I've added a brand new section to the Mental Gym today!

It's called Mental Division: Decimal Accuracy, and deals with converting division problems or fractions into their decimal equivalents as precisely as possible, without using a calculator.

The new tutorial starts by making sure you know how to divide by numbers from 2 through 11 by heart. Many fractions and division problems can be reduced to these problems, and become easier when you know the answers off the top of your head.

After that, I jump up to showing how to divide by several numbers near 100, including 100, 99, 98, 91, and 90. The answers are deceptively simple to work out.

Finally, the tutorial wraps up with showing how to cover a wider range of numbers, using just these strategies. For example, you'll learn how to handle dividing by 13 by turning it into division by 91. When you can't apply any existing division patterns, I even discuss estimating the decimal equivalent.

I wrote this tutorial because I've seen very few mental math tutorials that cover division with decimal accuracy, and those that do almost never cover numbers near 100.

Knowing how to convert a fraction to a decimal equivalent also works well with my recent tutorial on Estimating Square Roots. For example, let's say someone asks for the square root of 581.

Using the procedure taught in that article, you'd estimate that the square root of 581 is about 24 and 5/49ths. Using the new tutorial's conversion to decimal accuracy, you'd realize that 5/49ths is the same as 10/98ths, and so you can give the decimal equivalent of 24 and 5/49ths as 24.1020408163265306 and as much farther as you'd care to go!

The nice thing about the decimal accuracy addition to the feat estimating square roots is that you only have to use it when you can work it out. Even if you find you can't give an exact decimal, you simply leave them with an incredibly accurate fraction.

I'm open to adding more decimal conversion methods to this tutorial, so if you have any tricks for numbers not already covered there, let me know about them!

Lately, some excellent free memory courses have been showing up on YouTube.

The price is right, and the techniques are quite useful, so in today's post, I'll show you where to find these free video memory courses.

The first course is about one of the toughest, yet most useful, things to memorize - numbers! AEMind's YouTube account features videos on many aspects of memory, mostly geared for memory competitions. Here's his first video on memorizing numbers:



The links to the whole series (as of this writing, anyway) are below:

How to Memorize Numbers Part 1
How to Memorize Numbers Part 2 | Pictures 0-9
How to Memorize Numbers Part 3 | Pictures 10-19
How to Memorize Numbers Part 4 | Pictures 20-29
How to Memorize Numbers Part 5 | Pictures 30-39
How to Memorize Numbers Part 6 | Pictures 40-49
How to Memorize Numbers Part 7 | Pictures 50-59
How to Memorize Numbers Part 8 | Pictures 60-69

You'll also be able to keep up with any newer videos in the series via my YouTube playlist, Memory Technique 5: Phonetic Peg (Major) System.

As I write this, AEMind is also beginning a series on remembering names and faces. This, of course, is one of the most useful, and most-requested, memory techniques. Below is the introductory video of the names and faces series, and you can keep up with future releases through my Memory Feat: Names and Faces YouTube playlist.



Our next course comes from Kerin Gedge from New Zealand. Kerin's course teaches how to memorize long texts word-for-word. He teaches it from the standpoint of memorizing Bible scripture, but the techniques are equally useful for memorizing scripts, music lyrics, speeches, and more. Below is the introductory video of the series:



Here's the complete list of Kerin's text-memorization videos:

How To Memorize: An Introduction
How To Memorize Part 2: The Verbal Technique
How To Memorize Part 3: More on the verbal technique
How To Memorize Part 4: Fusing the Verbal Techniques
How To Memorize Part 5: The Handwriting Technique
How To Memorize Part 6: The Typing Technique
How To Memorize Part 7: Fusing all the techniques
How To Memorize: Closing Statements
How To Memorize: Bonus Video

My Memory Techniques: Misc. Techniques playlist contains these and several other memory techniques that are not often covered.

If you're especially interested in memorizing long texts word-for-word, you also might want to check out my free web app, Vertbatim 2.

Playing around with the math and graphics inGeogebra, the computer algebra system I mentioned in my previous post, I was reminded of a series of educational programs I used to watch on public television long ago.

I couldn't even remember the name, but a little digging eventually turned it up. It was called Project Mathematics!, and produced by Caltech from 1988 to 2000. I've found some of the episodes online, and will share them with you.

According to the Project Mathematics! homepage, there were 9 episodes total, plus a teacher's workshop tape. The teacher's workshop tape included brief segments of each of the episodes to give a general idea. You may have seen the Pi segment of the teacher's workshop tape on YouTube:



That's only a short clip (amusingly, it's about 3:14 long). The full Story of Pi episode is about 25 minutes long, and goes into much more detail about Pi.

The earliest episode I could find online was The Theorem of Pythagoras, at the Internet Archive's Moving Image Archive, courtesy of A/V Geeks.com. Even though you might not have seen the previous episode on similarity, the prerequisites on the video catch you up well. BetterExplained.com's post Understanding Why Similarity "Works" can help fill in the rest of what you might have missed.



After this episode cam the Story of Pi episode mentioned above. The next 3 episodes all deal with the nature of sines and cosines. Part 1 explores the reason sines and cosines are important when examining functions that repeat at regular intervals. Part 2, below, examines how sines and cosines are used in trigonometry. Part 3 delves into the nature and use of the addition formulas for sine and cosine.



All the episodes on YouTube are posted by NASA, so they should remain available there for the foreseeable future.

A site called GhanaTubes (presumably based in the West African nation of Ghana) hosts the Polynomials episode. Unfortunately, the sound quality isn't very good on this version.

If you have the RealPlayer plugin, you can watch some clips from the episodes, including the missing first episode Similarity, as well as the final two episodes, The Tunnel of Samos and Early History of Mathematics.

Besides Project Mathematics!, Caltech also used the same style of animation in their series The Mechanical Universe, which was a course in college-level physics. As mentioned in my science documentary post last year, Google Video still hosts the full series online.

Take the time to watch a few of these in full. Even if you're not interested in the subject matter itself, I think you'll find that they're fun and engaging, making them great lessons in how to teach effectively.

Exploring mathematical concepts is now easier and more impressive than ever before, thanks to computers and the internet.

In this post, we'll look at some incredible mathematical tools, all of which are available online for free!

• Wolfram|Alpha - If you're not already familiar with Wolfram|Alpha, especially considering the numerous mentions I've made of it here on Grey Matters, you're in for a treat. Instead of simply searching for sites that may or may not answer your question, it simply tries to answer your question directly. I suggest watching Stephen Wolfram himself introduce the concept via video (Part 1, Part 2). You can get a better idea of its capabilities through the Wolfram|Alpha blog, examples, and tumblr blog, as well. You can even use your queries to develop embeddable widgets for your own website!

(Click here to watch part 2)


• Google Calculator - Google's search engine has a built-in calculator. While it doesn't give you all the features of Wolfram|Alpha, it's still quite impressive. It can handle arithmetic, conversions, graphing, and more! Soople is a site with a graphical front end for Google calculator, so you can get a better idea of its capabilities.

• Instacalc - No, this isn't as big a name as either of the first two, but that doesn't mean it's any less valuable. Instacalc is an online calculator designed to let you work with multi-step calculations. You can even embed your calculations, as I did in my post on Gas Math.

• GeoGebra - GeoGebra is a VERY impressive CAS (Computer Algebra System). I include it here, as it runs primarily as a sort of Java web app, but is also available as an offline application. As seen in the video below, GeoGebra is primarily used to develop interactive “worksheets” that can help users explore mathematical concepts. For example, here's an interactive worksheet I developed to help Grey Matters readers explore the Estimating Square Roots technique I recently posted.

It's often also used to teach students mathematical concepts by having them develop interactive worksheets for themselves. If you're familiar with Wolfram's CDF format, as used in Wolfram|Alpha Pro and Mathematica, the concept is similar, except you can develop these for free, instead of paying hundreds of dollars. You develop your worksheet using graphics pages, an algebra view for managing variables, and even a spreadsheet.



The level of support for GeoGebra is awesome, especially considering it's a free program. There's a Wiki to help you learn how to use it, a site where you can download tens of thousands of worksheets (and upload your own!), a forum, and a wide variety of instructional videos to help you along.

• CodeCogs Online LaTeX Equation Editor - LaTeX is a language used to define the typesetting features of documents, in such a way as to give consistent results across different platforms. It's found favor with mathematicians (Wolfram|Alpha's output is rendered with LaTeX), as it allows a more effective display of formulas and equations. For example, the following display is difficult to achieve in standard HTML, but easily achieved in LaTeX, with help from CodeCogs (click to see the LaTeX markup behind it):



CodeCogs also offers a web interface for turning C or C++ code into online calculators (login required), and a LaTeX rendering engine for Excel.

• Mathway - Mathway offers several online calculators, each geared to a specific mathematical subject, such as algebra, trigonometry, or calculus. It's entry method similar to CodeCog's LaTeX Equation Editor, but Mathway will work through the problem, as opposed to simply displaying it. With a free account, you can even save your calculations as worksheets to be used over and over again (free accounts see only the results, while paid accounts allow you to see the intermediate steps).

There are many more tools I can post, but the ones above are the current stand outs. If you program in Javascript, you might want to check out ASCIIsvg and/or JSXGraph.

Wikipedia provides an online calculator comparison chart here, and a computer algebra system comparison chart here, if you'd like to explore even more resources.

July's snippets are fresh out of the oven, and ready for your enjoyment and amusement!

This month, the snippets are all about math-based magic.

• Back in January, I mentioned a TV show called Square One TV, including the fact that Harry Blackstone, Jr., used to teach math-based magic tricks on the show. I've taken a little extra time to find all of these Backstage With Blackstone segments available on YouTube, and have linked them below:

5 envelope spelling
20 cent trick
1089 Trick
Cardless card trick (Addition)
Cardless card trick (18)
Dime, Penny, Nickel
Fibonacci Dice
Heads and Tails
Imaginary Dice
Magic Safari
Name the Number

There are several more Backstage With Blackstone segments, but not all are available via online video.

• Over at Futility Closet, there was a recent post titled Blind Dates, about a calendar prediction routine developed by Mel Stover. Interestingly, a similar routine was performed on Square One TV, but not in the Backstage With Blackstone sement.

If that catches your interest, and has you puzzled, check out my Easy Magic Square Cheat post from 2010. There are several resources there to help you understand it, and even expand the possibilities. (Note: Werner Miller's Age Square has moved here.)

• Speaking of Futility Closet, just as I was writing this post, they put up an entry called Order and Chaos, with a link to a similar 2009 post called So Much for Entropy. My post on the Gilbreath Principle (as it's known) helps you understand why this works. My write-up was even recommended in episode 152 of Scam School. You can find several more discussions of this amazing principle here, as well.

• Professor Peter McOwan, of Queen Mary, University of London, has some excellent videos over at the Maths Careers site. Check out The Maths in Magic, a brief demonstration of mathematical magic without explanations of the routines. He also has a full lecture on how math is used to cheat people, titled Maths Hustle.

If you enjoy those, you'll probably also enjoy his other sites Illusioneering and CS4N (Computer Science For Fun).

That's all for now, but if you have any favorite links concerning math-based magic, feel free to share them in the comments!

In the previous post, you learned how to work out a close estimate of the square root of any number up to 1,000.

Building on that skill, this post will show you how to go a little farther with this. You'll learn some extra touches, how to display this skill in context, and more!

Simplifying

I've shown how to find square roots of perfect squares in past tutorials, but how do you handle numbers that aren't perfect squares?

In this tutorial, you'll learn how to quickly determine an approximate square root for any number from 1 to 1,000. The method is a little challenging, but the results are impressive and worth the work.

In the feat itself, you're going to have someone have someone enter the square root into their calculator in a way that's easy for the calculator to understand, and then have them square it. If they give you the number 269, you instantly tell them to divide 13 by 33, and then add 16, explaining that you've determined 16 and 13/33rds to be the approximate square root of 269. When they square that number, they'll see that the answer is quite close (roughly 268.761).

Before you learn this feat, there are a couple of other feats you should learn. You should be comfortable with squaring 2-digit numbers, and being able to find the square roots of perfect squares. You'll also need to know the squares of the numbers from 1 to 31 off the top of your head, in order to handle the numbers from 1 to 1,000.

During this feat, you'll be subtracting 3 digits numbers. You can brush up on your mental 3-digit subtraction with help from this video.

That's enough for the preparation, how do you actually do the feat?

Start by asking someone to take out their calculator, make sure it's cleared, and then ask them for any number from 1 to 1,000. As an example, we'll use 149, which we'll refer to as the given number.

Step 1: Find the closest perfect square that is less than or equal to the given number. We'll refer to as the reference square or ref. square, and the root of the reference square will be called the reference root or ref. root. If they happen to give you a perfect square, you can state the square root instantly (and impressively).

With 149, you should instantly recognize that the closest perfect square below it is 144 (12²). So our reference square is 144 and the reference root is 12.



Step 2: From the given number, subtract the reference square, and remember this result. This result will be known as the numerator.

Starting with 149, we subtract 144 (the reference square) to get 5.



Step 3: Ask the person who suggested the given number to enter the numerator into their calculator, and then press the division key (÷).

Continuing with our example, they would enter 5, and then press the ÷ key.



Step 4: While they're entering the information from step 3 into the calculator, double your reference root and then add 1. This total will be referred to as the denominator.

The reference root is 12, so we double that to get 24, then add 1, giving a total of 25. 25 will be our denominator.

Note: You might be curious as to why you're doubling the reference root and adding 1. This is a short cut for finding the differences between your reference square, and the next perfect square.

If you think of the reference root as x, then the next number must be x + 1. For any perfect square x², the next perfect square is (x + 1)². The long way to determine the difference between them would be to work through the equation (x + 1)² - x².

However, it turns out that (x + 1)² - x² simplifies to 2x + 1! Doubling our reference root and adding 1 is much quicker than working through exponents!

Step 5: Have them enter the denominator into the calculator, and press the equals (=) button. The answer displayed will now be a decimal equal to the numerator divided by the denominator.

In our example, they've divided 5 by 25, which is .2.

Note: What the calculator is displaying at this point has a very useful double meaning. Our reference square in this example is 144, which is 12² (as we've already determined). The next perfect square is 13², or 169.

Picture the range of 144 to 169 as a line, and 149 as a single point along that line, as in this Wolfram|Alpha diagram. The first meaning of the 5/25 is that our given number 149 is 5/25 of the way between two perfect squares.

Since 149 is 5/25 of the way between 144 and 169, then it's reasonable to assume that 149's square root would be about 5/25 of the way between 12 and 13. This is the second meaning: It's the fraction we need to add to the reference root.

What we've been doing up to this point, then, is finding out how far between two perfect squares we have to travel, and expressing that as a fraction. Because of the way squaring works, this won't be an exact square root, but will come very close.

Step 6: Have them enter the addition (+) key, then enter the reference root, and then the equals (=) key.

Continuing with the example, we'd have them enter + 12 (the ref. root) =, so the calculator should now display 12.2.



Step 7: To prove how good your mental estimate is, have them press the x² button on their calculator. If they don't have one, the same result can be achieved by pressing the × button, followed immediately by the = button.

With 12.2 displayed, they now press x², and see a number approximately equal to 148.8399, which is very close to the given number 149!



Just to lock it in, let's try with another example. Let's say you're given a much higher number, such as 806.

Working through the process as above, we find the reference square, the reference root, and work through the process from there:



Once again, the squared result of 805.763 is very close to the given number 806!

As mentioned above, this works because we're working out the distance between two squares, and then seeing how far along that distance is the given number. Working that out as a fraction allows us to scale this answer down to be used as part of the given number's root. You can use this online web app I've developed to understand this concept more completely.

You might be wondering how close your estimates, when squared, will be to the original given number. The range of numbers with the biggest divisors, of course, will be the numbers from 961 up to 1,000. Using the process I teach above, here's a list of the results you'll get for each of those numbers.

Notice that the results are all just under the given number. For example, when you're given 962, your estimated square root, when squared, will return an approximate result of 961.984. If we look at just the margins of error for each number from 961 to 1,000, you'll note that it never gets farther away than .25 (or ¼)!

If you're presenting this as a bet, you can include the proposition that you have to be within plus or minus ½ in your estimate. This is only a smoke screen, as you know the resulting square will always be less than the given number, and it will never be off by more than ¼.

Instead of verbally instructing someone to enter the numbers in the calculator, you could write the answer down first. In this case, you would work through the process almost exactly backwards. Let's use 638 as an example.

The reference square, in this case, would be 625, and the reference root would be 25. Write down the reference root on the paper first.



Next, work out the denominator by doubline the reference root, then adding 1 to it. Write this as the denominator of the fraction on the paper.

The ref. root in this example is 25. We double that to get 50, and add 1 for a denominator of 51.



Finally, subtract the reference square from the given number to get the numerator.

638 - 625 is 13, so 13 is the numerator.



Sure enough, 25 and 13/51sts, when squared, gives approximately 637.81! Note that this is within our established -¼ margin of error.

You can practice using the random number generators at Wolfram|Alpha or Random.org and any handy calculator.

Naturally, the more squares you memorize, the higher you can go. If you memorize the squares of numbers up to 100, then you'll be able to estimate square roots of any number from 1 to 10,000! And yes, even at that scale, the resulting square will still never vary from the given number by more than ¼.

You can find out more about presenting this feat by reading the next post, Estimating Square Roots: Tips & Tricks.

Yesterday (July 4, 2012), the big item in the news concerned CERN's announcement that they were 99.99995% sure they'd discovered a Higgs-like particle.

This is a very important discovery, but since it deals with particle physics, there's plenty of confusion among the general populace as to what this means exactly. Fortunately, there are some great resources out there that can help explain.

We can't talk about the Higgs boson without delving into the world of subatomic particles. If you need a quick refresher course on electrons, neutrons, and protons, check out the Venus Explains the Atom video in my Hunting the Elements post.

For a good, quick pre-discovery look at the Higgs boson itself, watch What is a Higgs Boson? below:



In short, the discovery of the Higgs boson is exciting, because it verifies a theory about why anything has mass, and not just energy. The discovery gets at the very heart of existence as we know it.

As mentioned in Plus Magazine's article The Higgs boson: a massive discovery, the particle itself is difficult to find, as it decays almost immediately. If that's the case, how do you find it?

In the following video, The Higgs Boson and Mass, there's a more detailed view of how the Higgs boson and the Higgs field are believed to work, as well as how the Large Hadron Collider (LHC) is being used to find it:



Yes, this all has to do with Einstein's Theory of Relativity. If you want to get a better idea of Einstein's theory and its impact, I recommend the documentary Einstein's Big Idea.

The final video I'll post is the most detailed, yet still quite clear, idea of the nature of the Higgs boson and how the LHC was used to find it. It explains everything in a simplified chalkboard-type talk, and is called The Higgs Boson Explained (full screen recommended):



Don't be afraid to look for further information on it. On YouTube's minutephysics channel, start with The Higgs Boson, Part I, and go from there!

I wrote of Nicolaas Govert de Bruijn's passing back in February, but only gave a brief idea of the powerful magic his work made possible.

Grey Matters favorite James Grime has just released an excellent video that will give you an idea of how mystifying the use of de Bruijn sequences can be.

In the following video, James Grime will describe the trick, and then explain the workings behind it:



Quick note: You should use 3 decks to make a 48-card deck, not 4 as stated in the video. 48 cards is close enough to a regular 52-card deck that very few people will notice there are missing cards.

If you think about it, this works much in the same way as a trick with marked cards, but the people themselves unwittingly become the markings!

While you could practice and perform the method just as in the video, this wouldn't be Grey Matters if I couldn't improve the method with some memory techniques.

First, the code you're going to get from your audience is effectively a binary code. We'll assume that you have people take cards from your left to your right. We'll also assume that you're thinking of black as 0 and red as 1.

For example, if the 2nd and 4th spectators from your left are the only ones to respond to the red card question, your code would be 0101.

What you need atthis point, is an easy way to memorize any group of 4 binary digits. Over in the Mentat Wiki, their page on Binary Number Systems has a section that works perfectly, called the Nybble (4-Bit) Method.

Once you know all the names for each binary group of 4, you'll need to be able to recall cards. In the Mental Gym, I teach Bob Farmer's ingenious playing card mnemonics. With the practice quiz, you can learn these mnemonics quicker than you think!

As with any memory technique, we now need to link the information we have to the information we need.

First, you need to memorize the complete sequence of cards, using the card mnemonics in the following way:

ace-fist-narc-Kosmo-together-sid-tenderheart-Adam-throwback-seahorse-jade-tombstone-fork-night-quasar-thursday-ace-fist-narc

Note that the first 3 are repeated at the end, so as to cover all possibilities.

To remember this, use the Link System approach of creating wild and crazy connection. Picture an ace coming out of your mouth (to remember it as the first card on the list), which then becomes a fist that starts hitting a narc who turns out to be someone or somthing named Cosmo, and so on. Make sure you memorize the entire story, including the repetition of -ace-fist-narc at the end.

Below is a chart of all the binary combinations, the code names given in the Nybble System, the mnemonic of the 1st card in that sequence, and the cards: