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- Mathematics In Europe - Mathematical Induction

that the truth of assertion A k for arbitrary k in N implies the truth of assertion A k 1 This is the induction implication So what does this process get us In fact a great deal If we have proved the base case and the induction implication we know that 1 A 1 is true 2 If A k is true for some k then A k 1 must also be true But from these two facts we can conclude that A 2 must be true But then from the induction implication and the truth of assertion A 2 we may conclude that A 3 must be true and so on so that we may in fact conclude that each of A 1 A 2 A 3 A 4 A 5 is true Note the Following In a proof by induction we are required to prove an implication that is we must prove that the given assertion holds with the parameter k 1 under the assumption that it holds for k We do not have to prove that the assertion itself is true for the given parameter k Both the truth of the base case of the induction and the induction implication must be satisfied Without a valid initial statement we may still assert that the truth of the assertion for k implies its truth for k 1 However that implication by itself does not suffice to prove the assertion for a single parameter k in N The initial step does not necessarily have to be proved for the parameter k 1 or k 0 It is sometimes the case that the assertion is false for small values of k and holds only from some larger value of k on However it is worthwhile to begin the induction from the smallest possible value of k To make an induction proof as clear as possible we emphasize the following steps Prove the base case State the induction hypothesis Prove the induction implication While we have chosen k as the name of our parameter it is just a variable and can be replaced with any other letter Many propositions have several variables and it is necessary to make clear which is the parameter for the induction Examples 1 Use induction to prove that the formula 1 3 5 2n 1 n 2 is true for all natural numbers n That is A n is the assertion 1 3 5 2n 1 n 2 Base case k 1 A 1 asserts simply that 1 1 2 which is clearly true Induction hypothesis We assume that A k 1 3 5 2k 1 k 2 holds for an arbitrary k in N Induction implication We must show that given the induction hypothesis that is the assumption that A k is true then A k 1 must also be true We prove this implication as follows 1 3 5 2 k 1 1 1 3 5 2k 1 1 3 5 2k 1 2k

Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/146-mathematical-induction (2013-11-18)

Open archived version from archive - Mathematics In Europe - Mathematical Induction (2)

few lines of print one can establish the truth of an infinite number of statements It is the key to almost all mathematical assertions that need to be proved for an infinite number of cases The Missing Induction Step Here we shall carry out the so called induction step for the proof of the summation formula given above We need to show that the sum of the first n 1 numbers that is 1 2 n n 1 is given by the above formula if we are given the induction hypothesis that the sum of the first n numbers is given by the formula 1 2 n n 1 Given the induction hypothesis we conclude that 1 2 n n 1 1 2 n n 1 n 1 But this expression is equal to 1 2 n 1 n 2 this is easily checked by algebraic manipulation and that is the summation formula for the case n 1 We have shown then that if one assumes the formula for n it must also be true for n 1 Where Does the Formula Come From Induction is the official way of validating assertions about an infinite number of natural numbers But before one can begin to prove an assertion one must have an assertion to prove How does one discover such things This is the creative aspect of mathematics One needs intuition experience luck and frequently a clever way of visualizing the problem Let us see how this works with our standard example the assertion about the sum of the first n natural numbers 1 2 n n n 1 2 We have seen the proof so now let us see where the formula came from There are many ways of arriving at the formula and we will look at two of them One possibility is to imagine the problem as a sum of areas We begin with the small square in the lower left hand corner We place two squares on top of it as shown then three all the way up to n squares in the top row The result is something like half a chessboard Now half a chessboard has area n n 2 but the area of our squares exceeds that of the half chessboard by half a square for each square on the diagonal We therefore need to add an area of 1 2 n and the result is that 1 n should equal n n 2 1 2 n And this expression is equal to n n 1 2 Alternatively we could proceed as the schoolboy Gauss did in the anecdote presented in Chapter We write the sum 1 n as 1 n 2 n 1 by collecting the first and last summands the second and second to last and so on Each term in the larger parentheses is equal to n 1 and the number of terms is n 2 if n is even If n is odd there are n 1 2 such

Original URL path: http://mathematics-in-europe.eu/tr/80-information/landscapes-concepts/586-mathematical-induction-2 (2013-11-18)

Open archived version from archive - Mathematics In Europe - Complex Numbers Are Not So Complex as Their Name Suggests

is 1 The spectacular part was this this new domain of numbers which contains the square root of every negative number has the surprising and pleasant property that in it all polynomial equations can be solved and it will not be necessary to expand into ever more complex number systems as the equations that we wish to solve become more complicated All of this was settled in the eighteenth and nineteenth centuries Since then mathematicians engineers and physicists have used complex numbers with the same ease and confidence that the ordinary person uses numbers like 3 and 12 Complex numbers can be pictured as points in the plane and in principle these numbers are no more problematic than the numbers that we work with every day So what good are they Complex numbers are as necessary to mathematicians engineers and physicists as negative numbers are to bookkeepers However it is true that it takes a bit of acclimatization before one is fluent with their manipulation and one certainly doesn t need them in solving the problems of daily existence It was certainly a marketing disaster calling such numbers complex and imaginary Such names give them an aura of mystical otherworldliness that they do not deserve However anyone who becomes confused by them is in good company Such irritation with the irrational is well described by Robert Musil in his novel The Confusions of Young Törless You did you understand that What The business about imaginary numbers Yes That s not so difficult You just have to realize that the square root of negative one is the unit of calculation But that s just the point There is no such thing Quite right but shouldn t one try in any case to apply the operation of taking the square root to negative numbers as well But how can you when you know and know with mathematical precision that it is impossible What You Really Need to Know about Complex Numbers You will get along just fine with complex numbers if you know the following facts They can be pictured in the plane Think of a point in the plane with the usual rectangular coordinates In Figure you can see depicted the point with coordinates 2 3 Figure 1 1 The complex plane We shall no longer speak of points in the plane but of complex numbers For a point with coordinates x y we write x y i For example the point in the figure is the number 2 3 i That may seem a bit mysterious but what is important is that one can write any point such as for example 12 14 as a number result 12 14 i and conversely one can determine the unique point associated with every complex number for example the number 3 2 5 i is associated with the point 3 2 5 It is easy to calculate with complex numbers Addition is defined as follows For example to add 2 3 i and

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/301-complex-numbers-are-not-so-complex-as-their-name-suggests (2013-11-18)

Open archived version from archive - Mathematics In Europe - Algebraic and transcendental numbers

more complicated a number z is algebraic if there is a polynomial of the form a 0 a 1 x a 2 x 2 a n x n with the following properties 1 The numbers a 0 a 1 a n are integers with a n not equal to zero 2 If z is substituted for x in this polynomial the result is zero Here is an example Let us denote the square root of 2 by y Then y is an algebraic number which can be seen by observing that y 2 2 0 That is x 2 2 is a polynomial with the required properties Fact If two algebraic numbers are added subtracted multiplied or divided the result is again an algebraic number We stated above that the algebraic numbers lie between the real numbers and the rational numbers in particular that every rational number is algebraic And indeed that is the case a rational number q can always be written as a fraction q m n where m and n are integers But then q is a root of the polynomial nx m and is thus algebraic Transcendental Numbers A real number that is not algebraic is called transcendental A counting argument can be used to show that transcendental numbers indeed exist there are uncountably many real numbers but the following argument shows that the set of algebraic numbers is countable Begin with a natural number n There are only countably many polynomials of degree n with integer coefficients since for each of the n coefficients one has only countably many choices This holds for every n and therefore there are altogether only countably many polynomials with integer coefficients One proceeds as in the proof of the countability of the rational numbers simply write down all first

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/157-algebraic-and-transcendental-numbers (2013-11-18)

Open archived version from archive - Mathematics In Europe - The number zero

Misyon Hoşgeldin Mesajları EMS Destekçiler Diller Iletişim Yasal Bilgi arama The European Mathematical Society Our Sponsor Munich RE The number zero Ayrıntılar Kategori landscape numbers For us there is no problem in calculating with the number zero just as we do with any other number The history of zero is rather involved Zero was recognized as a full fledged number only toward the end of the Middle Ages We shall not go into the historical question whether zero was invented in India or whether it is a Greek invention brought to India during the campaigns of Alexander the Great What is clear is that without zero our decimal system would not work For example the number 701 that is seven hundreds no tens and one one could not be so clearly represented without the number zero As a comparison the system of Roman numerals has no zero and consequently it can provide no simple methods of addition and multiplication Today one emphasizes that zero is the identity element for addition This means simply that 0 x x 0 x is always true In words adding zero to a number leaves the number unchanged There are several books devoted to the

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/150-the-number-zero (2013-11-18)

Open archived version from archive - Mathematics In Europe - The number Pi

is independent of the circle chosen the ratio of the circumference of a circle to its diameter is always the same And what is this ratio It turns out to be 3 1415926535 8979323846 where the dots indicate that the sequence of numbers continues indefinitely no matter how many decimal places you write out what you have will be only an approximation of the actual value It is possible to introduce the number without recourse to geometry There is an analytic definition which shows that also has something to do with oscillations Further information 1 By the definition of the circumference of every circle is equal to times its diameter or 2 times the radius As a formula C d 2 r Moreover the area A of a circle with radius r is equal to r 2 2 Since antiquity efforts have been made to express the number as precisely as possible Today more than 65 billion decimal digits of p are known 3 In the hierarchy of numbers belongs to the most complicated category To be precise in 1761 Lambert proved that is irrational and in 1882 Lindemann showed that it is in fact transcendental With Lindemann s result a two thousand year old problem was finally laid to rest given a line segment of unit length it is impossible using only straightedge and compass to construct a line of length Thus squaring the circle is impossible as well More precisely given a circle it is impossible to construct a square of the same area using only straightedge and compass 4 The number has taken on cult status Many people especially the young have become fascinated with everything having to do with this number you will find relevant Internet addresses in the literature cited below In 1999 there was

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/152-the-number-pi (2013-11-18)

Open archived version from archive - Mathematics In Europe - Euler

a limiting value namely the number 2 71828182845905 This number is so important that it has been given its own special symbol the letter e in honor of the Swiss mathematician Leonhard Euler Method 2 The number e arises in a natural way in growth processes Imagine a population of cells or organisms or anything else in which the rate of growth of the population is proportional to the existing population In particular if at time t there are f t individuals then the growth in the population over the next s time intervals will be a s f t here a is some constant that represents say the population s fertility We may write this as follows f t s f t s a f t And as s approaches 0 we have the formula f t a f t where f denotes the derivative of f It suffices now to consider the special case a 1 since the general situation differs only by a scaling factor We may consider then the following problem find a function f for which the relationship f f holds that is find a function whose derivative has the same value as the function itself and for which f 0 1 holds as well The latter condition is simply a practical normalization One can then prove that there exists a unique function with these properties It is called the exponential function and its value at a number x is denoted by exp x The number e is then the value of this function at the point x 1 We then have that exp x e x for all real numbers x Method 3 For concrete calculations it is often useful to know that exp x can be written as an infinite sum exp x 1 x x 2 2 x 3 3 recall that n n factorial stands for the product 1 2 n With the substitution x 1 we obtain e 1 1 1 2 1 3 Since the denominators in this sum become very large very quickly it is possible to approximate e very well using only a few terms Method 4 Those who understand what is meant by the area under a curve can define e as the value of x for which the area under the curve 1 x between 1 and x has the value 1 That this method yields the same value for e as in the previous methods follows from elementary rules of integration the indefinite integral of the function 1 x is the logarithm function whose value at x e is 1 Important information about e 1 The number e is undoubtedly one of the most important numbers in both pure and applied mathematics This has to do primarily with its significance in growth processes Properties of the exponential function including those in the domain of complex numbers were studied extensively by Euler and as we mentioned above the number e is so named in his

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/153-euler (2013-11-18)

Open archived version from archive - Mathematics In Europe - The complex number i

square root In this sense 1 414213 is the square root of 2 and 7 is the square root of 49 Since the square of every real number is positive or 0 no negative real number can have a real square root However it is possible to extend the domain of real numbers to the even larger domain of complex numbers in which all the real numbers do have square roots If you imagine the real numbers as points on a line say the x axis in a coordinate system then the complex numbers may be viewed as the set of all the points in the plane We may then define for complex numbers an addition which functions like vector addition and a multiplication which takes some getting used to so that all the usual properties of numbers are satisfied the commutative associative and distributive laws We note in particular that there is a complex number whose square corresponds precisely to the point 1 on the x axis The number that is associated with the coordinates 0 1 has this property It is called the imaginary unit and is denoted by the letter i Thus i is a square root of 1 Furthermore i this number corresponds to the point 0 1 is also a square root of 1 In contrast to the case of the real numbers here there is no obvious preference for one of the roots over the other Therefore mathematicians are not entirely happy about saying i is the square root of 1 just as in everyday speech we don t say that we ran into the brother of Irene if she in fact has two brothers Complex numbers are a fertile domain for the existence of square roots the equation x 2 a 0 always

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/154-the-complex-number-i (2013-11-18)

Open archived version from archive