As a follow-up to last week's tutorial on calculating powers of e in your head, I'm going to teach you how to do the same for our old friend Pi!
As an added bonus, calculating powers of Pi can be slightly easier than powers of e, so even if you passed up last week's tutorial, you should still give this one a look.
BASICS: In a manner similar to the previous tutorial, we'll request a number x, and solve for y in the equation πx = 10y.
This method, just like the previous one, is also based on turning this problem into a logarithm. It becomes log10(πx) = y, which simplifies to x × log10(π) = y. This works out to about x × 0.49714987... = y.
This time around, we can take advantage of the fact that this number is close to 0.5!
TECHNIQUE: We'll use π16 as our example requested power.
The first step this time involves setting up a subtraction problem with 2 numbers, both of which start with the given number (16, in our example).
BUILDING THE SUBTRACTION PROBLEM: To begin, take the given number an multiply it by 500. This can be made simpler, if you prefer, by simply tacking ,000 on the end of your number, and then dividing by 2. This is the first number we need.
Applying this step to our example number 16, we add a comma and 3 zeroes to it (16,000), and then divide by 2 to get 8,000. We have our first number.
Note: If you're given an odd number, you will always end up with a ,500 at the end. For example, if the given number was 15, this step would result in 7,500. Knowing this is a handy way to make sure you didn't misplace the comma.
To get the second number for the subtraction problem, simply multiply the given number by 3. This should be easy enough to do without any special tips.
Doing this, our second number is 16 × 3 = 48.
SUBTRACTION:Having set up the 2 numbers for the subtraction problem the next step, not surprisingly, is to perform the subtraction by subtracting the smaller number from the larger number.
We've worked out the numbers 8,000 and 48 in our 16 example, so the subtraction problem is 8,000 - 48.
If you're like most people, though, you remember writing down subtraction problems with lots of zeroes in school, and having to borrow over multiple places. That being the case, you're probably wondering how to deal with all this in your head! The following video teaches you how to deal with problems like these without ANY borrowing:
To work out 8,000 - 48 using the above technique, it's probably better if you think of the problem as 8,000 - 048. The first step, as in the video, is to round the leftmost digit up, from 0 to 1 in this case, and seeing that 80 - 1 = 79. We already know the answer begins with 79!
How far up would you have to go from 48 cents to get to a whole dollar? Getting the answer of 52 cents shouldn't be a problem here. That's the other half of the answer.
Your running total, at this point, is 7,952. After a little practice, subtracting from zeroes in your head will seem not only less scary, but nearly effortless.
ADJUSTING FOR APPROXIMATION: We're going to add a little now to improve the accuracy of our answer. How do we do that?
Take the number you just subtracted, and throw away the ones digit. Divide the remaining digits by 2. If that ends in a .5, just throw the .5 away, as well. This is the number you add to your running total.
We just subtracted 48 to get 7,952. We take 48 and throw away the ones digit, leaving 4. Dividing that by 2, we get 2. Finally, we add 2 to 7,952 to get 7,954 as our new running total.
DIVIDING BY 1,000: To divide by 1,000, I could tell you to move the decimal point three places to the left, but there's an even simpler technique this time. All you have to do is replace the comma in the total with a decimal point!
With this approach, 7,954 instantly becomes 7.954 with very little effort.
At this point, you're done! As you can verify on Wolfram|Alpha, π16 ≈ 107.954.
THE FULL PROCESS ALL AT ONCE: To run through this at once, and to better acquaint you with the full range of situations you'll run across, let's try to work out π33 = 10y.
- Multiply 33 × 1,000 to get 33,000, and divide by 2, getting 16,500.
- Multiply 33 × 3 to get 99.
- Subtract 16,500 - 99 = 16,401.
- Throw away the 9 (the ones place of 99), leaving 9, and divide that by 2 (4.5), throwing away the .5 to leave 4.
- Add 16,401 + 4 = 16,405.
- Replace the comma with a decimal point, resulting in 16.405.
TIPS: Most of the tips I gave for e apply for π, as well. I'll repost the relevant ones below for convenience, modified for our Pi examples.
• By looking at the whole number part of the answer (the significand) and adding 1 to that, you can state the number of digits the full answer would have. In our 7.954 example, we take the whole number part, the 7, and add 1 to get 8, so we can state that the answer is a 8-digit number. Having worked out π33 to be about 1016.405, we can state that the answer is a 17-digit (16 + 1) number!
• You can handle exponents with a decimal in them by working them out as if they were a whole number, and then adjusting for an appropriate number of additional decimal points when you're moving the decimal point. For example, π1.6 is the same as π16, but with the decimal moved one place more to the left. Since π16 ≈ 107.954, it's easy to see that π1.6 ≈ 100.7954.
• If you want to take this a step further, and be able to say that π16 is roughly equal to 9 × 107, check out Nerd Paradise's Calculating Base 10 Logarithms in Your Head, the video Calculating logarithms in your head, and the PDF How to Quickly Calculate Logarithms to Three Decimal Places in Your Head.