Estimate square roots and even higher-order roots by using simple processes.
It's often useful to compute the square root of a number, especially when you want to visualize areas or compute diagonal distances. There are a few methods for computing square roots on paper, some of which are not widely known. If all you have is your brain, however, it's still possible to come up with a quick estimate that's reasonably accurate.
To estimate the square root of a number, start by pairing up the digits of the number, beginning with the decimal point and moving away. So, for instance, to compute the square root of 500,000, we would pair up the digits like this:
50 00 00
Each pair of digits will, in fact, represent a digit in the square root. The leftmost pair of digits (or single digit, if there are an odd number of digits in your original number) is used to compute the leftmost (most significant) digit of the result. You find the result digit by determining the highest square that will fit into the pair without going over.
The biggest perfect square that fits into 50 is 49, which has a
square root of 7. So, we know our square root will have the form
7dd—that is, a 7 followed by two digits, or "seven hundred
and something." Further, since 50 is very close to 49, we can surmise
that the square root will be in the low 700s. Had it been close to the
next square (64, with a square root of 8), we would know that the
square root would be closer to 800. Unfortunately, computing the exact
value of the later digits requires pencil and paper, but our estimate
is in the ballpark.
What about numbers smaller than 1? Basically, the same method applies. You still pair up digits going away from the decimal point, and the most significant digit will still be the square root of the biggest square that fits into the leftmost pair of digits. So, to compute the square root of 0.0038234, you would pair up the digits like so:
00 . 00 38 23 40
Each pair of digits again represents a single digit in the answer, and the decimal place stays where it is. The biggest square that fits in 38 is 36, with a root of 6. So, your square root will be 0.06.... Since 38 is close to 36, we can also surmise that our approximation is already pretty close to the answer, so the next digit will be small.
Numbers in science and engineering are often expressed in scientific notation, which is a convenient way to express astronomically large or small numbers. With scientific notation, numbers are expressed in the form:
a × 10b
where a is usually in the
range 1 <= a < 10, and
b is an integer. Nearly all
modern calculators use scientific notation for representing
large or small numbers; typically, the 10 is omitted and the letter E for "exponent" is used (e.g., 5.2349 E+41). Since the exponent tells us
the number of digits, we can use a very similar method to estimate
the square root of some very large or very small numbers.
When the exponent b is
even, it's easy to come up with an estimate. Simply estimate the
square root of the a part and
halve the exponent. So, given a number like 5.8345 ×
1082, we can immediately tell that the
square root is going to be 2.d × 1041,
because:
4 is the biggest square that fits in 5.
4 has a square root of 2.
The exponent 41 is half of 82.
This works for both positive and negative even exponents.
When the exponent b is odd,
we need to borrow the first digit from the other side of the decimal
point of a. Estimate the square
root of that, and you have the a
part of the result. To get the exponent of the result, subtract 1 from the exponent
of your original number before halving it.
For instance, given 5.0234 × 1017, we would find the square root of the biggest square that fits in 50 (49, with a square root of 7) and subtract 1 from the exponent 17 to make it 16 before halving it, giving us a square root of 7.d × 108.
For a number with an odd negative exponent, such as 1.9123 × 10-43, you will still subtract 1 from the exponent before halving (-43 to -44); since 4 is the closest square root to 19, the answer will be 4.d × 10-22.
What about cube roots, fourth roots, and so on? There is a more general rule that will let you estimate these. Instead of pairing up the numbers, to compute the nth root of a number, divide its digits into groups of n digits each, going away from the decimal point. So, to compute the fourth root of 7,324,643,245, group it like this:
73 2464 3245
As before, each group represents one digit of the result. And
the first digit will be the number closest to the fourth root of the
first group, 73. While there is no easy way to compute or memorize
the fourth roots of numbers, remember that it can be only
one of nine possible digits, so it is fairly easy to find it via
trial and error. 2 × 2 × 2 × 2 is 16; 3 × 3 × 3 × 3 is 81, which is
a little too large. So, our answer will be 2dd, a number in the high 200s. As you've
probably realized, the method for computing square roots described
earlier is just a specific (n = 2) example of
this more general method.
Suppose you come across a real estate listing advertising a house with an area of 3,700 square feet. Breaking this up into pairs of digits gives you:
37 00
You have two groups, so it'll be a two-digit number. The largest square that fits into 37 is 36, with a square root of 6. And it's pretty close, so this house will have as much floor space as a square house a little longer than 60 feet on a side.
It was recently reported that the ozone hole over Antarctica has shrunk to 6,000,000 square miles. How big is this? Breaking this up into pairs of digits gives you:
6 00 00 00
The biggest square that fits into 6 is 4 (2 × 2), so you should be able to determine, with a moment's reflection, that this represents a square area measuring in the middle 2,000s on each side. Still a pretty big hole in the ozone!
Tip
If you'd rather picture a circle, it will always have a diameter that's about 13% bigger than the square root you just estimated.
Also in the news at this writing, Hurricane Katrina reportedly left an estimated 16 million cubic yards of debris littering the coastline of Mississippi alone. How big a warehouse would it take to hold all of that? Because it's cubic, we gather the digits in groups of three:
16 000 000
That's three digits, and the cube root of 16 is somewhere between 8 (2 × 2 × 2) and 27 (3 × 3 × 3), so this represents a cube that's about 250 × 250 × 250 yards. That's a warehouse wider, longer, and taller than two football fields, and that's just the debris littering Mississippi.
Wikipedia. "Square root." http://en.wikipedia.org/wiki/Square_root. A good summary of more traditional methods of estimation and exact computation of square roots, including the exact method that this estimation method is based on. Go there to learn how to compute a square root exactly, with the help of pencil and paper.
Doerfler, Ronald W. Dead Reckoning: Calculating Without Instruments. Gulf Publishing Company. Contains further methods for computing roots.
Mark Schnitzius
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