essay on real numbers

Real and Complex Numbers Essay

Introduction, real numbers, complex numbers.

All operations in mathematics depend on real numbers. It is, therefore, correct to assert that complex real numbers help in the derivation of complex numbers. A deep understanding of the real numbers and, indeed, complex numbers are crucial to carrying out any mathematical operations in any field.

To further understand the meanings attached to real and complex numbers, this essay will focus on brief but detailed comparisons between the two classes of numbers in mathematics. Additionally, the essay will use some graphical presentations of the theoretical explanations to enhance understanding.

A real number is any quantitative representation along a continuum (Denlinger, 2010, p. 3). The best demonstration of real numbers is the one-dimensional number line. Real numbers include integers, fractions that are rational numbers but not integers, decimals, square roots, and some algebraic expressions. The figure below illustrates some of the common real numbers.

Some of the most basic properties of real numbers include rationality and irrationality and algebraic or transcendental. Rational numbers can also be positive or negative, or zero. It is important to note that real numbers mainly measure continuous quantities. One of the distinctive properties of real numbers is the real number binomials. Through the binomial theorem, it is possible to prove that power ‘r’ is a real number using expressions such as the one below.

f (x) = (1+x) r = a 0 + a 1 x + a 2 x 2 + a 3 x 3 ….a k x k ……

Real numbers form an important part of complex numbers. According to Kasana (2005, p.2) complex numbers are pairs of ordered real numbers. Simply put, complex numbers consist of a real number and an imaginary number that needs mathematical solving for its completion. While real numbers are straight forward, complex number systems more often than not represent an expression of the form.

For instance, a+bi represents a complex expression where ‘a’ and ‘b’ are real numbers, and ‘i’ represents an imaginary unit whose value will help in solving the complex expression. Any number whose square is a real number less than zero is an imaginary number. The square of an imaginary number is always negative. When the imaginary number, for instance, ‘bi’ is added to a real number, the result is a complex number. It is therefore correct tom assert that imaginary numbers are non-zero complex numbers with zero as the real part. The following are some examples of complex numbers.

2 + 5 i , 6 +22 i , 12 – i

The ‘i’ in complex numbers are not real. The definition of the imaginary unit is as follows:

One of the most distinct differences between real numbers systems and complex number systems is the complexity of a complex number as perpetuated by its parts.

One of the most commonly used expressions in mathematics is a+bi=z. This expression makes the typical complex number. The real number part of the above expression is ‘a’, which mathematicians refer to as the real part of ‘z’. The other real number ‘b’ is the imaginary part of the complex number.

Despite their differences however, both real numbers and complex numbers take intractably interact in mathematical operations. While operations involving real numbers seem simple, operations involving complex numbers are quite involving. Operations involving both addition and subtraction of real and imaginary parts of the summand numbers make the basic procedure. The following expression illustrates the addition definition of operations involving complex numbers.

(a+bi) + (c+di) = (a+c) + (b=d)i

A graphical representation of for instance, the sum of 3 + i and –1 + 2i whose answer is 2 + 3i will be as follows;

The formula for multiplication of complex numbers mainly makes use of the square of the imaginary unit. The following expression shows the definition for the multiplication involving complex numbers (Kasana, 2005, p.2).

(a+bi)(c+di) = (ac-bd) + (bc+ad)i. For example,

The result of the multiplication between (3 + 2i) and (4 + 5i) will be as follows:

(3 + 2i)(4 + 5i) = (3 × 4) + (3 × (5i)) + ((2i) × 4) + ((2i) × (5i))

= 12 + 15i + 8i + 10i ²

= 12 + 23i -10 (Note that 10i ² = 10(-1) = -10)

Therefore, (3 + 2i)(4 + 5i) = 2+23i.

The following a graphical illustration of complex number multiplication of the equation (1+i)*(-1+i)=-2

One of the most dominant contrasts between real and complex numbers is their use and representation on the number line. Unlike real numbers, complex number systems enable the extension of the one dimension number line. While real numbers uses the conventional number line, complex numbers use the two-dimensional complex plane that the Argand diagram illustrates.

In a nutshell, any analysis of the two systems of numbers will yield many more differences and similarities whose origin is the operations involving the numbers in the two systems. This therefore was a theoretical approach that only gives a face value comparison.

Denlinger, C. (2010). Elements of Real Analysis . New York: Jones & Bartlett Learning.

Kasana, H.S. (2005). Complex Variables: Theory And Applications 2Nd Ed . London: PHI Learning Pvt.

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10.1: On the Nature of Numbers

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Learning Objectives

• On the Nature of Numbers: A Dialogue (with Apologies to Galileo)

Interlocuters: Salviati, Sagredo, and Simplicio; Three Friends of Galileo Galilei

Setting: Three friends meet in a garden for lunch in Renassaince Italy. Prior to their meal they discuss the book How We Got From There to Here: A Story of Real Analysis. How they obtained a copy is not clear.

Salviati : My good sirs. I have read this very strange volume as I hope you have?

Sagredo : I have and I also found it very strange.

Simplicio : Very strange indeed; at once silly and mystifying.

Salviati : Silly? How so?

Simplicio : These authors begin their tome with the question, “ What is a number? ” This is an unusually silly question, don’t you think? Numbers are numbers. Everyone knows what they are.

Sagredo : I thought so as well until I reached the last chapter. But now I am not so certain. What about this quantity \(\aleph _0\)? If this counts the positive integers, isn’t it a number? If not, then how can it count anything? If so, then what number is it? These questions plague me ‘til I scarcely believe I know anything anymore.

Simplicio : Of course \(\aleph _0\) is not a number! It is simply a new name for the infinite, and infinity is not a number.

Sagredo : But isn’t \(\aleph _0\) the cardinality of the set of natural numbers, \(\mathbb{N}\), in just the same way that the cardinality of the set \(S = \{Salviati,Sagredo,Simplicio\}\) is \(3\)? If \(3\) is a number, then why isn’t \(\aleph _0\)?

Simplicio : Ah, my friend, like our authors you are simply playing with words. You count the elements in the set \(S = \{Salviati,Sagredo,Simplicio\}\); you see plainly that the number of elements it contains is \(3\) and then you change your language. Rather than saying that the number of elements in \(S\) is \(3\) you say that the cardinality is \(3\). But clearly “ cardinality ” and “ number of elements ” mean the same thing.

Similarly you use the symbol \(\mathbb{N}\) to denote the set of positive integers. With your new word and symbol you make the statement “ the cardinality (number of elements) of \(\mathbb{N}\) is \(\aleph _0\). ” This statement has the same grammatical form as the statement “ the number of elements (cardinality) of \(S\) is three. ” Since three is a number you conclude that \(\aleph _0\) is also a number.

But this is simply nonsense dressed up to sound sensible. If we unwind our notation and language, your statement is simply, “ The number of positive integers is infinite. ” This is obviously nonsense because infinity is not a number.

Even if we take infinity as an undefined term and try to define it by your statement this is still nonsense since you are using the word “ number ” to define a new “ number ” called infinity. This definition is circular. Thus it is no definition at all. It is nonsense.

Salviati : Your reasoning on this certainly seems sound.

Simplicio : Thank you.

Salviati : However, there are a couple of small points I would like to examine more closely if you will indulge me?

Simplicio : Of course. What troubles you?

Salviati : You’ve said that we cannot use the word “ number ” to define numbers because this would be circular reasoning. I entirely agree, but I am not sure this is what our authors are doing.

Consider the set \(\{1,2,3\}\). Do you agree that it contains three elements?

Simplicio : Obviously.

Sagredo : Ah! I see your point! That there are three elements does not depend on what those elements are. Any set with three elements has three elements regardless of the nature of the elements. Thus saying that the set \(\{1,2,3\}\) contains three elements does not define the word “ number ” in a circular manner because it is irrelevant that the number \(3\) is one of the elements of the set. Thus to say that three is the cardinality of the set \(\{1,2,3\}\) has the same meaning as saying that there are three elements in the set \(\{Salviati,Sagredo,Simplicio\}\).

In both cases the number “\(3\)” is the name that we give to the totality of the elements of each set.

Salviati : Precisely. In exactly the same way \(\aleph _0\) is the symbol we use to denote the totality of the set of positive integers.

Thus \(\aleph _0\) is a number in the same sense that ’\(3\)’ is a number, is it not?

Simplicio : I see that we can say in a meaningful way that three is the cardinality of any set with . . . well, . . . with three elements (it becomes very difficult to talk about these things) but this is simply a tautology! It is a way of saying that a set which has three elements has three elements!

This means only that we have counted them and we had to stop at three. In order to do this we must have numbers first. Which, of course, we do. As I said, everyone knows what numbers are.

Sagredo : I must confess, my friend, that I become more confused as we speak. I am no longer certain that I really know what a number is. Since you seem to have retained your certainty can you clear this up for me? Can you tell me what a number is?

Simplicio : Certainly. A number is what we have just been discussing. It is what you have when you stop counting. For example, three is the totality (to use your phrase) of the elements of the sets \(\{Salviati,Sagredo,Simplicio\}\) or \(\{1,2,3\}\) because when I count the elements in either set I have to stop at three. Nothing less, nothing more. Thus three is a number.

Salviati : But this definition only confuses me! Surely you will allow that fractions are numbers? What is counted when we end with, say \(4/5\) or \(1/5\)?

Simplicio : This is simplicity itself. \(4/5\) is the number we get when we have divided something into \(5\) equal pieces and we have counted four of these fifths. This is four-fifths. You see? Even the language we use naturally bends itself to our purpose.

Salviati : But what of one-fifth? In order to count one fifth we must first divide something into fifths. To do this we must know what one-fifth is, musn’t we? We seem to be using the word “number” to define itself again. Have we not come full circle and gotten nowhere?

Simplicio : I confess this had not occurred to me before. But your objection is easily answered. To count one-fifth we simply divide our “ something ” into tenths. Then we count two of them. Since two-tenths is the same as one-fifth the problem is solved. Do you see?

Sagredo : I see your point but it will not suffice at all! It merely replaces the question, “ What is one-fifth? ” with, “ What is one-tenth? ” Nor will it do to say that one-tenth is merely two-twentieths. This simply shifts the question back another level.

Archimedes said, “ Give me a place to stand and a lever long enough and I will move the earth. ” But of course he never moved the earth because he had nowhere to stand. We seem to find ourselves in Archimedes’ predicament: We have no place to stand.

Simplicio : I confess I don’t see a way to answer this right now. However I’m sure an answer can be found if we only think hard enough. In the meantime I cannot accept that \(\aleph _0\) is a number. It is, as I said before, infinity and infinity is not a number! We may as well believe in fairies and leprechauns if we call infinity a number.

Sagredo : But again we’ve come full circle. We cannot say definitively that \(\aleph _n\) is or is not a number until we can state with confidence what a number is. And even if we could find solid ground on which to solve the problem of fractions, what of \(\sqrt{2}\)? Or \(π\)? Certainly these are numbers but I see no way to count to either of them.

Simplicio : Alas! I am beset by demons! I am bewitched! I no longer believe what I know to be true!

Salviati : Perhaps things are not quite as bad as that. Let us consider further. You said earlier that we all know what numbers are, and I agree. But perhaps your statement needs to be more precisely formulated. Suppose we say instead that we all know what numbers need to be? Or that we know what we want numbers to be? Even if we cannot say with certainly what numbers are surely we can say what we want and need for them to be. Do you agree?

Sagredo : I do.

Simplicio : And so do I.

Salviati : Then let us invent numbers anew, as if we’ve never seen them before, always keeping in mind those properties we need for numbers to have. If we take this as a starting point then the question we need to address is, “ What do we need numbers to be? ”

Sagredo : This is obvious! We need to be able to add them and we need to be able to multiply them together, and the result should also be a number.

Simplicio : And subtract and divide too, of course.

Sagredo : I am not so sure we actually need these. Could we not define “ subtract two from three ” to be “ add negative two to three ” and thus dispense with subtraction and division?

Simplicio : I suppose we can but I see no advantage in doing so. Why not simply have subtraction and division as we’ve always known them?

Sagredo : The advantage is parsimony. Two arithmetic operations are easier to keep track of than four. I suggest we go forward with only addition and multiplication for now. If we find we need subtraction or division we can consider them later.

Simplicio : Agreed. And I now see another advantage. Obviously addition and multiplication must not depend on order. That is, if \(x\) and \(y\) are numbers then \(x+y\) must be equal to \(y + x\) and \(xy\) must be equal to \(yx\). This is not true for subtraction, for \(3 - 2\) does not equal \(2 - 3\). But if we define subtraction as you suggest then this symmetry is preserved:

\[x + (-y) = (-y) + x\]

Sagredo : Excellent! Another property we will require of numbers occurs to me now. When adding or multiplying more than two numbers it should not matter where we begin. That is, if \(x\), \(y\) and \(z\) are numbers it should be true that

\[(x + y) + z = x + (y + z)\]

\[(x \cdot y) \cdot z = x \cdot (y \cdot z)\]

Simplicio : Yes! We have it! Any objects which combine in these precise ways can be called numbers.

Salviati : Certainly these properties are necessary, but I don’t think they are yet sufficient to our purpose. For example, the number \(1\) is unique in that it is the only number which, when multiplying another number leaves it unchanged. For example: \(1 \cdot 3 = 3\). Or, in general, if \(x\) is a number then \(1 \cdot x = x\).

Sagredo : Yes. Indeed. It occurs to me that the number zero plays a similar role for addition: \(0 + x = x\).

Salviati : It does not seem to me that addition and multiplication, as we have defined them, force \(1\) or \(0\) into existence so I believe we will have to postulate their existence independently.

Sagredo : Is this everything then? Is this all we require of numbers?

Simplicio : I don’t think we are quite done yet. How shall we get division?

Sagredo : In the same way that we defined subtraction to be the addition of a negative number, can we not define division to be multiplication by a reciprocal? For example, \(3\) divided by \(2\) can be considered \(3\) multiplied by \(1/2\), can it not?

Salviati : I think it can. But observe that every number will need to have a corresponding negative so that we can subtract any amount. And again nothing we’ve discussed so far forces these negative numbers into existence so we will have to postulate their existence separately.

Simplicio : And in the same way every number will need a reciprocal so that we can divide by any amount.

Sagredo : Every number that is, except zero.

Simplicio : Yes, this is true. Strange is it not, that of them all only this one number needs no reciprocal? Shall we also postulate that zero has no reciprocal?

Salviati : I don’t see why we should. Possibly \(\aleph _0\) is the reciprocal of zero. Or possibly not. But I see no need to concern ourselves with things we do not need.

Simplicio : Is this everything then? Have we discovered all that we need for numbers to be?

Salviati : I believe there is only one property missing. We have postulated addition and we have postulated multiplication and we have described the numbers zero and one which play similar roles for addition and multiplication respectively. But we have not described how addition and multiplication work together. That is, we need a rule of distribution: If \(x\), \(y\) and \(z\) are all numbers then \(x \cdot (y + z) = x \cdot y + x \cdot z\). With this in place I believe we have everything we need.

Simplicio : Indeed. We can also see from this that \(\aleph _0\) cannot be a number since, in the first place, it cannot be added to another number and in the second, even if it could be added to a number the result is surely not also a number.

Salviati : My dear Simplicio, I fear you have missed the point entirely! Our axioms do not declare what a number is, only how it behaves with respect to addition and multiplication with other numbers. Thus it is a mistake to presume that “numbers” are only those objects that we have always believed them to be. In fact, it now occurs to me that “addition” and “multiplication” also needn’t be seen as the operations we have always believed them to be.

For example suppose we have three objects, \(\{a,b,c\}\) and suppose that we define “ addition ” and “ multiplication ” by the following tables:

\[\begin{array}{c|c c c} + & a & b & c \\ \hline a&a&b&c\\ b&b&c&a\\ c&c&a&b\\ \end{array} \qquad \qquad \begin{array}{c|c c c} \cdot & a & b & c \\ \hline a&a&a&a\\ b&a&b&c\\ c&a&c&b\\ \end{array}\]

I submit that our set along with these definitions satisfy all of our axioms and thus \(a\), \(b\) and \(c\) qualify to be called “ numbers .”

Simplicio : This cannot be! There is no zero, no one!

Sagredo : But there is. Do you not see that a plays the role of zero – if you add it to any number you get that number back. Similarly b plays the role of one.

This is astonishing! If \(a\), \(b\) and \(c\) can be numbers then I am less sure than ever that I know what numbers are! Why, if we replace \(a\), \(b\) and \(c\) with Simplicio, Sagredo, and Salviati, then we become numbers ourselves!

Salviati : Perhaps we will have to be content with knowing how numbers behave rather than knowing what they are.

However I confess that I have a certain affection for the numbers I grew up with. Let us call those the “ real ” numbers. Any other set of numbers, such as our \(\{a,b,c\}\) above we will call a field of numbers, since they seem to provide us with new ground to explore. Or perhaps just a number field?

As we have been discussing this I have been writing down our axioms. They are stated below.

AXIOMS OF NUMBERS

Numbers are any objects which satisfy all of the following properties:

Definition of Operations : They can be combined by two operations, denoted “\(+\)” and “\cdot \).”

Closure : If \(x\), \(y\) and \(z\) are numbers then \(x + y\) is also a number. \(x\cdot y\) is also a number.

Commutativity : \(x + y = y + x\)

\(x \cdot y = y \cdot x \)

Associativity : \((x + y) + z = x + (y + z)\)

\((x \cdot y) \cdot z = x \cdot (y \cdot z)\)

Additive Identity : There is a number, denoted \(0\), such that for any number, \(x\), \(x + 0 = x\).

Multiplicative Identity : There is a number, denoted \(1\), such that for any number, \(x\), \(1 \cdot x = x\).

Additive Inverse : Given any number, \(x\), there is a number, denoted \(-x\), with the property that \(x + (-x) = 0\).

Multiplicative Inverse : Given any number, \(x \neq 0\), there is a number, denoted \(x^{-1}\), with the property that \(x \cdot x^{-1} = 1\).

The Distributive Property : If \(x\), \(y\) and \(z\) are numbers then \(x \cdot (y + z) = x \cdot y + x \cdot z\).

Sagredo : My friend, this is a thing of surpassing beauty! All seems clear to me now. Numbers are any group of objects which satisfy our axioms. That is, a number is anything that acts like a number.

Salviati : Yes this seems to be true.

Simplicio : But wait! We have not settled the question: Is \(\aleph _0\) a number or not?

Salviati : If everything we have just done is valid then \(\aleph _0\) could be a number. And so could \(\aleph _1, \aleph _2, \cdots\) if we can find a way to define addition and multiplication on the set \(\{\aleph _0, \aleph _1, \aleph _2, \cdots \}\) in a manner that agrees with our axioms.

Sagredo : An arithmetic of infinities! This is a very strange idea. Can such a thing be made sensible?

Simplicio : Not, I think, before lunch. Shall we retire to our meal?

Exercise \(\PageIndex{1}\)

Show that \(0 \neq 1\).

Show that if \(x \neq 0\), then \(0 \cdot x \neq x\).

Exercise \(\PageIndex{2}\)

Consider the set of ordered pairs of integers: \(\{(x,y)|s,y ∈Z\}\), and define addition and multiplication as follows:

Addition : \((a,b) + (c,d) = (ad + bc,bd)\)

Multiplication : \((a,b) \cdot (c,d) = (ac,bd)\).

• If we add the convention that \[(ab,ad) = (b,d)\] show that this set with these operations forms a number field.
• Which number field is this?

Exercise \(\PageIndex{3}\)

Consider the set of ordered pairs of real numbers, \(\{(x,y)|x,y ∈R\}\), and define addition and multiplication as follows:

Addition : \((a,b) + (c,d) = (a + c,b + d)\)

Multiplication : \((a,b) \cdot (c,d) = (ac-bd,ad + bc)\)

• Show that this set with these operations forms a number field.

The Real Numbers

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Toward the end of his distinguished career, the renowned British mathematician G.H. Hardy eloquently laid out a justification for a life of studying mathematics in A Mathematician’s Apology , an essay first published in 1940. At the center of Hardy’s defense is the thesis that mathematics is an aesthetic discipline. For Hardy, the applied mathematics of engineers and economists held little charm. “Real mathematics,” as he referred to it, “must be justified as art if it can be justified at all.”

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Module 1: Algebra Essentials

Introduction to real numbers, what you’ll learn to do: identify types of real numbers and use them in algebraic expressions.

Because of the evolution of the number system, we can now perform complex calculations using several categories of real numbers. In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions.

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• College Algebra. Authored by : OpenStax College Algebra. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution

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The Evolution and History of Numbers and Counting

History of numbers: essay introduction, the egyptians/babylonians number history, the hindu-arabic number history, the mayan number history, history of numbers: essay conclusion, works cited.

This paper explores the evolution of number system from ancient to modern. Here, you’ll find information on the development of number system of the Egyptians/Babylonians, Romans, Hindu-Arabics, and Mayans.

The evolution of numbers developed differently with disparate versions, which include the Egyptian, Babylonians, Hindu-Arabic, Mayans, Romans, and the modern American number systems. The developmental history of counting is based on mathematical evolution, which is believed to have existed before the counting systems of numbers started (Zavlatsky 124).

The history of mathematics in counting started with the ideas of the formulation of measurement methods, which the Babylonians and Egyptians used, the introduction of pattern recognition in number counting in pre-historical times, the organization concepts of different shapes, sizes, and numbers by the pre-historical people, and the natural phenomenon observance and universe behaviors. This paper will highlight the evolution history of counting by the Egyptians/Babylonians, Romans, Hindu-Arabic, and Mayans’ counting systems. Moreover, the paper will outline the reasons why Western counting systems are widely used contemporarily.

The need for counting arose from the fact that the ancient people recognized the measurements in terms of more or less. Even though the assumption of numbers based its arguments on archeological evidence about 50,000 years ago, the counting system developed its background from the ancient recognition of more and less during routine activities (Higgins 87). Moreover, ancient people’s need for simple counting in history developed odd or even, more or less, and other forms of number systems evolved into the current counting systems. The need for counting developed from the fact that people needed a way of counting groups of individuals through population increase by birth. In addition, Menninger asserts that the daily activities of the pre-historical people, like cattle keeping and barter trade led to the need for counting and value determination (105).

For instance, in order to count cows, prehistoric people used sticks. Collecting and allocating sticks to count the animals helped determine the total number of animals present. The mathematical history evolved from marking rows on bones, tallying, and pattern recognition, which led to the introduction of numbers. The bones and wood were marked, as shown below.

Moreover, the development of numbers evolved from spoken words by pre-historical people. However, the pattern of numbers from one to ten has been difficult to trace. Fortunately, any pattern of numbers past ten is recognizable and easily traceable. For instance, eleven evolved from ein lifon, which was used to mean ‘one left’ over by the prehistoric people. Twelve developed from the lif, which meant “two leftovers” (Higgins 143). In addition, thirteen was traced from three and four from fourteen, and the pattern continued to nineteen. One hundred is derived from the word “ten times” (Ifrah and Bello 147). Furthermore, the written words used by the ancient people, like notches on wood carvings, stone carvings, and knots for counting, gave a solid base for the evolution of counting.

The Incas widely used counting boards for record-keeping. The Incas used the “quip,” which helped the pre-historical people record the items in their daily lives. The counting boards were painted with three different color levels. These were the darkest parts, representing the highest numbers; the lighter parts, representing the second-highest levels; and the white parts, representing the stone compartments (Havil 127). In addition, the quip was used to do fast mathematical computations (Zavlatsky 154). Generally, the quip used knots on cords, which were arranged in a certain way to give certain numeral information. However, the quip systems of record keeping and information have been associated with several mysteries which have not yet been established. Examples of how the knots looked are shown below.

This form is the common system of counting and numbers used in the 21st Century. In India, Al-Brahmi introduced the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 (Menninger 175). The Brahmi numerals kept changing with time. For instance, in the 4th to 6th Century, the numerals were as shown below.

Finally, the numerals were later developed to 1,2,3,4,5,6,7,8,9 with time. The earliest system of using zero was developed in Cambodia. The evolution of the decimal points emerged during the Saka era, whereby three digits and a dot in between were introduced (Hays and Schmandt-Besserat 198). The Babylonians introduced the positional system, whereby the place value of the numerical systems was established. Moreover, the positional system by the Babylonians developed the base systems to the numerical, and the Indians later developed it further. The Brahmi numerals took different incarnations to develop, which resulted in the current number system (Higgins 204).

The Gupta numerals were one of the processes passed by the Hindu-Arabic number system to become the commonly used American number version. Currently, theories about the formation and development of the Gupta numerals remain debatable by researchers.

In addition, the Europeans adopted the Hindu-Arabic system through trading, whereby the travelers used the Mediterranean Sea for trade interactions (Havil 190). The use of the abacus and the Pythagorean dominated the European number evolution. The Pythagorean used “sacred numbers” even though the two systems diminished after a short while. With time, the Europeans borrowed the Hindu-Arabic number system to establish their mathematical number systems (Ifrah and Bello, 207). However, the process through which the Europeans adopted the Hindu-Arabic system has not been proven fully. It is believed that the Europeans adopted the Hindu-Arabic number system by relying heavily on it to build their current strong numerals (Higgins 210). For instance, the scope of the positional base system is quite large, which involves the conversion of different bases using the numerical number 10.

The Mayan civilization of counting and number systems developed in Mexico through ritual systems. The rituals were calendar calculations involving two ritual systems, one for the priests and the other for the ordinary civilians (Higgins 217). For instance, priestly calendar counting used mixed base systems involving numerical number multiples. The Mayan number systems form the base of mathematical knowledge. Moreover, the Mayan system of numbers used the positioning of numbers to allocate the place value of the combined digits (Havil 223).

The Mayans used the place value of numerical numbers, which were tabled to add and multiply numbers. Ultimately, the Hindu-Arabic and the Mayan number systems contributed highly to the evolution of numbers as opposed to the Egyptian/Babylonian number systems (Menninger 199). Nevertheless, the Western number system of counting and mathematics incorporated the strong features of all the other evolutions to get a standard solid number system. For instance, the American system, commonly used in most countries, uses decimal points, place values, base values, and Roman numbers from 1 to 10 (Ifrah and Bello 225). The figure below represents a sketch of the tabled digits by the Mayans.

The American version of numbers and counting used all the development features of the Mayans, Babylonians, Incas, Egyptians, and Hindu-Arabic systems to develop a reliable and universally-accepted number system (Hays and Schmandt-Besserat 214). This aspect is outstanding as it makes the American system stand out of all the number systems and counting. Nevertheless, the commendable work of the Mayans, Babylonians, Egyptians, and Indians cannot be underrated, as the historical trace of counting and number systems would be impossible without them.

The historical trace of number systems and counting covers a wide scope of pre-historical archeological evidence. Tracing ancient times by researchers poses a significant challenge in establishing counting and number systems. The research on number systems and counting has not yet been settled on the actual source information for evidence. Ultimately, the most effective number systems that led to the current dominant Western number system are the Mayans, Hindu, and Babylonian systems relying on the Incas’ developments. The prehistoric remains left mathematical evidence as stones and wood carvings, which led to the evolution of counting. Hence mathematical methodologies evolved. The methodology of research and arguments varies on the evolution of numbers. Consequently, there are no universally-accepted research findings on the mathematical and number systems evolution.

Havil, Julian. The Irrationals : A Story of the Numbers You Cant Count on, Princeton: Princeton University Press, 2014. Print.

Hays, Michael, and Denise Schmandt-Besserat. The History of Counting , Broadway: HarperCollins, 1999. Print.

Higgins, Peter. Number Story: From Counting to Cryptography, Gottingen: Copernicus, 2008. Print.

Ifrah, Georges, and David Bello. The Universal History of Number: From Pre-history to the Invention of Computer , Hoboken: Wiley, 2000. Print.

Menninger, Karl. Number Words and Number Symbols; Cultural History of Numbers, Mineola: Dover Publications, 2011. Print.

Zavlatsky, Claudia. Africa Counts; Number and Pattern in Africa Cultures, Chicago: Chicago Review Press, 1999. Print.

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Why Teaching Properties of Real Numbers is Important

The structure is all there in the lessons, but they're not over scripted.  Remember, I believe the majority of a lesson should be spontaneous.  It should be anticipated and prepared for, but how the lesson really unfolds depends on the audience.

Below you will find an overview of how and why I teach real numbers as well as two PowerPoint icons you can download and use as your own.  I only ask that you share where you found them.

Anything you purchase from Amazon.com through the banner below goes to producing more materials, and at no cost to you.

What Good Is It?

The Real Number Line has always been one of the dullest lessons I have to teach.

  Natural Numbers are the set of numbers you can count on your fingers, beginning with one.  The Whole Numbers are the Natural Numbers and Zero...Integers are ...

Blah Blah Blah

I have to teach it because it's in the curriculum.  And I always wonder, what use is it if a student knows the difference between a whole number and a natural number?

It is hypocritical of me to complain in such a fashion because I laud the virtues of education being greater than a set of skills or a body of knowledge.  Education is about learning to think, uncovering something previously unknown that ignites excitement and interest.  Education should change how you see yourself, how you think about the world.  It should enrich our lives.

Teaching the Real Number Line can be a huge first step in that direction, if done properly.

Math is About Ideas, Not Just Computation

There are some rich, yet entirely approachable, mathematical ideas that can be introduced with the Real Number Line (RNL).  For example, a series of questions to be posed to students could be:

•  The Natural Numbers are infinite, meaning, they cannot be counted entirely.  How do we know that?
• The Integers are also infinite.  How do we know that?
•  Is infinity a number?
• Which are there more of, Natural Numbers of Integers?  How can you know, if they're both infinite?

The idea of an axiom can be introduced.  Most likely, students assume math is true, or entirely made up, but correct or incorrect, because it is written in a book and claimed to be such by a teacher.  The idea of how we know what we know and if math is an invention or a discovery can be introduced by talking about axioms.  For example:

• Is it true that 5 + 4 = 4 + 5 ?
• If  a and  b  are Real Numbers, would it always be true that  a  +  b =  a  +  b ?  (What if they were negative?)
• Is it also true that  a  -  b = b - a ?  How do we know that?
• Is the following also true:  If  a = b,  and  b = c,  then  a = c ?  How do we know?

The idea here is not to teach students the difference between the Associative Property and the Commutative Property, but to use these properties to introduce students to math as a topic that can be discussed, and that it is not about answer getting, but instead about ideas.

For more on this topic and a few other related items, visit this page .

Why Are Some Rational Numbers Non-Terminating Decimals?

How it works is sometimes very clear and clean.  For example, 0.7 is said, "Seven tenths." And "Seven tenths," can also be written as the ratio of seven and ten.  And the number seven tenths is of course equal to itself, regardless of how it is written.  The number 0.27 is said, "twenty seven hundredths," which can easily be written as the ratio of twenty seven, for the numerator, and one hundred, as the denominator.  And this can continue so long as the decimal terminates.  But try the same thing with the a repeating decimal and you do not end up with things that are equal.

The algorithm to convert a repeating, but non-terminator decimal into a fraction is pretty straight forward .

But that does not address why a rational number would be a non-terminating decimal.

The simple reason that some rational numbers cannot be expressed as terminating decimals has to do with our numbering system.  We use base 10 numbers.  Our decimal system provides us an easy way to write fractions with denominators that are powers of 10.

That means that we have ten numbers, including zero, that fill up one column, like a car’s odometer.  When you travel 9 miles the odometer will read 000009.  When you travel the tenth mile the odometer will read 000010.

Not all of our number systems are base ten. While the metric system is base 10, or at least translates into powers of ten, the Imperial system is base 12 from inches to feet, but 5,280 feet to the next unit of miles, and so on.

Time is another great example of bases other than ten.  Seconds and minutes are base sixty.  You need sixty seconds before you have an hour, not ten.  But hours are base 24 because 24 hours are needed to make one of the next category, which is days.

But in base ten this is 0.4166666666666666... Our decimal system does math in base ten, not base sixty.  This is not 41 minutes!  A typical mistake would be two say 25 minutes is 0.25 of an hour.

Back to our original example of 1/3.  Not all numbers can be cleanly divided into groups of ten, like 3.  If we had a base 3 numbering system, where after the third number we moved, then 1/3 would just be 0.1.  But in our numbering system, 0.1 is one tenth.

Other numbers, like four, translate into ten more easily.  Consider the following:

The only issue remaining is that 2.5/10 is not a rational number because 2.5 is not an integer and rational numbers are ratios of two integers.  This can be resolved as follows:

Let's try the same process with 1/3.

As you can see, we will keep getting ten divided by three, forever.

This is a great example of how exploring a question can uncover many topics within the scope of the course being taught.

I hope this has caused you to pause and think of how exploring questions, relationships and properties in mathematics can lead to greater understanding than just teaching process and answer getting.

Here is a PowerPoint presentation you can download and use in your class.

The video below is a fun way to explore some of the attributes of prime numbers in a way that provides insight into the nature of infinity.   All of the math involved is approachable to your average HS math student.

Here is a link to the blog post that goes into a little more detail than offered in the video:   Click Here .

Click here to download a Power Point you can use in class.

Click here here download a PDF of the information covered in Real Numbers.

If you find these materials valuable, you could help me create more.

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Real Numbers Essay Example

• Pages: 3 (614 words)
• Published: May 1, 2017
• Type: Research Paper

In mathematics, a real number is a value that represents a quantity along a continuum, such as 5 (an integer), 3/4 (a rational number that is not an integer), 8. 6 (a rational number expressed in decimal representation), and π (3. 1415926535..., an irrational number). As a subset of the real numbers, the integers, such as 5, express discrete rather than continuous quantities. Complex numbers include real numbers as a special case. Real numbers can be divided into rational numbers, such as 42 and ? 23/129, and irrational numbers, such as pi and the square root of two. A real number can be given by an infinite decimal representation, such as 2. 4871773339... , where the digits continue indefinitely. The real numbers are sometimes thought of as points on an infinitely long line called the number line or real line.

Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in, ca. 600 BC, include what may be the first 'use' of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 50–690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. The Middle Ages saw the acceptance of zero, negative, integral and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects, which was made possible by the

development of algebra.

Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. The Egyptian mathematician Abu Kamil Shuja ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots. In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that here is no difference between rational and irrational numbers in this regard.In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that ? cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that ? s not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. Evariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory.Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873).

Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that ? is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David

Hilbert (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan.The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

In 1874 he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.

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Example of a Great Essay | Explanations, Tips & Tricks

Published on February 9, 2015 by Shane Bryson . Revised on July 23, 2023 by Shona McCombes.

This example guides you through the structure of an essay. It shows how to build an effective introduction , focused paragraphs , clear transitions between ideas, and a strong conclusion .

Each paragraph addresses a single central point, introduced by a topic sentence , and each point is directly related to the thesis statement .

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Other interesting articles, frequently asked questions about writing an essay, an appeal to the senses: the development of the braille system in nineteenth-century france.

The invention of Braille was a major turning point in the history of disability. The writing system of raised dots used by visually impaired people was developed by Louis Braille in nineteenth-century France. In a society that did not value disabled people in general, blindness was particularly stigmatized, and lack of access to reading and writing was a significant barrier to social participation. The idea of tactile reading was not entirely new, but existing methods based on sighted systems were difficult to learn and use. As the first writing system designed for blind people’s needs, Braille was a groundbreaking new accessibility tool. It not only provided practical benefits, but also helped change the cultural status of blindness. This essay begins by discussing the situation of blind people in nineteenth-century Europe. It then describes the invention of Braille and the gradual process of its acceptance within blind education. Subsequently, it explores the wide-ranging effects of this invention on blind people’s social and cultural lives.

Lack of access to reading and writing put blind people at a serious disadvantage in nineteenth-century society. Text was one of the primary methods through which people engaged with culture, communicated with others, and accessed information; without a well-developed reading system that did not rely on sight, blind people were excluded from social participation (Weygand, 2009). While disabled people in general suffered from discrimination, blindness was widely viewed as the worst disability, and it was commonly believed that blind people were incapable of pursuing a profession or improving themselves through culture (Weygand, 2009). This demonstrates the importance of reading and writing to social status at the time: without access to text, it was considered impossible to fully participate in society. Blind people were excluded from the sighted world, but also entirely dependent on sighted people for information and education.

In France, debates about how to deal with disability led to the adoption of different strategies over time. While people with temporary difficulties were able to access public welfare, the most common response to people with long-term disabilities, such as hearing or vision loss, was to group them together in institutions (Tombs, 1996). At first, a joint institute for the blind and deaf was created, and although the partnership was motivated more by financial considerations than by the well-being of the residents, the institute aimed to help people develop skills valuable to society (Weygand, 2009). Eventually blind institutions were separated from deaf institutions, and the focus shifted towards education of the blind, as was the case for the Royal Institute for Blind Youth, which Louis Braille attended (Jimenez et al, 2009). The growing acknowledgement of the uniqueness of different disabilities led to more targeted education strategies, fostering an environment in which the benefits of a specifically blind education could be more widely recognized.

Several different systems of tactile reading can be seen as forerunners to the method Louis Braille developed, but these systems were all developed based on the sighted system. The Royal Institute for Blind Youth in Paris taught the students to read embossed roman letters, a method created by the school’s founder, Valentin Hauy (Jimenez et al., 2009). Reading this way proved to be a rather arduous task, as the letters were difficult to distinguish by touch. The embossed letter method was based on the reading system of sighted people, with minimal adaptation for those with vision loss. As a result, this method did not gain significant success among blind students.

Louis Braille was bound to be influenced by his school’s founder, but the most influential pre-Braille tactile reading system was Charles Barbier’s night writing. A soldier in Napoleon’s army, Barbier developed a system in 1819 that used 12 dots with a five line musical staff (Kersten, 1997). His intention was to develop a system that would allow the military to communicate at night without the need for light (Herron, 2009). The code developed by Barbier was phonetic (Jimenez et al., 2009); in other words, the code was designed for sighted people and was based on the sounds of words, not on an actual alphabet. Barbier discovered that variants of raised dots within a square were the easiest method of reading by touch (Jimenez et al., 2009). This system proved effective for the transmission of short messages between military personnel, but the symbols were too large for the fingertip, greatly reducing the speed at which a message could be read (Herron, 2009). For this reason, it was unsuitable for daily use and was not widely adopted in the blind community.

Nevertheless, Barbier’s military dot system was more efficient than Hauy’s embossed letters, and it provided the framework within which Louis Braille developed his method. Barbier’s system, with its dashes and dots, could form over 4000 combinations (Jimenez et al., 2009). Compared to the 26 letters of the Latin alphabet, this was an absurdly high number. Braille kept the raised dot form, but developed a more manageable system that would reflect the sighted alphabet. He replaced Barbier’s dashes and dots with just six dots in a rectangular configuration (Jimenez et al., 2009). The result was that the blind population in France had a tactile reading system using dots (like Barbier’s) that was based on the structure of the sighted alphabet (like Hauy’s); crucially, this system was the first developed specifically for the purposes of the blind.

While the Braille system gained immediate popularity with the blind students at the Institute in Paris, it had to gain acceptance among the sighted before its adoption throughout France. This support was necessary because sighted teachers and leaders had ultimate control over the propagation of Braille resources. Many of the teachers at the Royal Institute for Blind Youth resisted learning Braille’s system because they found the tactile method of reading difficult to learn (Bullock & Galst, 2009). This resistance was symptomatic of the prevalent attitude that the blind population had to adapt to the sighted world rather than develop their own tools and methods. Over time, however, with the increasing impetus to make social contribution possible for all, teachers began to appreciate the usefulness of Braille’s system (Bullock & Galst, 2009), realizing that access to reading could help improve the productivity and integration of people with vision loss. It took approximately 30 years, but the French government eventually approved the Braille system, and it was established throughout the country (Bullock & Galst, 2009).

Although Blind people remained marginalized throughout the nineteenth century, the Braille system granted them growing opportunities for social participation. Most obviously, Braille allowed people with vision loss to read the same alphabet used by sighted people (Bullock & Galst, 2009), allowing them to participate in certain cultural experiences previously unavailable to them. Written works, such as books and poetry, had previously been inaccessible to the blind population without the aid of a reader, limiting their autonomy. As books began to be distributed in Braille, this barrier was reduced, enabling people with vision loss to access information autonomously. The closing of the gap between the abilities of blind and the sighted contributed to a gradual shift in blind people’s status, lessening the cultural perception of the blind as essentially different and facilitating greater social integration.

The Braille system also had important cultural effects beyond the sphere of written culture. Its invention later led to the development of a music notation system for the blind, although Louis Braille did not develop this system himself (Jimenez, et al., 2009). This development helped remove a cultural obstacle that had been introduced by the popularization of written musical notation in the early 1500s. While music had previously been an arena in which the blind could participate on equal footing, the transition from memory-based performance to notation-based performance meant that blind musicians were no longer able to compete with sighted musicians (Kersten, 1997). As a result, a tactile musical notation system became necessary for professional equality between blind and sighted musicians (Kersten, 1997).

Braille paved the way for dramatic cultural changes in the way blind people were treated and the opportunities available to them. Louis Braille’s innovation was to reimagine existing reading systems from a blind perspective, and the success of this invention required sighted teachers to adapt to their students’ reality instead of the other way around. In this sense, Braille helped drive broader social changes in the status of blindness. New accessibility tools provide practical advantages to those who need them, but they can also change the perspectives and attitudes of those who do not.

Bullock, J. D., & Galst, J. M. (2009). The Story of Louis Braille. Archives of Ophthalmology , 127(11), 1532. https://​doi.org/10.1001/​archophthalmol.2009.286.

Herron, M. (2009, May 6). Blind visionary. Retrieved from https://​eandt.theiet.org/​content/​articles/2009/05/​blind-visionary/.

Jiménez, J., Olea, J., Torres, J., Alonso, I., Harder, D., & Fischer, K. (2009). Biography of Louis Braille and Invention of the Braille Alphabet. Survey of Ophthalmology , 54(1), 142–149. https://​doi.org/10.1016/​j.survophthal.2008.10.006.

Kersten, F.G. (1997). The history and development of Braille music methodology. The Bulletin of Historical Research in Music Education , 18(2). Retrieved from https://​www.jstor.org/​stable/40214926.

Mellor, C.M. (2006). Louis Braille: A touch of genius . Boston: National Braille Press.

Tombs, R. (1996). France: 1814-1914 . London: Pearson Education Ltd.

Weygand, Z. (2009). The blind in French society from the Middle Ages to the century of Louis Braille . Stanford: Stanford University Press.

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An essay is a focused piece of writing that explains, argues, describes, or narrates.

In high school, you may have to write many different types of essays to develop your writing skills.

Academic essays at college level are usually argumentative : you develop a clear thesis about your topic and make a case for your position using evidence, analysis and interpretation.

The structure of an essay is divided into an introduction that presents your topic and thesis statement , a body containing your in-depth analysis and arguments, and a conclusion wrapping up your ideas.

The structure of the body is flexible, but you should always spend some time thinking about how you can organize your essay to best serve your ideas.

Your essay introduction should include three main things, in this order:

• An opening hook to catch the reader’s attention.
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• A thesis statement that presents your main point or argument.

The length of each part depends on the length and complexity of your essay .

A thesis statement is a sentence that sums up the central point of your paper or essay . Everything else you write should relate to this key idea.

A topic sentence is a sentence that expresses the main point of a paragraph . Everything else in the paragraph should relate to the topic sentence.

At college level, you must properly cite your sources in all essays , research papers , and other academic texts (except exams and in-class exercises).

Add a citation whenever you quote , paraphrase , or summarize information or ideas from a source. You should also give full source details in a bibliography or reference list at the end of your text.

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Real Numbers Questions

Real numbers questions are given here to help the students of Classes 9 and 10 for their exams. Real numbers is one of the important concepts from the examination point of view. This article will get real numbers questions based on the latest NCERT curriculum. Practising these questions will help you to improve your problem-solving skills.

What are Real Numbers?

Real numbers are the collection of all rational and irrational numbers . A unique real number corresponds to every point on the number line. Also, there is a unique point on the number line corresponding to each real number.

Also, check: Real numbers

Real Numbers Questions and Answers

1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:

(i) x 2 = 5 (ii) y 2 = 9 (iii) z 2 = .04 (iv) u 2 = 17/4

(i) x 2 = 5

On simplifying, we get;

Therefore, x is an irrational number.

(ii) y 2 = 9

Therefore, y is a rational number.

(iii) z 2 = .04

Therefore, z is a rational number.

(iv) u 2 = 17/4

u = ± √17/2

As we know, √17 is irrational.

Therefore, u is an irrational number.

2. Locate √9.3 on the number line.

Below are the steps to locate √9.3 on the number line.

Step 1: Draw a line AB = 9.3 cm

Step 2: From the point B, add 1 cm and mark it as C

Step 3: Mark the point of bisection by a compass and say it as ‘O’

Step 4: Taking AO as the radius, draw a semi-circle.

Step 5: From B, draw a perpendicular AB touching the semi-circle and mark as D

Step 6: Draw an arc on the number line by taking compass pointer on B and pencil on D

Step 7: The point which intersects the number line is the square root of 9.3, i.e. BE = √9.3

3. Find the value of a, if 6/(3√2 – 2√3) = 3√2 – a√3.

6/(3√2 – 2√3) = 3√2 – a√3….(i)

Consider LHS,

6/(3√2 – 2√3)

Rationalizing the denominator, we get;

= [6(3√2 + 2√3)]/ (18 – 12)

= 6(3√2 + 2√3)/6

= 3√2 + 2√3….(ii)

From (i) and (ii),

4. Simplify the following:

(i) (17) ⅕ (3) ⅕

(ii) (1 3 + 2 3 + 3 3 ) ½

= (17 × 3) ⅕

= (1 + 8 + 27)½

5. If a = 5 + 2√6 and b = 1/a, what will be the value of a 2 + b 2 ?

a = 5 + 2√6

= 1/(5 + 2√6)

= (5 – 2√6)/ (25 – 24)

= 5 – 2√6

a 2 + b 2 = (a + b) 2 – 2ab

= (5 + 2√6 + 5 – 2√6) 2 – 2(5 + 2√6)(5 – 2√6)

= 100 – 2(1)

6. Using Euclid’s division algorithm, find which of the following pairs of numbers are co-prime:

(i) 231, 396

(ii) 847, 2160

Let us find the HCF of each pair of numbers.

(i) 396 > 231

By Euclid’s division lemma,

396 = 231 × 1 + 165

231 = 165 × 1 + 66

165 = 66 × 2 + 33

66 = 33 × 2 + 0

Here, the remainder is 0.

Therefore, HCF(396, 231) = 33.

Hence, numbers are not co-prime.

(ii) 2160 > 847

2160 = 847 × 2 + 466

847 = 466 × 1 + 381

466 = 381 × 1 + 85

381 = 85 × 4 + 41

85 = 41 × 2 + 3

41 = 3 × 13 + 2

3 = 2 × 1 + 1

2 = 1 × 2 + 0

Therefore, HCF(2160, 847) = 1

Hence, the numbers 2160 and 847 are co-prime.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

From the given,

Time taken by Sonia to drive one round = 18 minutes

Time taken by Ravi to drive one round = 12 minutes

Suppose they both start at the same point and at the same time and go in the same direction, then the time taken by them to meet again at the starting point is the LCM of 18 and 12.

Prime factorization of 18 = 2 × 3 × 3 = 2 × 3 2

Prime factorization of 12 = 2 × 2 × 3 = 2 2 × 3

As we know, LCM is the product of the greatest power of each prime factor involved in the numbers.

So, LCM of 18 and 12 = 2 2 × 3 2 = 4 × 9 = 36

Hence, Sonia and Ravi will meet again at the starting point after 36 minutes.

8. Prove that one of any three consecutive positive integers must be divisible by 3.

Let n, n + 1, and n + 2 be the three consecutive positive integers, where n is an integer.

By Euclid’s division lemma, we have a = bq + r; 0 ≤ r < b

Let a = n and b = 3 such that n = 3q + r …(i)

Where q is an integer and 0 ≤ r < 3, i.e. r = 0, 1, 2.

Substituting r = 0 in (i), we get;

⇒ n is divisible by 3.

n + 1 = 3q + 1

⇒ n + 1 is not divisible by 3.

n + 2 = 3q + 2

⇒ n + 2 is not divisible by 3.

Substituting r = 1 in (i), we get;

⇒ n is not divisible by 3.

n + 1 = 3q + 1+ 1 = 3q + 2

n + 2 = 3q + 1 + 2 = 3q + 3 = 3(q + 1)

⇒ n + 2 is divisible by 3.

Substituting r = 2 in (i), we get;

n + 1 = 3q + 2 +1 = 3q + 3 = 3(q + 1)

⇒ n + 1 is divisible by 3.

n + 2 = 3q + 2 + 2 = 3q + 4 = 3(q + 1) + 1

Therefore, for each value of r such that 0 ≤ r < 3 only one out of n, n + 1 and n + 2 is divisible by 3.

Hence proved.

9. Show that 5 + √3 is irrational.

Let us assume that 5 + √3 is a rational number.

We can find coprime a and b (b ≠ 0) such that 5 + √3 = a/b.

√3 = (a/b) – 5

√3 = (a – 5b)/b

Since a and b are integers, we get (a/b) – 5 is rational, and so √3 is rational.

But this contradicts the fact that √3 is irrational.

This contradiction has arisen because of our incorrect assumption that 5 + √3 is rational.

Therefore, we conclude that 5 + √3 is irrational.

10. Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Let a be an odd integer and b = 6.

By Euclid’s division algorithm,

a = 6m + r for some integer m ≥ 0

Here, r = 0, 1, 2, 3, 4, 5 since 0 ≤ r < 6.

So, a = 6m or, 6m + 1 or, 6m + 2 or, 6m + 3 or, 6m + 4 or 6m + 5

Let us consider a = 6m + 1 or, 6m + 3 or 6m + 5 since we know that 6m, 6m + 2 and 6m + 4 are even integers.

For a = 6m + 1,

(6m + 1) 2 = 36m 2 + 12m + 1

= 6(6m 2 + 2m) + 1

= 6q + 1, where q is some integer and q = 6m 2 + 2m.

For a = 6m + 3,

(6m + 3) 2 = 36m 2 + 36m + 9

= 6(6m 2 + 6m + 1) + 3

= 6q + 3, where q is some integer and q = 6m 2 + 6m + 1

For a = 6m + 5,

(6m + 5) 2 = 36m 2 + 60m + 25

= 6(6m 2 + 5m + 4) + 1 = 6q + 1, where q is some integer and q = 6m 2 + 5m + 4.

Therefore, the square of an odd integer is of the form 6q + 1 or 6q + 3, for some integer q.

Video Lesson on Numbers

Practice Questions on Real Numbers

• If a = 2 + √3, find the value of a – 1/a.
• Show that 0.142857142857… = 1/7.
• Find three rational numbers between –1 and –2.
• Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
• Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628, leaving remainders 1, 2 and 3, respectively.

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Free Essays Science Mathematics Arithmetic Real Numbers And Number Sets

Essay About Real Numbers And Number Sets

• Algebra Arithmetic

Gmat Quantitave ExercisesFundamentals of the Quantitative SectionConcepts you must review and questions you must answerBEFORE the corresponding lessonNUMBER SETS1) Z+ = positive integers = { 1, 2, 3, 4, 5, …}2) = whole numbers = { 0, 1, 2, 3, 4, 5, …}4) Q =rational numbers={ : , }5) I = irrationals = { : }6) R = real numbers = Q U IDef: is called positive ifis called negative ifQuestions: a) Is the number 0 positive or negative?b) If x is a nonnegative number, what are all the possible values for x?Real number linex-1 0 1Exercise: Analyze the behavior of the powers and roots in the different intervals of the real number line indicated above.Students notesINTEGERSDef: is calledeven if , with , or(ii) odd if , withQuestions: a) Is 0 even?b) Is 3.5 odd?Consequences:Even + even = evenEven + odd = oddOdd + odd = evenEven even = evenEven odd = evenOdd odd = oddQuestion:The sum of six consecutive odd integers is

Consequences: If, if, then, or (1) is an odd number, then how does the sum of integers after those preceding (1) get truncated (see above question)Quizzes: The real numbers after one decimal point are not truncated (see Question above).The numbers after zero (in binary form) are not truncated.

Gmat Quantitave ExercisesFundamentals of the Quantitative SectionConcepts you must review and questions you must answerBEFORE the corresponding lessonNUMBER SETS1) Z+ = positive integers = { 1, 2, 3, 4, 5, …}2) = whole numbers = { 0, 1, 2, 3, 4, 5, …}4) \[ \int{Q^{-1}-4}} \rangle{\frac{(h^5/H)}{h^7} – 2} = \frac{\frac{(h^6/H)}{h^7}} 2.03 \]

[This number is exactly 7^7.7^8 . (The difference in the real numbers is a zero on first step as indicated by the parentheses for 1, 2, 3, 4, 5, …-3-6.)]Gmat Quantitave ExercisesFundamentals of the Quantitative SectionConcepts you must review and questions you must answerBEFORE the corresponding lessonNUMBER SETS1) Z+ = positive integers= { 1, 2, 3, 4, 5, …}2) = whole numbers= { 0, 1, 2, 3, 4, 5, …-3-6.)]

A2 | A3 | A4 | A5 | A6 | A7 | B8 = A[A7, C1, D2] (1) x : x (x = [x – 1.14] (1) x : x = [x [d(3)] = [f(3) x]) x (z = A[x2(-1)] (1) x : x = [x … (x – 1)] = [A[x2(3)] …(x2(-1)] = | | (z x [f(3))))/= C [F1, F2, F3, | (f[x[c((0 .. x)] (1 – 2)) x]) (f[x[c(1 .. x])(2 .. x)]](x1 : x = [x .. m[0-z] (2) x : ((0 .. x) ^ 5.75)] = & [F0, F1, F2,…, …, (0 .. x – 1)] (x2 : x (x [f(3) x]) (f[x[c(0 .. x)](1 .. x)]](x4 : x = [x2[f(3)] (4) x : ((0 .. x) ^ 5.75)] = & [F0, F1,, …, …, …, (

2)  (1 – (1 – c-1)).5)  0/10 (20 – 10).2)  (1 – (1 – c-1)).6) R=(20-30).1) (10 – 20).0) = 50  I=(40x)  (IxIx) = 25  (Ix – 2x).2)  I= (20x)  (R) * 8x = (xIx – 4x).9)  (20x) = 23x*(A+B-IxIxIxIxIxIxIxIxIxIxIxIx) + IxIxAxIx.6) Questions: Is there a positive or negative?How are all possibilities different?Exercise: Identify the first, and last possible real numbers from different times and find their values.The answers may be different and the examples may be different. (I)  (IxIx).8)  IxA = {IxA,0}; IxA2= {IxA2,1}; IxA * 11 = -11x; R=11.8 = R =11.6  (R + 1)  (IxA – 1);  (IxA2 – 11);  (-11 * IxA + 11)  (-11 * R – 1)  (A =   – 1);  (A2 =   – 16);  (-16 * IxA – 16)  (-16 * R – 1)  (A = IxA.6) Exercises: Evaluate the permutations of all possible real numbers and determine the odd combinations and the most likely of values.The results may not be identical.The permutations of, are the same regardless of what they are for the number being evaluated.Exercise: Evaluate two permutations (I and II) and find the result with 1.6. The results may not be the same depending on what the permutations of may be.It is possible to use the same permutation (is the number being evaluated and is the next iteration the same?)To figure out the total numbers and the average of their results, check out this video. If the answer should be an odd number, try the following.Get one answer for the prime number. Find the prime number for each of the permutations (I + I-II, etc.). Use the sum of that integer and the sum of the two numbers if the answer should be the mean prime number. If no answer should be specified, use “plus, minus, or even” as appropriate. The original question may simply have been incorrect.Exercise: Determine the sum of any of the permutations of the series.The result may not be exactly equal or different depending on which one the permutations should be for the numbers (I, II).Answer: If both integers should be equal, then use the same sum. The final

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Free Educational Essays. (October 3, 2021). Real Numbers And Number Sets . [Online]. Available at: https://www.freeessays.education/real-numbers-and-number-sets-essay/ [Accessed: 4-April-2024 ]