First things first, let us see clearly what is a square root and what its algebraic properties are. Suppose that there is the equation: x2=4, which, normally, has two solutions: x=2 and x=-2. These are numbers whose square equals 4. However, if you consider the equation x2=3 and draw a graph of the function y=x2 you will see for sure that in this case we also have two solutions, one positive and the other negative, which are not integers. Moreover, these solutions are not rational. In order to deal adequately with such irrational solutions, we need to introduce a special symbol of the square root. Again, what is a square root ? Arithmetic square root √a is a non-negative number which square equals a≥0. In case of a<0, we have an undefined mathematical expression as there does not exist real number whose square is a negative number.

What is, precisely, a mathematical problem and how come that it differs from mathematical questions? Of course, students who are already familiar with such advanced academic things as a case study template have perceived so far, probably, what is the main difference between a problem and a question. The latter may be asked in full expectation that the interlocutor will give the right answer after a short delay and little mental stretch. Hence, ‘Is it raining outside?’ represents a classic question that implies a fast and simple solution. However, we start with a problem only if we expect that it could entail a mental stretch or adequate computational resources that are required to cope with the problem and come up with a suitable and irrefragable answer. Indubitably, it is not possible to apply any simple and easy-going categorization to problems. They are so much different from the system of straightforward question-asking that first was coined by Aristotle in his Metaphysics. The posing of problems could take an awful lot of inventiveness and imagination, even if one of them bears a resemblance of simplicity; ‘What is a square root ?’, nevertheless, cannot be ascribed to the system of ‘How?’, ‘What?’, ‘Why?’ and ‘By what means?’ questions. Mathematical problems appertain to the variegated class of riddles, to the mainstays of the lecture room, and quite a big pile of toilsome labor waits for every student who is in need to work out standard academic problems bot set by himself or herself and by others.

Usually, the best and most precisely formulated mathematical problems appear as just solve this task. That is, you might be bending over your desk trying to do what looks like a simple requirement – and an hour later it becomes your worrisome plight and burden for the rest of your life. That the simplicity of the statement of problem does not come along with the simplicity of the answer would be your first-rate life lesson; that is a gentle mathematical irony afoot. Sometimes, even your most sophisticated knowledge, such as how to write a philosophy paper cannot be much of an asset to you – all the terseness of an equation does not promise you an analogous short term for finding its solution. If the problem is especially hard and good, a solution that crowns it represents nothing more than a letter of introduction to another level of thinking and interaction with this problem. It means that solving the problems allows you to shift your attention to something that you just has not achieved so far – and the solution takes you to an even deeper ground of question-asking and answer-seeking.

Predominantly, the square root represented a big mystery and an unsolvable dilemma to the Pythagoreans. They knew for a fact that √2 cannot be expressed as a fraction – and today we can give √2 to any degree of accuracy that pleases us (actually, it is not a problem to give it to 999 decimal places). Nevertheless, the Pythagorean believed that Number is the basis and the very building block in the hands of cosmic powers. Moreover, there is a very important proportion in geometry – to be exact, the ration of the diagonal of a square to its side – which also cannot be expressed as a ratio of whole numbers. In order to understand what is a square root, we need to provide an elegant and brief proof why √2 is not expressible as the ratio of one whole number to the other.

Ultimately, the five steps of our proof are as follows:

- step #1. Suppose that √2 is expressible as a ratio of whole numbers all the same. Using A and B as whole numbers, we can put an equation into the following form: √2=A/B. In addition, we can assume that A and B in our equation have no common factors (this still leads us to an obvious contradiction as long as this equation cannot hold); there surely are no common factors because else-wise it may be possible to simplify the said fraction by dividing both denominator and numerator. One way or another, it is necessary for us to expose and prove a contradiction in order to show what is a square root.
- step #2. As it has been said, we cannot divide the fraction by the common factor; moreover, it is surely not the case that both A and B are even numbers. Therefore, we take our second step by squaring both sides of the equation: (√2)2=A2/B2. As the continuation, we can rewrite the equation in the following way: 2=A2/B2. All-in-all, the aforementioned arithmetic is simple enough to be understood by a student who knows what is a lab report format.
- step #3. Multiplying the equation in the last form by the quantity B2 we can show that A is even: 2B2=A2. Thus, the square of A is even (because we know that the square of any odd number is obligatorily odd). This should make us think that A itself is even too.
- step #4. Now we can demonstrate that B is, analogically, even. It is possible to express the evenness of B by writing A as twice a whole number, which we will call C: A=2C. Consequently, this gives us the following equation: 2B2=4C2 and both sides of the latter can be divided by 2: B2=2C2. The puzzling part of this outcome is the similarity of the resulting equation to 2B2=A2. We should be convinced that B2 is even, nonetheless.
- step #5. Here is where steps 2 and 3 taken together demonstrate the sought contradiction. Speaking more precisely, the very assumption that √2 can equal A/B has as a result, a nonsensical conclusion: that both denominator and numerator are even numbers. Assuming the square root of 2 is fully expressible as the ratio of one whole number to the other we have revealed that both A and B cannot be even in an equation √2=A/B.

As it has been mentioned on the beginning of the article, any positive number has a square root alias SQRT (especially, when you type commands for a computer program). Several decades ago, more or less, students in schools and higher institutions were taught to compute the square root of numbers by hand. Nowadays we can be relieved from this burden thanks to calculators and computer software. Thus, first and foremost, our main definition and the answer to what is a square root is as follows: the square root of a positive number N in the equation X2=R is the positive number that is the solution to the said equation. Although, there exists also another number whose square equals N and this number is the negative of √N. Turning back to what was previously pointed out, we can denote that √N is positive by providing the following equation: (√N)2=N.

It will be, undoubtedly, very easy for any student who has worked at least once on a research proposal to comprehend perfectly all the particularities of the second solution, which is a negative one. Negative numbers, those peculiar quantities invented by Indian mathematicians over a millennium ago, are, of course, "symbolized debits." If you owe 5 dollars to someone, you set down in your ledger book: -5. To put it briefly, we can say that: (-√N)2=N. This should remind you of the quadratic formula that was well known to people as ancient as Babylonians and that you, probably, met in high school math classes. The purported aim of this formula is to help you find that number whose square equals any specific multiple of itself minus (or plus) any specific number. However, it will be much easier for understanding to put the definition of the quadratic formula as the problem of finding the unknown X: X2+bX+c=0. The man who invented this kind of equation is known as François Viète. This sixteenth-century mathematician was the first who came up with the idea of explicitly differentiating between X, Y… which are unknown values and values b, c… that are not yet specified. That is, in order to check what values of X you could put into the equation you should employ the quadratic formula, and each X will confirm that the answer is zero.

Giving way to our imagination, we could say that algebra itself resembles a huge house and this architectural analogy will present us with an overview of the main problems that are a hurdle for our understanding of what is a square root. Thus, we can see how some problems form the walls, joists and the beams and that not all of them are load-bearing to maximum capacity. Here we are presented with the problem of finding a square root, and, indeed, the quadratic formula shapes this problem as load-bearing one. All this is somehow reminiscent of writing an informative essay : you just need to solve the equation according to the keys that the formula provides you with, and the solution will support significantly more than its own weight – that is, having solved the equation it will be possible to generate solutions to other quadratic equations. Of course, if all the mathematical problems represent an utter and inexplicit mystery how to solve this or that equation, the knowledge how to obtain square roots alone will be of relatively little use. No wonder, in this case, that finding square roots and solving formulas are viewed as a highly valuable enterprise since all universal answers are of a great asset to seeking solutions to more general problems in the first place. The famous Italian mathematician Girolamo Cardano used to call the rules of algebra: ‘The great art… that surpasses all human… perspicuity of mortal talent…’ In these generous and grandiloquent words, Girolamo describes the science of math as a wonderful, universally egalitarian occupation. That is true, all mathematicians try to embrace everything that can be learned and the substance of this science can certainly be learned in the most demanding of senses. All math teachers have days that radiate with the optimism of knowing that mathematics is as accessible to everyone as the summer breeze and, even better than that, is already in everyone's repertoire. Mathematics is – and, we believe, always will be available to anyone who is eager to seek the truth and expand the worldly knowledge.

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