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袁萌  陈启清  12月30日

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Hyperreal Calculus MAT2000 –– Project in Mathematics

Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen

Abstract This project deals with doing calculus not by using epsilons and deltas, but by using a number system called the hyperreal numbers. The hyperreal numbers is an extension of the normal real numbers with both innitely small and innitely large numbers added. We will rst show how this systemcanbecreated,and thenshowsomebasicpropertiesofthehyperreal numbers. Then we will show how one can treat the topics of convergence, continuity, limits and dierentiation in this system and we will show that the two approaches give rise to the same denitions and results.

Contents

1 Construction of the hyperreal numbers 3

1.1 Intuitive construction . 3

1.2 Ultralters . . . . . . . . . . . 3

1.3 Formal construction . . . . . . . . . . . . . . . . 4

1.4 Innitely small and large numbers . . . . . . . 5

1.5 Enlarging sets . . . . . . . . . . 5

1.6 Extending functions . . .. . . . . 6

2 The transfer principle 6

2.1 Stating the transfer principle . . . . . . . 6

2.2 Using the transfer principle . . . . .  . . . . . 7

3 Properties of the hyperreals 8

3.1 Terminology and notation . .. . 8

3.2 Arithmetic of hyperreals . .  . . 9

3.3 Halos . . . . . . . . . . . . . . . . . 9

3.4 Shadows . . . .  .  . . . . . . . . . 10

4 Convergence 11

4.1 Convergence in hyperreal calculus. . . . .. . . . 11

4.2 Monotone convergence . . . 12

5 Continuity 13

5.1 Continuity in hyperreal calculus . . . . . . . . . . . . 13

5.2 Examples . . . . . .  . . . . 14

5.3 Theorems about continuity.  15

5.4 Uniform continuity . . ... 16

6 Limits and derivatives 17

6.1 Limits in hyperreal calculus . .17

6.2 Dierentiation in hyperreal calculus . . . . . . .. . 18

6.3 Examples . . . . . . . . . . 18

6.4 Increments . . . . .  . . . 19

6.5 Theorems about derivatives . 19

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