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.

Why’s this important?

Seeing arithmetic as a type of transformation prepares you to make sense of seemingly-weird concepts, like the square root of -1, and visualize problems in a new way. Let’s take a look.


Addition is simple, yet it can mean several things:

  • Accumulate: Count up similar quantities (often for tangible items).
  • Slide: Shifting a number along a scale (for less tangible things, like temperature).
  • Combine: Make a new quantity out of two different ones (like notes in a music chord).

Addition viewed as accumulate, slide or combine

What’s the right meaning? It depends on the context. When adding apples, we count up similar items (3 apples + 4 apples = 7 apples). When measuring temperature, we add heat to move along a scale (3 degrees + 4 degrees = 7 degrees).

When adding vectors, a combination makes the most sense: 3 blocks east + 4 blocks north = 5 blocks in a new direction (“as the crow flies” distance). In this case, you must track the component parts, keeping “North” and “East” separate. When adding apples, you can combine everything: once you have 7, you don’t care that it was once 3 and 4.

A single operation (addition) can take on several intuitive meanings. This list isn’t exhaustive — they are the interactions I’ve noticed, and I’m sure you have others.


Multiplication can also be interpreted in several ways:

  • Repetition: Performing multiple additions.
  • Scaling: Making a number grow or shrink all at once.

Multiplication viewed as repetition or scaling

Context determines our meaning. With apples, “4x” means turning an order of 2 apples into an order of 8 (4 groups of 2). With photo software, “4x” means expanding a 2-inch photo to be 8 inches long. Each meaning is different: you’d be annoyed if I gave you a giant, 5-lb apple or 4 separate photos.

In a narrow sense, multiplication is “repeated addition”. Sure. But that’s not always the easiest interpretation — care to “repeatedly add” 7.3 times?

Negatives and Inverses

Negatives and inverses both represent the idea of “reverse” or “opposite”. But that’s ambiguous: What’s the opposite of multiplying by two?

Negative and inverse multiplication

“Opposite” can mean a few things:

  • Multiply by 1/2: Turn a profit of 1 into a profit of 1/2 (“unscale” it)
  • Multiply by -2: Turn a profit of 1 into a loss of 2 (flip it)

Yet again, our context determines meaning. When a company “reverses a gain” it implies a loss, aka multiplying by a negative. When we “reverse a zoom” in photo software, we want to shrink the photo (not mirror-image it), so we multiply by 1/2. Context, context, context (tired of that word yet?).

When adding, there’s only one type of opposite: the reverse of +8 is -8. But the trick is to know that -8 really means “0-8″ or even “0 + (-8)”: you’re moving backwards relative to some reference point. Moving “in reverse” means different things depending if your original direction was East or North.

What’s in an equation?

Equations ask questions. When you see

\displaystyle{x^2 = 9}

It’s more than just a plug and chug problem. Think about the question like this:

\displaystyle{1 * x^2 = 9}

What transformation (“times x”), when applied twice, will turn 1 into 9?

Visual diagram of squares, 3 and -3

We have two answers:

  • Scale by 3 (times 3): Do it twice and you’ll get 9: 1 * 3 * 3 = 9
  • Scale by 3 and flip (times -3): Done twice, you get 9 also: 1 * -3 * -3 = 9

Nifty. I included “1″ to show what is being transformed. Sure, it’s optional, but it’s not something we think about. What is the “times 3″ acting on?

Stepping back this way, we can see arithmetic as a method to push, pull, tug and squeeze one number into another. We’ve managed to turn one large transformation (“times 9″) into two equal, smaller ones (“times -3″ or “times 3″).

Real-World Example: Random Numbers

Enough theory — let’s show this mindset in action. Most programming languages offer a random() function that gives a number from 0 to 1. But what if you want something from 5-10?

The question is: how do I transform my range of 0-1 into a range from 5-10?

Diagram of scaling random numbers to new ranges

Arithmetic to the rescue!

  • First, you stretch 0-1 into 0-5 by multiplying by 5
  • Next, slide 0-5 to 5-10 by adding 5
  • And tada. You have a range from 5-10.

Try it out below. You start with a number “r” and transform it into the proper range.

By the way, this range could be the ages 18-65, the years 1960-2007, or the temperatures 30F – 80F for use in your simulation (everyone runs simulations, right?). No matter your range, you can start with the “0 to 1″ building block and modify it.

Sure, you can figure this out without a diagram, but sometimes it’s nice to visualize what’s happening. Our brain is a vision-processing supercomputer, so let’s use its strengths.

What’s next?

This post introduces the idea that arithmetic is a transformation. You bend numbers into other ones, and each transformation has a meaning. Some fit a situation better than others: use the one you like most.

The goal isn’t to turn multiplication into a cumbersome diagramming process. It’s a technique, a mindset, a new weapon to use against seemingly complex operations.

When studying linear algebra (matrices), you can view multiplication as a type of transformation (scaling, rotating, skewing), instead of a bunch of operations that change a matrix around. This approach will help when we cover imaginary numbers, that foul beast which has befuddled many students.

Little insights help bigger ideas click. Happy math.

Other Posts In This Series

  1. Rethinking Arithmetic: A Visual Guide
  2. Learning How to Count (Avoiding The Fencepost Problem)
  3. Techniques for Adding the Numbers 1 to 100
  4. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  5. A Quirky Introduction To Number Systems
  6. Another Look at Prime Numbers
  7. Fun With Modular Arithmetic

Questions & Contributions


  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.


  6. 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”

  7. @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.

  8. 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.)

  9. 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.

  10. 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.

  11. 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?

  12. @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.

  13. Wouldn’t it make more sense to use lines than blocks? A block would be a number with a width value a height value, and a starting coordinate (x and y values minimum) to have four values in all. A line only has two values, being a starting coordinate (only x minimum) and a magnitude. A point itself is a value (or equal to multiplying times 1 or adding by 0). Your random number would be a line also, but one that acts differently than a transformation line in that you can transform to either the magnitude of the random number or the location of the random number.

  14. Here’s a fighting aspect to it: throwing a combination of punches like 1-2-3 is really repeating a natural rhythm of punches, which gives the opponent three times the rhythm information and three times the chances of counterattack. Using this as a starting point can help explain why there is a point of diminishing returns for the effectiveness of long combinations, aside from just the muscle deterioration or energy reserve depletion theories. (mostly, because this holds true in video games as well, even video games that ignore fatigue!)

  15. You’re splitting , my head, I know I should probably put this on that page but you’re view of calculus doesn’t really help me (i only read it once though) I’ll keep reading to see if I can make something click! :) Awesome site, it helped a tiny bit so far :/

  16. Hey man nice article!
    I never thought that you could teach a concept like “transformation” with arithmetic. I never learned about arithmetic until upper div. math in college (linear algebra)…
    Yeah, I’m studying to become a math teacher, and I’ve read plenty of articles on how most of how math is taught in America (at least) doesn’t actually teach much. Students (like myself) aren’t learning and are turned off from math…

    I appreciate the work you’re doing with this site

  17. @Seamus: Thanks for the note — sorry the calculus article didn’t help, my view is being refined over time, and there may be even better analogies down the line.

    @Frank: I appreciate the comment! Yes, unfortunately a lot of math education is about memorization and moving on, vs. really getting the concepts we’re talking about. It’s really awesome that you’re going into the profession with a desire to change it :).

  18. i love math, I want to learn every lessons in math, someday i want to be a good student for the teacher in college… “success lies beyond sacrifices”

  19. Addition and multiplication are way to combine two numbers. Also, they’re commutative.

    Threrefore, I think the perfect visualization would start with both numbers lying on the line, and them flipping ans spinning and tugging at each other until they settle into the new configuration.

    I’m sure you know how to visualize the commutativity of multiplication of positive integers using rectangles and rotation. That’s the kind of stuff that I have in mind.

    So I’m not completely satisfied with the visualizations in this post. I’m gooing to have to keep looking for new ways.

  20. these are really good things…keep them running with jumps to higher level…and please please keep them free…u’ll earn your name only that way…many of us hackers ..or real mathematics insight seekers don’t want to buy a book…they prefer to read for free…support open and free source movement from heart…THANKS A LOT FOR THESE…I’LL PASS THESE ON TO SCHOOLS I KNOW…

  21. Thanks for the good program. I sent in a question on the 1st Oct 12 and since then have been looking for real basic stuff on convention and stuff like that as that’s my only stumbling block, to be able to understand the words, and now I have found rethinking arithmatic: a visual guide and that’s what I have been looking for, something along the lines of basic arithmetic. So hopefully I will gradually find such things as basic as realising asterisks have the meaning of multification, etc. Without going any further afield. Thanks.

  22. Hi,
    Thanks for saving my life , by the way can you update this post with different meaning of division please i cant really get it.

  23. Hi Satheesh, thanks for the note. I try to see division as just a type of multiplication. I.e., instead of “divide by 3” think “multiply by 1/3”. Then, division just becomes a different type of scaling [which will normally make you smaller… but if you divide by 1/2, you make yourself larger].

  24. Sateesh, I love the explanations that you and the other commentators have offered. I used many of these when I was teaching in the inner city, where a significant minority of our students used fingerplay for addition and subtraction, and even for multiplication (they just left division questions blank).

    I am blogging for a site whose philosophy is that it you kick math facts out of the prefrontal cortex and down to the automated recall areas of the brain like the parietal sulcus, the higher brain is available for conceptual work. At, you’ll find an article with hot links to the brain science articles. I’m curious what you think of a confluence of conceptual and automated thinking. I think it’s like strategic planning meeting operations in a company.

  25. Did you really mean in the email you sent, that “The opposite of the opposite of 1 is 1”? Or did you mean “The opposite of the opposite of -1 is 1”. Wasn’t sure about that part.

  26. Hi Brett, great question — yep, it was meant to be 1.

    “The opposite of 1” is -1. And if we take the opposite again (“the opposite of the opposite”) we get the original back. Hope that helps!

  27. Really like your weekly emails. I am doing some work with R and I was curious to predict how many distance results would the dist() function produce. I started by summing: (n-1) + (n-2) + (n-3) … (n- (n-1)). Then using the insights presented in newsletter: “A quick intro to intuitive learning” I arrived at: (n-1) * n / 2 gives the answer. the dist() function compare each element with every other element but removes the duplicates. For a set of 4 {1,2,3,4} there would be 6 distances generated.
    > a
    [1] 1 2 3 4
    > dist(a)
    1 2 3
    2 1
    3 2 1
    4 3 2 1
    > length(dist(a))
    [1] 6 ## Result from R

    Using my new formula: 4 * (4-1) / 2 = 6. ## It works

    Now, for 40 elements:
    > b = 1:40 # create a set of 40 elements 1 to 40
    > length(dist(b))
    [1] 780 ## Result from R

    Using my new formula: 40 * (40 – 1) /2 = 780 ## Works again!

    This really great after just reading the first news letter… so thanks and keep them coming!!!

  28. Dear Khalid,

    I am 45. I am an engineer,
    But I never really understood some basic concepts in maths, like multiplication of negative number by a negative number. I used to memorize many concepts, which ultimately have been internalized. But there was always a kind of vacuum, a kind of unease.
    Your ideas and explanations are great. It really opened a different perspective not only in understanding maths, but many other things in life.
    I feel the site on calculus needs some rework.

    Keep it up Khalid keep it up.

  29. I’m a very basic guy and I ask myself why on earth would I want to transform a random number in the range 0 -1 to the range 5 – 10. Without any concreteness this is meaningless.

    Love the rest of this article though.

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