Mental math

For my future me. This is how I do mental math in 2013.

So me and Cong and Nam was staying together. And we came up wit a wild idea of challenging each other mental math. Here are the questions in chronological order:

1. 27^2

2. 27^3

3. 13^13 (me)

4. sin 1 (Nam)

5. tan 2 (me)

6. root of 12345 (me)

7. fifth root of a billion (Cong)

8. 1.76^893 (Thanh)

9. the pi root of e, the whole thing to the power of one and a half.

Here is how I do it, everything without a pencil and paper.

1. 27*27 = 30*24 + 3^2 = 720 + 9 = 729

2. 27 ^ 3 = 729*27 = 729*30 - 729*3 = 21870 - 2187 = 19870 - 187 = 19770 - 87 = 19700 - 17 = 19683

3. (approximate) 13^2 = 169, 169^2 = 170^2 = 28900, 28900^2 = 29000^2 = 29^2 * 1000000 = (30*28+1)*1000000 = 841 with 6 zeros.

So that is 13^8. For 13^5 = 28900*13 = 30000*13 = 390000 = 4 with 5 zeros

So 13^13 = 8*4 with 13 zeros = 32 with 13 zeros = 3.2 * 10^14.

The correct answer is 3.028*10^14. Hm close enough.

4. sin 1. Use Taylor series

5. tan 2 = tan 2pi/3.

6. root of 12345 = root of 12300. shoud be 110 a bit more. so should be like 112. Correct answer is 111.1

7. fifth root of a billion. First cut off 5 zeros out of a billion and try calculate fifth root of 10000. Try 9^5. It is 81^2 * 9 = 6400 * 9 = 54000 so too big.

Try 8^5 = 64^2 * 5. 64^2 = 65^2 = 4225, 4225*5 = 20000 still too big.

Try 7^5 = 49^2 * 7 = 50^2 * 7 = 2500 * 7 = 14000 still slightly too big.

So make a wild guess 65.

Correct answer is 63.

8. My favourite one. 1.76^893. Idea is we keep squaring 1.76 9 times. We will have 1.76^(2^9)=1.76^512. Since 893 is a bit more than 1.5 of 512. Let's say if to the power of 512 has 100 zeros. to the power of 893 should have 160 zeros.

Ok 1.76^2 = 1.7^2 = 2.89 = 3.

3^2 = 9

9^2 = 81

81^2 = 6400

6400^2 = 6500^2 = 42250000, so 4 with 7 zeros

4 with 7 zeros squared is 16 with 14 zeros so 1.6 with 15 zeros

1.6 with 15 zeros squared is 2.56 with 30 zeros.

2.56 with 30 zeros squared is 6.25 with 60 zeros.

6.25 with 60 zeros squared is 40 with 120 zeros. 

So 1.76^893 should be like be around 10^190 or 10^200.

True result is 1.76^219.

9. The pi root of e. e is 2.71828. We calculate third root of 2700 then. Should be more than 10. 11^3 = 1331. Try 12. 12^2 = 144, 144*12 = 1600. Not enough. Try 13. 13^2 = 169 = 170. 170*13 = 10*(15^2 - 4) = 2250 still a bit too small. So it should be 13.5. So fifth root of e is like 1.35. Whole thing to the power 1.5. Root of 1.35 is like 1.15. So final result is 1.35*1.15 = 1.3*1.2 = 1.56. Right answer is 1.61.

Alright a bunch of numbers probably make you feel sick. You can actually make some adjustment to these calculation and have a closer result. For example in question 8. We know root 3 is 1.73, so when we round up 1.76^2 to be 3. We have lower our result. And since this is a power question this error is significant. So at the end we should add a small pernaty, 10% of the number of zero we found. So let say we found 200 + 10% of 200 = 220. Yay exact answer. Just kidding this is data-snooping :). More on data snooping on a later post.

While I was busy working with numbers, there is a broken love affair.

Ok let's stay positive. We'll fix it: