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.
In the TED video Benjamin, in a very entertaining style, performs a number of impressive feats. All numbers are supplied by the audience and Benjamin does all of the calculations entirely in his head:
- He quickly multiplies together two-digit numbers.
- He quickly squares some two-digit numbers.
- He quickly squares some three-digit numbers, slowing down only a little for one of the harder calculations.
- He squares a four-digit number with some mental effort, but all in his head.
- He selects a four-digit number that was one of the squares he computed earlier, 8649. He then asks volunteers from the audience to each multiply that four-digit number by any 3-digit number, unknown to Benjamin. Each of the volunteers is asked to tell Benjamin how many digits are in the product and to tell him all of the digits of the product in any order, leaving out one of the digits. Benjamin is able to determine the missing digit in each case. He's likely using the fact that the digits of 8649 add up to 27, which is a multiple of 9. So, any multiple of 8649 will also be a multiple of 9 and its digits will add up to 9. So, if he is told all of the digits of the product except one he can quickly determine the missing digit by knowing that all of the digits add up to a multiple of 9. You can read about this property of numbers and nines in this Wikipedia article: Casting Out Nines.
- He tells people the day of the week they were born given the month, day, and year.
- For his grand finale, Benjamin squares a five-digit number. To perform this feat mentally Benjamin makes use of a mnemonic device to be able to hold partial products in his head. He converts numbers to words using this device. The number he squares is 57,683. He explains generally how he does the calculation and he talks out loud while he's doing the arithmetic, giving insight into his approach. He makes use of the fact that 57,683 squared is (57,000+683)^2. Then, making use of the fact that (a+b)^2 = a^2+2ab+b^2 he goes on to compute 57,000^2+2(57,000)(683)+683^2. If you watch the video enough times you can glean more information about how does the pieces of this calculation in his head.
I very much enjoyed this video because it gets people to think that mathematics can be fun, interesting, and impressive. To me, mental arithmetic is a great gateway into seeing Math as something accessible to everyone, regardless of their background. This is why I've written so many articles and made some videos about it.