Mental Math

There are a few math stories that I tell rather often, and this is my favorite of the bunch. Other people seem to enjoy it as well. Given that I’ve never committed the story to writing, I thought that it was time to do so.

During my undergraduate experience, we had an optional two week winter session that allowed for individuals to take simple introductory courses in order to meet various graduation requirements. One particular winter, I signed up to take a psychology 101 course, and the second day of the class we learned about behaviorism and conditioning. I started thinking about various ways that I had conditioned myself, and I realized that there was a single recurring thought that oftentimes pass through my mind with no discernible pattern or regularity. The thought started when I was around 17 years of age, and at the time of this story I think I was 22. It didn’t matter if I was brushing my teeth, out with friends, driving my car, etc. It didn’t matter if I was daydreaming or having a serious conversation. The thought was this:

“Nineteen squared is three hundred sixty one.”

Now, I have no recollection of learning this fact. I never memorized it, and I cannot think of any practical reason that this thought would stick so strongly in my brain somewhere (I learned later that a full size Go board was 19×19, alas). But the thought came to me nonetheless.

During my senior year of undergraduate work, I had a number theory class, where the professor wouldn’t allow us to use calculators for arithmetic problems we would work through. One day, in class, we were working on a rather extensive arithmetic calculation when the professor suddenly hesitated at the blackboard. He turned to us, and asked, “Does anyone know what nineteen squared is?”

Well, I instantaneously answered, “Three hundred sixty one.” Not even one second later. Everyone in the class sort of turned and looked at me with expressions that said something like, “Who in the world keeps track of the square of nineteen?” or “How did he calculate that so fast?”

It was as if the whole of my life had led up to that one moment. If was as if the math gods had prepared me for this solitary moment, where the haunting thought that appeared for no reason whatsoever would ACTUALLY be useful.

Needless to say, when class ended, I was slightly afraid to leave the room. Given that I had obviously fulfilled my purpose in life, I felt that an anvil would imminently fall from the sky and put an end to my misery. But to my great benefit, no anvil fell. And ever since that day, the recurring thought has left my mind. Well, except when I tell this story.

(Photo by rexhammock)

I must confess that I’ve never learned to use an abacus (or a slide rule, for that matter). I came across the following video, and thought that it would act as another great view into the wonderful world of mental math. It’s quite tremendous what the human brain is capable of. Check it out:

This is perhaps one of the most amazing videos I’ve every watched on the Internet. I was literally left speechless. I can accomplish certain small feats of mental math, but this is absolutely unbelievable. Arthur Benjamin shows us some inspiring abilities of the human mind. I’m sure he has spent a fair amount of time learning his methods, and that to him his abilities are perfectly normal (in some sense!), but it’s great to watch someone with this talent. I highly recommend watching the video in its entirety.

Is it possible to teach a horse how to do math? Around the year 1900 it looked like the answer was a resounding yes. My friend Trevor suggested that I write something up about Clever Hans, who was a horse capable of performing mathematical feats on par with a young human teenager. Hans could add, subtract, multiply, divide, work with fractions, differentiate musical tones, and understand the German language. At least, Hans could apparently do all of these things. Several other pages have described the Clever Hans case in great detail, so I’ll refer you to them. Please check out one or more of the following links to learn about an interesting phenomenon that has less to do with math and more to do with psychology:

• Damn Interesting Link
• Wikipedia Link
• Skeptic’s Dictionary Link

A mental math installment…

While walking to Carnegie Mellon the other day I started thinking about squaring two digit numbers. So instead of hoarding my findings and short amount of research I thought I would write a tutorial on how to square two digit numbers, and then some.

Method 1: If You Know the Previous Square

This method is only marginally helpful, but will come in handy if you know how to easily square the number previous to the number you’re trying to square. Let’s say you’re trying to find x². If you know what (x-1)² is already, all you have to do is add x and (x-1) to (x-1)² to find x².
Proof:

(x-1)² + x + (x-1) = (x² - 2x + 1) + x + x - 1 =
(x² - 2x + 1) + 2x - 1 =
x² + (2x – 2x) + (1 – 1) =
x²

q.e.d.

Example:

Let’s find 31² knowing that 30² = 900.

31² = 30² + 31 + 30 = 900 + 31 + 30 = 961.

This method is mostly helpful for squaring numbers which are one more than a multiple of ten, since humans can square multiples of ten without much thought (more on this later). Also, you’re not restricted to squaring two digit numbers with this method, which is quite fantastic.

Method 2: Multiplying One Up, One Down, and then Adding One

In my opinion this method is a bit more fun, but it definitely requires some mental multiplication. In fact, you’ll almost be doing as much work (or more) using this method as you would be in outrightly squaring a number, but it is quite an amusing trick.

Let’s say you want to square a number x. If you multiply (x-1) and (x+1) together, and then add 1, you’ll find x².

Proof:

(x-1)(x+1) + 1 =
(x² + x - x – 1) + 1 =
x² + (x – x) + (1 – 1) =
x²

q.e.d.

Example:

Let’s find 31² again using this trick.

31² = (30)(32) + 1 = 960 + 1 = 961.

Like I said, if you can’t quickly calculate that (30)(32) is 960, then this method isn’t saving you much mental energy, but this averaging method could be a shortcut for some.

Method 3: Squaring Two Easy Numbers Instead of One Hard Number (Plus a Step)

In my opinion, this method is best practice. This explanation will be a bit lengthier, and also a bit harder to write out in plain English, but it’s the most fruitful by far of the three methods.

Like I said in my previous post on mental math, perhaps the most extensive repository most folks have for math is their times tables. With some exception, people are able by their mid-teens to multiply together two one digit numbers with relative ease, all the way up to (9)(9) = 81. Let’s make use of this fact, plus the fact that folks can easily multiply multiples of ten, to square any two digit number easily.

First, one needs to realize that any two digit number is at most five numbers away from a multiple of ten. For instance, 34 is 4 numbers away from 30, and 65 is 5 numbers away from both 60 and 70. So when we’re thinking of squaring large two digit numbers it’s best to think of it as a multiple of ten (e.g. 10, 20, 30, 40, …) plus or minus a number no greater than five.

So when we go to square a number like 74 mentally (yikes!), it’s better for one to imagine this number not as 74 but as (70 + 4), or to imagine the number 66 not as a single number, but as the difference between two, i.e. (70 – 4). It’s a matter of breaking down a difficult single process into several easier ones.

So let’s find 74². Instead of looking at the single number 74, let’s break it down into (70 + 4). Now let’s square this number.

(70 + 4)² =
(70 + 4)(70 + 4)

At this point in the game we’re multiplying two binomials together. Remember the FOIL (Front, Outer, Inner, Last) method from way back when? I thought so. Continuing with this method we have

70² + (70)(4) + (4)(70) + 4²
4900 + 280 + 280 + 16 (a)

We’ve broken the process down into more or less multiplying single digit numbers together and then adding zeroes at the end. We know 7² = 49, (7)(4) = (4)(7) = 28, and 4² = 16, and it’s then easy to multiply these numbers by powers of ten.

Adding these numbers up we get 5476. We’ve taken a difficult multiplication problem and turned it into an easier addition problem (though some would beg to differ, I’m sure). It works the same for a number like 66, which looks like (70 – 4) when we break it down. The only difference is that we’ve traded our plus sign for a negative one. The only part of the arithmetic that changes is that instead of adding 280 twice to expression (a) above, we subtract it twice. So similarly to 74² we have that

66² =
4900 - 280 - 280 + 16 =
4356

And this method, in my opinion, is by far the easiest way to mentally square two digit numbers. A similar process will work with three digit numbers. Perhaps I’ll write on that later.

There you go. Hopefully the next time you need to square a number quickly you’ll be more equipped for usual. So for now, happy squaring, and let me know if anyone has any suggestions or additions.

Note: For all you nitpickers out there let it be known that when I’ve used the word “number” at any point in this post I’ve actually meant this word to mean “positive integer”.

Find a great article on math lessons as well as online mathematics programs at Ericae.net We also have resources on a variety of education related subjects such as scuba diving schools and dance lessons.

As many of you know part of my work as a student was and is in mathematics. So you shouldn’t be surprised to learn that I’ll be posting some thoughts in this space every now and again of a mathematical nature. Here’s my introduction on my mathematical posts…

I love the calculator. I love the idea that a machine can perform computations faster than I can. Calculators (or computers if you will) vastly outdo humans in both speed and accuracy when it comes to computational ability, just as humans outdo computers when it comes to other varieties of tasks. In fact, as an aside, much of the aim of artificial intelligence is to teach a computer how to perform actions that humans have mastered, but at which computers are currently mediocre. One common example is the task of face recognition in human beings, which in part is being developed as a security measure and in part is being developed because why the hell not?

And in case you were wondering, the origin of our modern use of the word “computer” comes from our historic human “computers”, or those men and women who were employed to compute various tedious and repetitive calculations by hand. Those were in the days when there were such things as slide rules and logarithmic tables. Those days are now over. This is at least what I’ve been told.

I remember taking a course in college called Linear Algebra, which is concerned mostly with giving its learners near death levels of frustration, but which is also concerned here and again with matrices. I remember spending over an hour performing some stupid computation by hand. It was a homework problem and we had to show all our work. When I checked my answer via computer, I was supposing that it would take the computer program at least a minute or so to answer. Nope. Its answer was instantaneous, and its answer, oddly enough, was correct.

Having said all this, though, I regret to see that human capacity for mental math has decreased in these present times. In fact, the greatest feat of mental math that many of us perform with any sort of frequency is related to our simple times tables. And generally as adults we don’t “perform” any mental math when we use them because we memorized the facts years ago. Cognitively we generally don’t mull over how or why 6 x 5 = 30. We learned the mechanics of how simple multiplication worked in grade school, so we take it for granted that 6 x 5 = 30. Indeed.

It amuses me that many of our cell phones come with a built in “tip calculator”, allowing us to quickly generate what 15% of $23.15 is. What amuses me even more is how often I actually see people using this calculator. And in case you were wondering, the answer rounded to the nearest cent is $3.47. And no, I did not use a calculator to figure that out. 10% of $23.15 is $2.315 so to speak and half of that (5%) is $1.1575. Add those together to get $3.4725, which is 15%. Round to $3.47.

I don’t support mental math because of the math part of it. I support it for the puzzle part of it. Oftentimes tremendous ideas come out of puzzles, and I espouse mental math for this art of thought. More on this idea at a later date.

I realize that I’m in the vast minority of people who walk around this earth actually thinking, for fun, about numbers. And I realize that this fact is a strange and scary fact for many, if not all, of you. What I want to try to do in my mathematical posts is try to give you all a glimpse of something beautiful in math. It’s the beauty that drove me to it in the past, and this same beauty is driving me back to a type of math (applied logic) for graduate school at Carnegie Mellon University. Here and there I’ll try to provide a helpful tip as well.