Brian Foley runs a web-site, Math Mojo and a blog, The Math Mojo Chronicles. The web-site aims to make, in Brian's words, "Math meaningful." While I enjoy the site, I struggled to explain what Math Mojo was about until I found this description in the What is Math Mojo page:
Math Mojo is a way of looking at math that fosters a sense for numbers. The more new ways you learn and practice, the more of a feel you will get for manipulating and understanding how numbers work. They will become less of a mystery, and you will feel better about your ability to do math. These methods are based on several different speed-math techniques. They all work at least as well as the methods that you were taught in school. In fact, schools that teach these methods do much better than the national average.
Last November I wrote about the Trachtenberg system of speed mathematics and showed one of the techniques - multiplying an arbitrarily large number by 12 with great ease. In this article I want to show you why the technique for multiplying by 12 works, share two more of the Trachtenberg multiplication techniques, and give you some direction as to how you can develop your own multiplication techniques.
In that first article about Trachtenberg multiplication I taught you to multiply by 12 by "doubling the digit and adding its neighbor (on the right)." I gave an example of multiplying 346 by 12:
Last month I blogged about mathemagician Arthur Benjamin and his amazing mental Math feats. Benjamin is a master of doing arithmetic in his head with lots of digits involved. In particular, he's able to square a 5-digit number without writing down partial results. How does he do it? I picked up a copy of Benjamin's Secrets of Mental Math to learn how. Here are the steps Benjamin provides for squaring 46,792.
1. First, Benjamin breaks the number into 46,000 + 792.
2. Then he does a little algebra. If a=46,000 and b=792, then (a+b)^2 = a^2 + 2ab + b^2 = (46,000)^2 + 2(46,000)(792) + 792^2.
3. This simplifies a bit to 1,000,000(46^2) + 2,000(46)(792) + 792^2.
4. Then Benjamin sets out to do the middle product: 2,000(46)(792).
There's an interesting web-site, brainetics.com, that is all about doing mental Math quickly. I have to confess that while I know quite a few mental Math tricks and while I've written quite a number of posts and made several videos about mental Math tricks I'm not particularly fast at applying these tricks. Doing Math quickly in one's head is all about knowing techniques, having a strong memory and maintaining focus. I know techniques. Memory and focus are currently a challenge for me.
Brainetics sells a $180 product geared to improving mental Math abilities. I'm not rushing to spend $180 to see how helpful Brainetics might be but I'd love feedback from anyone who has used the product.
There are numerous introductions to Vedic mathematics on the web. I won't be doing a general introduction to Vedic Math now. In this article I want to explore one particular Vedic mathematics technique, using something called bases, to optimize certain multiplication problems.
This technique is extremely powerful and it takes getting used to. It's partially a cookie cutter technique but there's also some thinking involved in selecting proper bases for performing multiplication. Don't get frustrated if you can't understand this technique in one reading. It took me a fair amount of focused attention and practice to understand and appreciate the power of this approach. If there's enough interest I'll produce some videos explaining this Vedic technique.
In case it's not enough that young chimps have better numerical memory than adult humans it now seems that monkeys can perhaps impress their friends with mental Math tricks. A new study shows that monkeys can do mental Math. The article states:
"Rhesus macaque monkeys performed nearly as well as college students at quick mental addition, researchers reported Monday, adding to the evidence that non-verbal math skills are not unique to humans."
The competition involved two sets of dots that flashed briefly onto a computer screen. The humans and the monkeys had to mentally determine the sum of the two sets of dots and pick the right answer on a different screen.
The humans won this time. Whew!
Check out this amazing video of Mental mathemagician Arthur Benjamin performing calculation feats in front of an audience.
The video comes from the TED.com blog. TED is an organization dedicated to changing the world by spreading important ideas. TED, which stands for "Technology, Entertainment, Design", but whose scope is much broader than when it was founded in 1984, makes their best talks and performances available for free via the Internet. These videos are made during the annual TED conferences which TED describes as follows in its about page:
The annual conference now brings together the world's most fascinating thinkers and doers, who are challenged to give the talk of their lives (in 18 minutes).
Benjamin is a Math professor at Harvey Mudd College, a college very well known for its mathematical talent among both professors and students. He is also coauthor of Secrets of Mental Math, which I've not yet read, where he apparently reveals many of his mental Math techniques.
You Tube also has a number of highly rated video clips of Benjamin.
Here's a video on how to quickly square a 2-digit number. The technique is based on this algebra:
If you have a number with digits "ab" then the number is 10a+b.
(10a+b)^2 = 100a^2+20ab+b^2.
If you enjoy this video check out all of the Wild About Math! mathcasts.
I can't prove this but I bet that doing lots of mental Math can help slow down aging. In studying a number of techniques for doing fast arithmetic I notice a number of mental skills at play:
- Pattern recognition. Many tricks involve recognizing and exploiting a pattern.
- Visualization of processes. Multiplying large numbers together requires you to hold a mental image of multiple steps.
- Alertness. Falling asleep will cut way down on your efficiency in every technique!
- Mental speed. Impressing your friends will require you to develop speed in your technique.
- Memory. Efficient arithmeticians memorize more Math facts than others. Some people, for example, find memorizing the squares of all numbers up to 25 to be helpful in certain techniques. Also, you often need to maintain a running total in your head and keep track of carries..
- Concentration. When multiple steps need to happen in a sequence your powers of concentration will improve.
A friend and I were discussing how mental Math could help maintain the plasticity of the brain, its ability to reorganize itself in response to new information. She thinks that doing a number of cross-multiplications every day can keep the brain active as we age. I agree.
A few people were underwhelmed with my video on how to multiply together a pair of 2-digit numbers without writing down partial results. They didn't see a time savings. Fair enough, although not needing to write anything but the three (or four) digits of the answer can be a nice time savings and a source of smugness for many.
Next week I'm going to produce some more mathcasts. The first one will show how to multiply together a pair of 3-digit numbers using the cross multiplication approach. I think that will impress more people.
There's a general approach to these multiplications I want to teach you. If you understand the algebra behind this approach you can derive the steps for multiplying together numbers containing any number of digits. However, I'll warn you that beyond 5 digits this approach, and any other approach that works for any numbers and not just special cases, will really work your mental muscles.