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).
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This article is different from others as it discusses my early experiences with microcomputer programming. The article is, however, very relevant to this Math blog because it hopes to inspire children of all ages to take a very hands-on approach to learning how calculators do arithmetic, and gives suggestions for pursuing a self-directed and very fruitful exploration into the subject. Teaching yourself how computers do arithmetic will give you a lifelong foundation for programming computers or for just understanding how computers work.
I was fortunate enough to grow up in the 1970's when microcomputers were just starting to become popular. As someone with a lifelong love of Math and an interest in computer programming I bought my first computer in the late 1970's. It was an Ohio Scientific Challenger 1P computer. It came standard with 4K of RAM. That's kilobytes, not megabytes or gigabytes. I remember plunking down a few hundred dollars to double the memory to 8K. A fun little exercise is to compare the cost of the extra 4K of memory (let's say it was $300) to the cost of 4 gigabytes of memory for a typical PC today, considering that 4GB is roughly 1 million times larger than 4KB. I say roughly, because 4GB is actually 1024^2 times larger than 4kB.
I know that this blog is doing really well when I'm attracting some very high caliber comments, many of which deserve their own posts to address.
Here are some comments worthy of mention:
1. Efrique made the connection between nomograms and the odd way a parabola can be used to perform multiplication regarding the post A clever use of a parabola to perform multiplication. Efriuqe has, in fact, two helpful comments about the post. I know only a little about nomograms but will learn more and post about them. My sense is that nomograms were forgotten when the slide rule was replaced by the calculator. You can read about nomograms at Wikipedia.
2. Clueless posted a comment to Seven old Wild About Math! posts that I wish had been more popular noting that if you modify the graph of the parabola referenced in item #1 above to y=|x| then the same approach you used with the parabola yields a line that crosses the y axis at a value that is the harmonic mean of the y-values of the two points you drew the line through. Great observation and a great exploration for students. If the harmonic mean is new to you then you can get a background at this Wikpiedia article.
3. Eric has a really interesting way to square large numbers. It's not the approach I am going to document in my followup to How to square large numbers quickly (part 1) but Eric's approach is quite interesting. Can you figure out what is going on in the example Eric provided looking at just his example and not at his other comments? Can you prove why his approach works?
4. The Mad Hatter commented on A fun little counting puzzle to start 2008. He proposes a closed-form formula for the sum of the digits of consecutive integers starting from 1 and going up to (and including) abcd (a 4-digit number). I've not made the time to explore this proposed formula but I'm very curious as to how this was derived. Look at the formula. It's intense! The Mad Hatter is obviously mad about Math!
5. Mathmom took the time to document a solution to A fun little counting puzzle to start 2008 in a great amount of detail.
In a twist on reminiscing on old popular posts from the previous year I present some of my early posts that I thought were particularly interesting but you didn't. I like to believe that's because not many of you knew about this blog when I started it October 6th.
Just think of this post as my personal Carnival of Mathematics, with all submissions coming from my blog. Anyway, check out some of these posts:
1. Have you ever considered how early U.S. area codes were determined? Not the most math-y post but I found the information for that post in a Math encyclopedia. This post should be of interest to curious folks.
2. If you have any interest in dissections and tessellations check out Is it a triangle or a square? I was absolutely fascinated to discover a simple hinged dissection of an equilateral triangle that becomes a square.
3. Anagrams are fun. I found a web-site that will take your name or any other words and permute the letters to form other words. One of the permutations of my name is Male Nerd Sol. How fitting!
4. I did a mini review of Proofs Without Words: Exercises in Visual Thinking in the post titled A picture is worth ... This is one of several books out there that give very visual demonstrations of mathematical theorems.
6. I reviewed Mark Wahl's Math for Humans book. This book more than any has sparked my interest in how to reach students who think and learn differently than how schools in the U.S. traditionally teach.
7. In a clever use of a parabola to perform multiplication I gave an interesting little problem that completely delighted me when I saw it and solved it. This is one of those problems that gets me to wonder how anyone ever came up with it. Brent wrote about this odd problem in his blog, The Math Less Traveled.
I imagine that many of you are familiar with this remarkable mathematical equation that incorporates what are arguably the five most important mathematical constants into one equation. Yes, phi is missing from this equation. I've been reading this fascinating book, Where Mathematics Comes From. Wikipedia has an article about the book. The book seeks to found a cognitive science of mathematics. While the book is very philosophical and abstract in many places, what fascinated me were the very rich metaphors discussed for a number of common mathematical operations and concepts. These metaphors really helped me to see the conceptual basis for some mathematical processes and operations I took for granted and really helped to ground and deepen my understanding.
Some examples of metaphors:
1. Addition and subtraction. Moving to the right or left along the number line
2. Multiplication of a positive number by a positive number. Moving to the right along the number line by a factor.
3. Multiplication by -1. Rotation to the symmetry point of n.
4. A simple fraction (1/n). "Starting at 1, find a distance d such that by moving distance d toward the origin repeatedly n times, you will reach the origin. 1/n is the point-location at distance d from the origin."
5. Exponential function. The mapping of sums onto products. This explains what it means to raise numbers to non-integer and to negative powers.
In 1956 Robert Abbott invented a game he called Eleusis. In 1959 Martin Gardner popularized the game in his Scientific American column. Eleusis is what's called an "induction game" which means that players need to try to induce the rules of the game. Inductive games are compared to deductive games, where players know what the rules are and try to deduce their moves. Yes, I find the terms confusing but, nonetheless, they're worth knowing because if you find yourself enjoying Eleusis you'll know how to refer to that type of game.
Eleusis is played with an ordinary deck of cards. The idea of the game is that one player creates a secret rule and other players have to play cards that satisfy that rule. Ultimately, of course, the goal is to determine what the rule is.
A couple of Carnival things:
1. The Carnival of Homeschooling came out a few days ago with two Math posts:
HappyCampers presents a delightful way to learn mathematics; Here's a fun group Math activity using the shoes on your feet! Happy Campers said, "We did this with our 4 & 5 year olds, but you could modify this activity for older children as well!" - Join in on the fun at the Wednesday CoOp posted at Reese's View Of The World.
Sol Lederman shows us great easy to learn games that kids and adults will really enjoy. All require only paper and pencil, so here's 8 really fun paper and pencil Math games posted at Wild About Math!.
2. The Carnival of Mathematics is coming up at Walking Randomly. Get those submissions in.
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.
Here's what you've all been waiting for, the third episode of Math Girl, fresh from YouTube.
I wrote about "Math Girl" when I first discovered her.
Without further ado, here's Math Girl.
For your convenience, here are episodes 1 and 2. Enjoy.
Math Girl 1 - Differentials Attract
Math Girl 2 -Zero's Dis-Continuity