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Download e-book for iPad: A concise introduction to pure mathematics by Martin Liebeck

By Martin Liebeck

ISBN-10: 1498722938

ISBN-13: 9781498722933

Accessible to all scholars with a valid heritage in highschool arithmetic, A Concise advent to natural arithmetic, Fourth Edition provides the most basic and gorgeous principles in natural arithmetic. It covers not just general fabric but additionally many attention-grabbing subject matters now not frequently encountered at this point, akin to the idea of fixing cubic equations; Euler’s formulation for the numbers of corners, edges, and faces of a high-quality item and the 5 Platonic solids; using top numbers to encode and decode mystery info; the speculation of the way to check the sizes of 2 limitless units; and the rigorous thought of limits and non-stop functions.

New to the Fourth Edition

  • Two new chapters that function an advent to summary algebra through the idea of teams, overlaying summary reasoning in addition to many examples and applications
  • New fabric on inequalities, counting tools, the inclusion-exclusion precept, and Euler’s phi functionality
  • Numerous new workouts, with options to the odd-numbered ones

Through cautious causes and examples, this renowned textbook illustrates the facility and sweetness of uncomplicated mathematical ideas in quantity conception, discrete arithmetic, research, and summary algebra. Written in a rigorous but available type, it maintains to supply a strong bridge among highschool and higher-level arithmetic, permitting scholars to review extra complex classes in summary algebra and analysis.

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Extra resources for A concise introduction to pure mathematics

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It is also possible to subtract complex numbers: (a + bi) − (c + di) = a − c + (b − d)i and, less obviously, to divide them: provided c, d are not both 0, a + bi (a + bi)(c − di) ac + bd bc − ad = = 2 + i. c + di (c + di)(c − di) c + d2 c2 + d 2 For example, 1−i 1+i = (1−i)(1−i) (1+i)(1−i) = −2i 2 = −i. We write C for the set of all complex numbers. Notice that if a and b are real numbers, then (a + 0i) + (b + 0i) = a + b + 0i, and (a + 0i)(b + 0i) = ab + 0i, 39 40 A CONCISE INTRODUCTION TO PURE MATHEMATICS so the complex numbers of the form a + 0i add and multiply together just like the real numbers.

However, proving the existence of a cube root and, more generally, an nth root of any positive real number x is much harder and requires a deeper analysis of the reals than we have undertaken thus far. We shall carry out such an analysis later, in Chapter 24. 2, and state it here. 1 Let n be a positive integer. If x is a positive real number, then there is exactly one positive real number y such that yn = x. If x, y are as in the statement, we adopt the familiar notation 1 y = xn . 1 1 Thus, for example, 5 2 is the positive square root of 5, and 5 7 is the unique positive real number y such that y7 = 5.

1 > 0), a contradiction. Therefore x ≥ 0. 1(1) that 1x > 0. 6 Let x1 , x2 , . . , xn ∈ R, and suppose that k of these numbers are negative and the rest are positive. If k is even, then the product x1 x2 . . xn > 0. And if k is odd, x1 x2 . . xn < 0. 33 INEQUALITIES PROOF Since the order of the xi s does not matter, we may as well assume that x1 , . . , xk are negative and xk+1 , . . , xn are positive. 1, −x1 , . . , −xk , xk+1 , . . , xn are all positive. By (4), the product of all of these is positive, so (−1)k x1 x2 , .

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A concise introduction to pure mathematics by Martin Liebeck


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