Calculate Powers of 2 In Your Head!

Published on Sunday, October 26, 2014 in , , ,

Ptkfgs' Doubling Cube imageEarlier this year, I posted about calculating powers of e in your head, as well as powers of Pi.

This time around, I thought I'd pass on a method for calculating powers of a much more humble number: 2. It sounds difficult, but it's much easier than you may think!

BASICS: For 20 up to 210, you'll memorize precise answers. For answers to 211 and higher integer powers, you'll be estimating the numbers in a simple way that comes very close.

First, you must memorize the powers of 2 from 0 to 10 by heart. Here they are, along with some simply ways to memorize each of them:

Problem   Answer    Notes 
  20    =     1     Anything to the 0th power is 1
  21    =     2     Anything to the 1st power is itself
  22    =     4     22 = 2 × 2 = 2 + 2
  23    =     8     3 looks like the right half of an 8
  24    =    16     24 = 42
  25    =    32     5 = 3 + 2
  26    =    64     26 begins with a 6
  27    =   128     26 × 21
  28    =   256     Important in computers
  29    =   512     28 × 21
  210   =  1024     210 begins with a 10
Take a close look at 210, which is 1024. It's very close to 1,000, so we're going to take advantage of the fact that 210 ≈ 103!

When multiplying 2x × 2y, remember that you simply add the exponents together. For example, 23 (8) × 27 (128) = 27 + 3 = 210 (1024). Similarly, you can break up a single power of 2 into two powers which add up to the original power, such as 29 (512) = 26 + 3 = 26 (64) × 23 (8).

TECHNIQUE: We'll start with 215 as an example.

Start by breaking up the given power of 2 into the largest multiple of 10 which is equal to or less than the given power, and multiply it by whatever amount is leftover, which will be a number from 0 to 9.

Using this step, 215 becomes 25 + 10, which becomes the problem 25 × 210.

For an powers from 0 to 9, you should know by heart, so you can convert these almost instantly. In the example problem we've been doing, we know that 25 is 32, so the problem is now 32 × 210.

Now we deal with the multiple of 10. For every multiple of 10 involved, you can replace 210 with 103. With our problem which is now 32 × 210, there's only a single multiple of 10 in the power, so we can replace that with 103. This turns our current problem into 32 × 103.

At this point, it's best to represent the number in scientific notation. In this feat, that simply refers to moving the decimal point to the left, so that the left number is between 0 and 10, and then adding 1 to the power of 10 for each space you moved the decimal. Converting to scientific notation, 32 × 103 becomes 3.2 × 104.

That's all there is to getting our approximation!

How close did we come? 215 = 32,768, while 3.2 × 104 = 32,000. I'd say that's pretty good for a mental estimate!

EXAMPLES: Over 6 years ago, I related the story of Dr. Solomon Golomb. While in college, he took a freshman biology class. The teacher was explaining that human DNA has 24 chromosomes (as was believed at the time), so the number of possible cells was 224. The instructor jokingly added that everyone in the class knew what number that was. Dr. Golomb immediate gave the exact right answer.

Can you estimate Dr. Golomb's answer? Let's work through the above process with 224.

First, we break the problem up, so 224 = 24 + 20 = 24 × 220.

Next, replace the smaller side with an exact amount. In this step, 24 × 220 becomes 16 × 220.

Replace 210x with 103x, which turns 16 × 220 into 16 × 106.

Finally, adjust into scientific notation, so 16 × 106 becomes 1.6 × 107.

If you know your scientific notation, that means your estimated answer is 16 million. Dr. Golomb, as it happened, had memorized the 1st through 10th powers of all the integers from 1 to 10, and new that 224 was the same as 88, so he was able to give the exact answer off the top of his head: 16,777,216. 16 million is a pretty good estimate, isn't it?

Below is the classic Legend of the Chessboard, which emphasizes the powers of 2. In the video, the first square has one (20) grain of wheat placed on it, the second square has 2 (21) grains of wheat on it, with each square doubling the previous number of grains.

The 64th square, then, would have 263 grains of wheat on it. About how many is that? I'm going to run through the process a little quicker this time.

Step 1: 263 = 23 + 60 = 23 × 260

Step 2: 23 × 260 = 8 × 260

Step 3: 8 × 260 ≈ 8 × 1018

While 263 is 9,223,372,036,854,775,808, our estimate of 8,000,000,000,000,000,000 works.

TIPS: If you're really worried about the error, there is a way to improve your estimate. Percentage-wise, the difference between 1,000 (103) and 1,024 (210) is only 2.4%. So, for every multiple of 10 to which you take the power of 2 (or every power of 3 to which you take 10), you can multiply that by 2.4% to get a percentage difference. You can then multiply that percentage difference by your estimate to improve it.

Just above, we converted 263 into 8 × 1018. Since we started with six 10s, our percentage difference would be 6 × 2.4%, or 14.4%. In other words, our estimate of 8 × 1018 could be made closer by adding 14.4% to 8.

Assuming your comfortable with doing percentages like this in your head, 8 increased by 14.4% is 8 + 1.152 = 9.152, so our improved estimate would be 9.152 × 1018. Considering the actual answer is roughly 9.223 × 1018, that's quite close!

Practice this, and you'll have an impressive skill with which to impress family, friends, and computer geeks!


100 Years of Martin Gardner!

Published on Tuesday, October 21, 2014 in , , , , , ,

Konrad Jacobs' photo of Martin Gardner“Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.” - Ron Graham

100 years ago today, Martin Gardner was born. After that, the world would never again be the same.

His life and his legacy are both well represented in David Suzuki's documentary about Martin Gardner, which seems like a good place to start:

As mentioned in the snippets last week, celebrationofmind.org is offering 31 Tricks and Treats in honor of the Martin Gardner centennial! Today's entry features a number of remembrances of his work in the media:

Scientific American — “A Centennial Celebration of Martin Gardner”

Included in the above article is this quiz: “How Well Do You Know Martin Gardner?”

NYT — “Remembering Martin Gardner”

Plus — “Five Martin Gardner eye-openers involving squares and cubes”

BBC — “Martin Gardner, Puzzle Master Extraordinaire”

Guardian — “Can you solve Martin Gardner's best mathematical puzzles?”, Alex Bellos, 21 Oct 2014

Center for Inquiry — “Martin Gardner's 100th Birthday”, Tim Binga
There are quite a few other ways to enjoy and remember the work of Martin Gardner, as well. The January 2012 issue of the College Mathematics Journal, dedicated entirely to Martin Gardner, is available for free online! The Gathering 4 Gardner YouTube channel, not to mention just searching for Martin Gardner on YouTube, are both filled with enjoyable treasures to be uncovered.

Here at Grey Matters, I've written about Martin Gardner quite a few times myself, as I have great respect for him. Enjoy exploring the resources, and take some time to remember a man who has brought joy, wonder, and mystery to the world over the past 100 years.


Even More Quick Snippets

Published on Sunday, October 12, 2014 in , , , , , ,

Luc Viatour's plasma lamp pictureIt's time for October's snippets, and all our favorite mathematical masters are here to challenge your brains!

• I'm always looking for a good mathematical shortcut, in order to make math easier to learn. More generally, I'm always looking for better ways to improve my ability to learn. I was thrilled with BetterExplained.com's newest post, Learn Difficult Concepts with the ADEPT Method.

ADEPT stands for Analogy (Tell me what it's like), Diagram (Help me visualize it), Example (Allow me to experience it), Plain English (Let me describe it in my own words), and Technical Definition (Discuss the formal details). This is a great model for anyone struggling to understand anything challenging. This is one of those posts I really enjoy, and want to share with as many of you as I can.

• If you enjoyed Math Awareness Month: Mathematics, Magic & Mystery back in April, you'll love the 31 Tricks and Treats for October 2014 in honor of the 100th anniversary of Martin Gardner's birth! Similar to Math Awareness Month, there's a new mathematical surprise revealed each day. It's fun to explore the new mathematical goodies, and get your brain juices flowing in a fun way!

• Over at MindYourDecisions.com, they have a little-seen yet fun mental math shortcut in their post YouTube Video – Quickly Multiply Numbers like 83×87, 32×38, and 124×126. As seen below, it's impressive, yet far easier than you might otherwise think:

They've also recently posted three challenging puzzles about sequence equations that you might want to try.

• If that's not enough, Scam School's latest episode (YouTube link) at this writing also involves three equations. If you have a good eye for detail, you may be able to spot the catch in each one before they're revealed:

That's all for this October's snippets, but it's more than enough to keep your brain puzzled through the rest of the month!


Convert Decimal to Any Base 2 - 9

Published on Sunday, October 05, 2014 in , , , ,

LaMenta3's binary pillow photoAbout 2 years ago, I posted about Russian/Egyptian multiplication, and included a technique for mentally converting decimal (base 10) into binary (base 2).

Recently, Presh Talwalkar covered this same technique on his Mind Your Decisions blog. I've only just realized that with a little modification, this technique can be used to quickly and mentally convert decimal to any base 2 through 9!

We'll use the Mind Your Decisions binary conversion video as a starting point. It's less than 3 minutes long, so it's a quick study:

In both my original Power of 2 post and the above video, the idea of ignoring the remainder is emphasized. Funnily enough, changing the technique to focus on the remainder makes this basic idea much more usable. If you remember division problems with answers like, “22 ÷ 6 = 3 remainder 4”, that's the type of division we'll be using in this post.

The first step is simply to take the given number and divide it by whatever base you're using, so that you have a quotient and a remainder. For a starting example, we'll convert the decimal number 84 into base 5. 84 ÷ 5 = 16 (the quotient), remainder 4.

The second step is to write down the remainder. In our example, we'd simply write down the 4.

Step 3 is to divide the quotient by the base again. This time, we'd work out 16 ÷ 5 = 3 remainder 1.

Step 4 is to write down this remainder to the immediate left of the previous remainder. Writing down the 1 to the immediate left of the 4 gives us 14.

Repeat steps 3 and 4 until you get a quotient of 0, at which point, you've got your answer! Finishing up our example, we'd use our current quotient of 3, divide that by 5, getting an answer of 0 remainder 3, write the 3 down to the left of the previous remainders, giving us 314. Since our quotient is 0, we also know we're done! Checking with Wolfram|Alpha, we see that 84 in base 5 is indeed 314!

TIP #1: Once your quotient is a number less than your base, you can simply write that to the left of the remainders and know you're done. In the above example, once we got down to 3, and we realize this is less than 5, we know this is the final step. Because of this, we can simply write the 3 down and stop.

In short, as long as you're given a decimal number and a base by which you're comfortable dividing that number, you can convert that number to that base in your head with little trouble. Not surprisingly, knowing division shortcuts and divisibility rules can be of great help here.

What about 147 (in base 10) to base 4? As long as you realize that the closest multiple of 4 is 144, and that you can handle 144 ÷ 4 in your head, the rest of the conversion shouldn't be a problem. 147 ÷ 4 = 36, remainder 3. Write down the 3, and then work with 36. 36 ÷ 4 = 9, remainder 0, so write the 0 to the left of the 3 (03), and work with 9. 9 ÷ 4 = 2, remainder 1. Write down the 1 to the left of the previous remainders (103). Tip #1 above tells us that, since 2 is less than 4, we can just write down that 2 to the left of the other numbers (2103) and know we're done. Sure enough, 147 in base 4 is 2103!

TIP #2: If the given number is less than the square of the base to which you're converting, you can do everything in a single step. All you have to do is work out the quotient and the remainder, write the quotient to the left of the reminder, and you're done! For example, what is 59 in base 8? 59 ÷ 8 = 7 remainder 3. Write down the 3 as before. Thanks to tip #1, we know that we can write the quotient down to the left, since 7 is less than 8, so we just write the 7 down next to it!

For base 8, this will work for any number less than 8 × 8, or 64. Similarly, for base 5, this will work for any number less than 25 (5 × 5), and so on. 44 in base 7? 44 ÷ 7 = 6 remainder 2, so we can quickly give the answer as 62!

Being able to convert to base 2 and base 8 in your head can be a great asset when working with computers. Practice this skill and have fun with it. You'll not only have a useful skill, but something with which to amaze and amuse others, as well!