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What is a Square Root?

Do you remember learning about square roots in middle school and high school?
Were they ever really confusing or some parts were always left unexplained. Like,
why can’t you take the square root of a negative number? These questions are
actually quite common and relatively simple to answer.

One good way to understand square roots is to first look at squares. In a square, to
find the area you can simply take one side times itself to get to the total area of the
square. In the picture we have the side of the square is 5 and
the area is therefore 5 x 5 which is 25. We then say that 25 is
the “square” of 5. The square of any number is simply the
number times itself. We write this as for y being an
integer. For example, 4, 7, and 13 as values of y give you
squares of 16, 49, and 169 for values of x.

 

Now we would like to look at this situation backwards. Given a number like 64 for
x, what is the value of y so that . This value is the square root of x. Which
we use the sign , also called a radical. The radical also implies that this
result is positive. For example, when we take the square root of 9 we actually get
two numbers, 3 and (-3), that are roots of 9. We recognize this as taking a square
root as meaning . Taking the square root of a number always gives you
two distinct answers.

Historically, the square root was one
of the most studied objects,
especially in terms of irrationability.
Around 2000 BC the Babylonian
tablet YBC 7289 contained to 9
significant decimals . This tablet
gives the number 1, 24 51 10 which,
in base 60, is very close the square
root of 2. The Pythagoreans proved
that is irrational, despite their
belief that all number were rational
and could be represented as a ratio
of two integers. Legend has it that
the Pythagorean Hippasus made this
discovery while out at sea. He caused such an uproar that they decided to throw
him off the boat rather than let his discovery be known. However, Theodorus later
proved that the numbers 3-17 were irrational, except for 4, 9, and 16.

The radical was not always used as a sign for the square root. In 1220 Leonardo of
Pisa used the symbol Rx, similar looking to a prescription symbol. The emergence
of the radical happened later around 1525 as what looked like a checkmark without
the line above . In 1629 Albert Girard suggested placing the index in the small
opening of the checkmark part. Thus a cube root would be written as . It was
not until Rene Descartes in 1637 did we have our traditional symbol for the radical
with the checkmark part and the line over the top. The last development for the
square root was that in 1895 it decided that would represent one value rather
than two.

Proving that is irrational

One of the most interesting properties of square roots is for integers, the result is
either another integer or irrational. The Greeks believed that all numbers could be
represented as the ratio of two integers . However, it was eventually proved that
the square root of 2 cannot be represented in this way. Here are two short proofs
that show
is irrational.

If
were rational then it can be represented by the ratio of integers p and q in
this way; . Using algebra we can square both sides and multiply by to
get . By the fundamental theorem of algebra we know that p and q have a
unique factorization. Looking on the left-hand side we can see that no matter
what, factors down to an odd power of 2, while the RHS factors down to an
even power of 2. This contradicts that they must be equal and therefore our
assumption that
is rational is incorrect.

Another way is by using an isosceles-right triangle. Showing two sides are
commensurable means that there is some integer length
that divides both sides which implies that the ratio of
sides is a rational number. If we show that the sides are
incommensurable it is equivalent that those sides cannot
be expressed as a rational number. Assume that we have
some integer m so that m|AB and m|BC. We then make
DB=AB so m|DB. It then follows that BC=CD+DB so
m|CD. Segment DE is constructed so that DE is perpendicular to BC and we have
similar triangles ABC and DEC by AA. We then have <EAD+<DAB=90 and
<EDA+ADB=90, and since ABD is isosceles we have <ADB=<DAB so it follows
that <EAD and <EDA are supplementary to the same angle, therefore they must
also be equal. So triangle EDA is isosceles and we have that m|CD and
CD=DE=EA so m|EA. It follows that CA = CE+EA and m|CE. We have
therefore proved that m|CE and m|CD therefore we have an isosceles-right triangle
with smaller dimensions than before whose lengths are divisible by m. We can
continue this construction for a smaller yet isosceles-right triangle, but eventually
we will reach a value for a side that is less than m. Since m cannot divide a
number less than itself we can see that our original assumption m|AC and m|CB is
false and sides AC and CB are incommensurable. By the Pythagorean theorem we
can show this ratio is .

Calculating Square Roots

Now some of you might be wondering, how did they find the values of square
roots, if they didn’t have calculators? There are actually many different ways that
they had available, using either algebra or geometry.

The Greeks were very interested in geometry, therefore their way of finding square
roots involved constructing them on paper using a compass and straightedge. Let’s
use their method to find the square root of
7. First we draw a line AB of length 7.
We then make C so that AC = 1. Either by
calculation or construction we make D
which is the midpoint of BC and then draw
a circle with D as the center and the radius
as DB or DC. Lastly, we make a
perpendicular line from A which intersects
the circle at E. The length of AE is now
the square root of 7. This method actually
comes from the geometric mean of two
numbers. Usually the geometric mean of 2
and 5 for example is . In the picture
it would be expressed as . We had CA = 1 so we were left with just
the square root of length AB.

 

Around 1655 the popular method of computing the square root involved a
continued fraction. In order to calculate the square root of 2
you would take compute the expression on the right. This
infinite fraction would get closer and closer to the square root
of 2 as you continued on the calculation. This is usually
represented as {1, 2, 2, 2…}. It can be shown that all integer
square roots can be represented in this way and their form with be periodic, that is
the number sequence in the chain of fractions will repeat. For example the square
root of 3 is represented as {1, 1, 2, 1, 2, 1, 2…}. This method worked very well,
but we not near as efficient as the next method.

Isaac Newton developed a method that is very commonly used today called simply
Divide and Average. This is a recursively defined function where
is some guess. This method is very efficient in quickly calculating a square
root. The guess is not so important in coming up with the final result. Often it
takes only 3 or 4 iterations before the fraction covers 8 or 9 digits of accuracy.

Next is the question that many people wonder quite frequently. How does my
calculator do the square root? Does it use one of these methods? Actually your
 calculator has its own way. Your calculator has very well defined functions for
and . We combine the identities and to get

Lastly we have roots of imaginary numbers. The imaginary number i is defined as .
We use this definition to take the square root of negative numbers, for
example -36. We initially learn in school that we cannot evaluate , but by
using our definition we can rewrite this as. Another
way you can think of it is to simply take the square root of the absolute value and
then multiply it by i. We can use any of the methods we already know to do this.

These methods cover how to take the square root of any real number or imaginary
number in a simple way, but how do we take the root of a complex number? I
asked a group of college mathematicians and they did not have an answer for me.
This is rarely covered in secondary or even post-secondary education, but it still
make you wonder “How do you do it?” We actually go back to our original
definition of the square root as the inverse of the function . Suppose we want
to find the square root of 5 + 6i. We want to find some y = a + bi such that

Expanding we get . Our real part is
and the imaginary part is. We can then solve these two
equations for a and b which gives us our solution(s). We can also do this
computation in polar coordinates by knowing that and by a theorem we
get. Polar coordinates also makes it easier to extend this idea
into cube, quartic, and nth roots.

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