EASIER THAN YOU THINK...

It seems so perfectly natural to square numbers to find areas, and to cube them to calculate volumes. It even seems reasonable to ask the questions in reverse … “If I have a square of area 16 square metres, or a cube of 27 cubic metres, what are their dimensions?”

Despite the innocent sounding nature of those questions, they opened up a whole new world of mathematics. Two and a half thousand years ago, the Pythagoreans asked, “If a square has an area of 2 square units, how long are its sides?” The answer astounded them! They had thought that everything in the universe was based on whole numbers, or ratios of whole numbers (fractions). They were able to prove/deduce that this new number was not even a fraction! It was a number of a kind that they could not understand.

Since it was not a fraction (a ratio), it was called an irrational number. Ir-rational simply means not rational or ‘not a fraction.’ In time, mathematicians discovered many other kinds of irrational numbers … but that is another story.

Obviously, since a new kind of number has been discovered, we needed to know more about its properties (just what is it?), if we could do anything with it, and whether it was useful or not. That quest continues. We are still learning, and there are surprises/epiphanies. It transpires, for example, that understanding these numbers and their connection with polynomials helps us prove why it is impossible to trisect an angle using a straight edge and compass.

The Radical Sign (Origin and Usage)

Since the days of the Pythagoreans, mathematicians have explored and studied this new kind of number.

Arabic scholars thought of these numbers as roots, meaning that these numbers were the orgin or source of the areas and volumes in much the same way as the roots of a tree are the source of its strength and nutrients and the root of a word is the origin of its spelling, structure and meaning. Scholars in the middle ages used the Latin term radix for these roots and, before symbols had been developed, simply wrote the word radix before a number to show that they were examining its root.

Eventually, in 1525, Christoff Rudolff (1499-1545) first used the symbol √ in his book Die Coss. Although no explanation remains as to why he did this, it appears likely that his symbol was a simplified way of writing the letter "r" for radix. His symbol was not universally accepted for some time, however, as many preferred using the letter "l" for the Latin latus (meaning the side of the square or cube).

For some time, if the root of a longer expression was required, parentheses were used.  For example the root of 2+4 would be written √(2+4). It was the famous French mathematician René Descartes (1596-1650) who introduced the idea of drawing a line over the expression as a grouping symbol ... in his book, La Geometrie (1637). This line is called the vinculum (the same name that is used for the line in fractions).

We still use this notation today.

We use roots as inverse functions.  In other words, finding a root is the inverse of finding a power ...

In each case, we speak of the "sixth root of 64," or the "fifth root of 32," etc.

Note that there are just two exceptions to this pattern.  Because multiplying a number by itself is how we find the areas of squares, we refer to the second root more commonly as "the square root" and, because we find the volumes of cubes by multiplying a number by itself three times, we refer to the third root more commonly as "the cube root."

Also, because the most common roots that you will be finding are square roots, the small superscript "2" is usually omitted from the radical sign (as I have shown in the illustration above).  Therefore, if you see a radical with no superscript, it means a square root!

Finally, we need to explain the terminology that we use.

The √ sign is called a radical sign.  The number or expression contained within it is called the radicand.  The bar across the top that acts as a grouping symbol and identifies the radicand (the expression that the radical applies to) is called the vinculum.  When we speak of "a radical," we may be referring to the √ sign only, or to the entire expression (the radical sign with the radicand included).

When we write a radical, such as √4, we mean the positive root only (if there is a negative root also).  Hence √4 = 2.  If we wish to indicate both roots, we must use the ± sign.  So x² = 4 would become x = ±√4 = ±2 since (2)(2)=4 and (-2)(-2)=4.  Just remember that all radicals indicate the positive root only and that any other signs must be added.

When you learn about complex numbers, and search for the nth root(s) of numbers, you will discover that there are n different such roots of any number (apart from zero).  In this case, the use of the radical indicates the principal (real) root of the radicand.

What is a surd?

In the mathematics syllabuses for schools in NSW the term surd is used in preference to the term radical. What is a surd and why would the school authorities use this term in their official documents?

First, the origins of this expression are, if anything, more ancient than the origins of radical.

I have mentioned that the Pythagoreans were the first to discover/identify the irrational numbers.  In subsequent years, Greek scholars like Euclid wrote of rational numbers using the term rhetos (ῥητῶς) which has the sense of explicit, stated, or spoken.  They wrote of irrational numbers using the term alogos (ἄλογος) which meant unreasonable, senseless, irrational, absurd (although it also has the sense of speechless as well).

During the ninth century, Arabic translators translated the Greek rhetos using the Arabic muntaq (made to speak), and the Greek alogos using the Arabic term jaðr açamm (deaf, dumb root).  The Persian astronomer, geographer and mathematician, Mohammed ibn-Musa al-Khwarizmi (c.780-c.850) wrote of rational and irrational numbers as sounded and unsounded.

When his (and other) works were translated into Latin during the 12th century, the word surdus (deaf, mute, dumb) was used.  It is interesting that this Latin word, surd, is still used in phonetics today for unvoiced consonants such as p, k, s and t (where the vocal chords are not used).  The meaning that is conveyed by the term is that a surd is a root that cannot be expressed (or spoken) as a rational number.

It appears that the Italian Gherardo of Cremona (c.1114–1187) was the first European to adopt this terminology.  The records show that he did so around 1150 while he was in Toledo, Spain, translating Arabic texts from local libraries into Latin.

Not many years later, in his Liber Abaci (1202) another Italian, Fibonacci, adopted the same term to refer to a number that has no (rational) root.

The English mathematician, Robert Recorde, used the term in his book The Pathwaie to Knowledge (1551) in which he taught Euclid's Elements: "Quantitees partly rationall, and partly surde."

Despite all this, there appears to be no general agreement concerning what a surd is (although there is good agreement over what a surd is not)!  A surd is not rational.  Therefore, not all radicals are surds.  For example, √4 is a radical, simply because it uses the radical (sign) ... yet it is rational because it is equal in value to 2.  Therefore, √4 is not a surd.  Similarly, the cube root of 27 is not a surd, because it is equal to three.

The confusion that we do have arises over two matters in particular ... whether a surd had to be irreducible or not (some scholars considered √6 not to be a surd since it could be written as √2⋅√3) ... and whether the radicand had, itself, to be rational.  Concerning the second matter, the Scottish mathematician George Chrystal (1851-1911) wrote in Algebra, 2nd ed. (1889) that "... a surd number is the incommensurable root of a commensurable number." He went on to explain that √e is not a surd and neither is √(1 + √2) since e and (1 + √2) are both irrational.

What are we to make of it all?  The schools in NSW do not really enter this debate since all their work appears to be done with rational radicands.  They speak of evaluating some radicals (where the result is rational) but of simplifying surds.  When used in this context (rational radicands), the use of the term surd seems quite reasonable.

For those who prefer to stay on less debatable ground, you might simply speak of radicals that are rational (can be evaluated) and those that are irrational (and can only be simplified).

How Do We Know That √2 is Irrational?

There are a number of proofs for the irrationality of 2.  A good summary of 28 different proofs can be viewed at the Cut the Knot site.

The classic proof that I am going to share with you is the same one that I was introduced to when I was in junior high school.  I remember being fascinated by its potency and how clever the method was.  It was my first encounter with this method called reductio ad absurdum.  It is a Latin term meaning reduction to absurdity or reduced to the absurd.  It is otherwise known as a proof by contradiction.

We begin by assuming the opposite of that which we wish to prove.  So, we assume that √2 is, in fact, a perfectly rational number.

Now, let's see where this thinking will lead us ... in other words, let's explore the consequences.

First of all, if it is rational, that means that we can find some fraction that is exactly equal in value.  Not only that ... because we know that every fraction has an infinite number of equivalent fractions, we will assume that we have found the simplest fraction that is exactly equal to √2.  Let's call it p/q.  Now, before we go on, we must think about p and q.  Because of what we have just shared, we know that p and q must both be whole numbers (integers).  We also know that they have no factors in common since, if they had common factors, they would have formed one of the equivalent fractions about which we spoke.  Therefore, if this fraction is in its simplest form, then we must agree that p and q have no factors in common (apart, of course, from one).  We say that p and q are relatively prime and can express this as (p,q)=1.

This means that we have now agreed that p/q = √2.

If we square both sides, we obtain p²/q² = 2

and, multiplying both sides by q², we obtain p² = 2q²!

Now, this is interesting.  2q² is obviously an even number.  Therefore, p² must be even.  This means that p must be even ... since the square of an odd number cannot be even.  So, we now know that one of the factors of p is 2.  Therefore, we can write p = 2k where k is also a whole number.

This interlude was vital, because it means that we can replace the p in our equation with 2k (since we know that p must be even).

Hence, p² = 2q² becomes (2k)² = 2q² and, therefore, 4k² = 2q² and 2k² = q².

Do you see what we have here?  We have now deduced, quite rationally, that q² = 2k².

This means that q must (also) be even!

So, we started by assuming that √2 could be written as p/q where p and q were whole numbers with no common factors.

We have now concluded that p and q must both be even, which means that they must have common factors!  These statements are obviously contradictory.  Either our argument was false (it wasn't ... we were very careful about that), or our initial assumption was false!

Our assumption must have been false!

Therefore √2 cannot possibly be written as a fraction (a ratio), and it is therefore irrational (not a ratio).

You have now seen that we have a new kind of number ... and this deserves further investigation.

How to Approximate the Square Root of a Number by Hand

If you are a student who is interested in becoming a well rounded mathematician I believe you should, at least once, calculate the square root of a number by hand. In this way you will see how the principles of algebra can help us understand and manipulate numbers better ... and that is always a worthwhile thing!

You will also be treading in the footsteps of mathematicians in history who had to make their calculations using techniques like these because they did not have access to electronic or mechanical calculators.

In this video I show you how to estimate the square root of a number and then calculate a second approximation that is even closer to the real value. This process can be repeated as many times as you wish to get even better approximations. Be warned, however ... the calculations get more difficult each time. I think doing it just once (or, maybe, twice) will be sufficient for you to satisfy yourself that the procedure works.

How to Find Cube Roots of Perfect Cubes With Ease

This is a fun activity. It is designed to help you develop a 'number sense' and to reduce your fear of dealing with large numbers.

I must confess, it is only rarely that you might be able to use this skill in a PRACTICAL way. This is because it is very necessary (for it to work) that you have a perfect cube to start with ... and how often are you likely to face that situation? This is not the point of the exercise, however.

By watching this video (and learning the skill that I explain and demonstrate) you will gain insights into how to analyse large numbers. You will also see how our understanding of place value helps us decide which parts of large numbers provide us with the information that we need!

Because students who depend on calculators usually fail to develop a good 'number sense' I think this is a very worthwhile activity ... along with estimation skills and the like. I hope you enjoy developing an active and analytical mind with this activity.

How to Calculate an Approximate Cube Root for Any Number

This is a more serious activity. It is a long video but, if you have a genuine interest in how the concepts of algebra and calculus can help us better understand and manipulate numbers, I think you should watch it. I certainly wish I had been shown this when I was at school.

Actually, I had encountered a lot of it in extracurricular reading and some of it in an advanced course during my last two years of school. Here I demonstrate three different, but related, ways of discovering the same approximation method and then (at the 22:33 mark, if you wish to skip the theory) demonstrate how to calculate the cube root of 31,217 to one decimal place fairly quickly. The reason this video is so long is that I believe it is important for students to understand how a concept/theory can be developed in a variety of different ways.

After you have watched this video, I challenge you to use the same principles to calculate a square root or a fourth or fifth root of some number.

Because calculators are so readily available now, this skill is no longer used in any practical sense. Historically, it was one of the methods used to calculate roots for the mathematical tables that were used before mechanical calculators were easy to obtain.

The reason for my presenting it is to help better students gain insights into how algebra, graphing and calculus can help us understand how to manipulate numbers. So much theory of numbers is developed this way that this is a very worthwhile video for the serious mathematics student.

How to Calculate a Cube Root Using the Division Method

In the first part of this video I demonstrate the method that I prefer for calculating cube roots.  There are other methods, but they are all "variations on the theme."  I then explain how the method works by discussing the binomial expansion of

(a + b)³ = a³ + 3a²b + 3ab² + b³.

If you have watched my earlier videos, you will see that the "theory" behind the method is essentially the same as that which lies behind the iterative method(s).

Because calculators are so readily available now, this skill is no longer used in any practical sense. Historically, it was one of the methods used to calculate roots for the mathematical tables that were used before mechanical calculators were easy to obtain.

The reason for my presenting it is to help better students gain insights into how algebra, graphing and calculus can help us understand how to manipulate numbers. So much theory of numbers is developed this way that this is a very worthwhile video for the serious mathematics student.

How to Approximate the nth Root of Any Number

In this video I explain the principles for approximating a fifth root and then generalise them to the finding of any root. I then calculate an approximation to the fifth root of 67 by hand. I recommend that you try this out for yourself (at least once).

From the beginning of my schooling I had struggled with Mathematics. Highly discouraged by my results, I grew to eventually hate Maths and lost all motivation to improve. Then part way through year seven my parents decided to send me to Crystal Clear Mathematics. Graeme was able to quickly identify areas I needed to improve and explain Maths in a way that finally made sense to me. I was able to understand Maths. This in turn then affected my confidence in my own ability.

Since working with Graeme I have gone from failing to significantly improved results. I now get As & Bs both in examinations and for assignment tasks. Graeme is patient, kind and his tutoring is individualised. He was so helpful and I am grateful for his help.

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