Earlier 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 10Take 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!
Posts
