2007 to make the diagrams.

[…] We’ve seen that regular addition can be thought of as “sliding” by a number. Addition with complex numbers is similar, but we can slide in two dimensions (real or imaginary). For example: […]

[…] Rethinking Arithmetic: A Visual Guide | BetterExplained (tags: toread math learning programming reference visual guide ** mathematics) […]

[…] Scale or stretch a number […]

[…] To most of us, it’s “the number in the middle” or a number that is “balanced”. I’m a fan of taking multiple viewpoints, so here’s another interpretation of the average: […]