System to extract the 2nd, 3rd, 4th and 5th root by heart
Carl Tyteca
by heart.
Can you tell me how to do this?
Thank you in advance.
Steve Monson
By heart, or by hand? I don't know how to do it by heart, but if you can
do the 2nd root, the 3rd and higher roots are done similarly. Just a
word of warning -- be prepared to do a *LOT* of long number arithmetic!!
below are a couple of articles I have used in the past:
Steve Monson
Use a monospace font, like Courier, to view these notes.
Extraction of roots, just like long division, is basically a guessing
game. You take your best shot, and try to see if you are as close as
you can get without refining your guess.
______
For instance, if you want to divide 23)16399
you inspect things and say, well, 20*700 is 14000, less than 16399,
so let's try 7 for our first digit. We can see that carries will
preclude 8. That is, you hope that 16399 = 23*(700 + n), with n
less than 100.
So, you plug and chug, getting
___700
23)16399
and subtract your guess... -16100
-----
299
Luckily, the remainder is less than 2300 (or we could have used 8
for our first guess). So now you assume, 16399 = 23*(700 + b), and
try b=10.
___710
23)16399
-16100
-----
299
-230
---
69
and again our remainder was less than 230, and we make one last guess,
assuming 16398 = 23*(710 + b), and wisely pick b=3. The point is,
that at every step, we have D = d(a + b) where d is the divisor, D is
the dividend, and a is what we have found so far, and b is what we
guess to be the next digit.
Now, to extend division to root extraction, just as we extend addition
to multiplication, and multiplication to raising of powers, we note that
2 2 2
(a + b) = a + 2ab + b
2
= a + b(2a + b)
Again, we use "a" as what we know so far, and "b" as our next digit's
guess. That's why we double our previous digits and add the new one:
2a+b. We then multiply by our new digit (b), and iterate the process.
Consider
______________
\/29 45 67 89 12
2
By inspection, we see that 50000 = 25 00 00 00 00, less than our
number. So, we pick a=0 (we don't know nuthin' yet), b=50000 and
proceed (supressing the extra zeros for readability)
_5____________
2 \/29 45 67 89 12
b =25 5 |25
-------
4 45
Now, we have a=50, 2a=100, and our next guess will be 4,
since 4*100=400 < 445
_5__4_________
\/29 45 67 89 12
5 |25
-------
2a 100 | 4 45
b +4 |
2a+b 104 | 4 16 multiply b(2a+b)
----------
29 67
Now we have a=540, and we choose b=2 (since 2*1080 is just less
than 2967)
_5__4__2______
\/29 45 67 89 12
5 |25
-------
100 | 4 45
+4 |
104 | 4 16
----------
2a 1080 | 29 67
b +2 |
2a+b 1082 | 21 64 multiply b(2a+b)
--------------
8 03 89
Keep on in like wise, till you have as many digits as you need...
_5__4__2__7__4
2 \/29 45 67 89 12
subtract a 5 |25
-------
a=50 100 | 4 45
b=4 +4 |
2a+b 104 | 4 16
----------
a=540 1080 | 29 67
b=2 +2 |
2a+b 1082 | 21 64
--------------
a=5420 10840 | 8 03 89
b=7 7 | 7 59 29
------------------
a=54270 108540 | 44 60 12
b=4 4 | 43 41 76
----------------------
1 18 36 ......
I hope you play around with the calculations and have a good time!
------------------------------------------------------------------
If you recall how to do square roots, you know that it's just
an extension of long division. You make your best guess,
subtract off the part that's right, and refine your answer,
digit by digit.
With that in mind, recall that
3 3 2 2
(a+b) = a + b(3a + 3ab + b )
At each step of the process, "a" is what we know, and "b" is
what we think the next digit will be. (It helps to suppress
trailing zeros, as they just clutter up the calculations.)
You group the radicand by threes, since each zero you add to
"a" adds three zeros to its cube.
Let's say you want to find
3 __________________
\/83 490 284 139 329
No one ever actually would, but then this is just math, eh?
By inspection, you find that the biggest cube less than 83 is 64,
so we start out with:
3 _4________________
\/83 490 284 139 329
64
--
19 490
You start out with a=0, and your first guess is b, since that
gives you your first cube guess. Now, things get a bit rougher.
It's quite common to pick a "b" that's too big, because your
new divisors grow unexpectedly large sometimes. Anyway, the next
step is with a=40, and we proceed from there:
3 _4___3____________
\/83 499 293 950 912
2 --
a=40: 3a 4800 |19 499
b=3: 3ab 360 |
b*b 9 | 2 2
5169 |15 507 <---- b(3a +3ab+b)
-------------
3 992 293
3 _4___3___7________
\/83 499 293 950 912
--
4800 |19 499
360 |
9 |
5169 |15 507
2 -------------
a=430 3a 554700 | 3 992 293
b=7 3ab 9030 |
b*b 49 | 2 2
563779 | 3 946 453 <-------b(3a +3ab+b)
-------------------
45 840 950
3 _4___3___7___0____
\/83 499 293 950 912
64
--
4800 |19 499
360 |
9 |
5169 |15 507
-------------
554700 | 3 992 293
9030 |
49 |
563779 | 3 946 453
-------------------
45 840 950
Now, we note that our new a=4370, and 3a*a = 57290700 which
is greater than 45840950, so our next b=0, and we
just go on from there to our last digit:
3 _4___3___7___0___8
\/83 499 293 950 912
64
--
4800 |19 499
360 |
9 |
5169 |15 507
-------------
554700 | 3 992 293
9030 |
49 |
563779 | 3 946 453
------------------- 2
a=4370 57290700 | 45 840 950 3a > divisor
b=0, so |
a=43700 5729070000 | 45 840 950 912
b=8 1048800 |
64 |
5730118864 | 45 840 950 912
-------------------------------
Don't try this at home, kids. Not without "bc" or some other
arbitrary-precision package. Of course, in high school,
all I had was a pencil and paper. Yeah, and it was a mile to
school, and uphill both ways, in the snow!
-------------------------------------------------------------
You can, of course, extend your reach to other bases.
For example, if you want to waste some time, try working
out something like this 5th root in base 7!
2 1 5 2 3. 6
5_________________________________
\/111 65165 30240 21363 65043.00656
44
---
1430000 |34 65165
143000 |
5500 |
130 |
1 |
------ |
1611631 |16 11631
|--------
21026550000 |15 53234 30240
1301661000 |
32266500 |
362220 |
1552 |
----------- |
22364534602 |14 55526 43213
--------------
240662153630000 |64404 54024 21363
442160221000 |
405403600 |
155540 |
22 |
--------------- |
241435053043462 |51320 31361 20254
|-----------------
2422113612416630000 |13054 22333 01106 65043
666023321502000 |
122063043300 |
6125640 |
144 |
------------------- |
2423113061143634414 |10602 34224 34645 36545
|-----------------------
24241123620123041360000 | 2151 55105 33131 25165 00656
1666145645262363000 |
52154641611600 |
1004306460 |
3531 |
----------------------- |
24243123151260313311221 | 2151 55105 33131 25165 00656
|-----------------------------
Russell Baker
method?
If you can get the root between consecutive integers or better by
inspection, the root of a reasonably sized number can be found pretty
nicely.
Interpolation assumes that the graph of the root function is nearly
linear between two points rather close together. Here is an example:
find the cube root of 10
2^3 = 8, and 3^3 = 27 so the cube root of 10 is between
2 and 3.
diff top to center y 2.00 8.00
2.00
diff top to bottom 1.00 x
10.00 19.00
3.00 27.00
y / 1 = 2/ 19
y = .1052
so x = 2.1052 (actual value: 2.15443...)
Rbaker