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Exploring Infinity in Real Arithmetic

Understand properties of infinity in real arithmetic, explore algebraic operations with infinity, and define extended real numbers. Learn about the Set of Real Numbers R, its definitions, and implications on arithmetic operations.

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Exploring Infinity in Real Arithmetic

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  1. Infinity and Beyond

  2. Lecture 1 Infinity in Real Arithmetic

  3. What is infinity()? • It’s a “number”  that is larger than all natural numbers, i.e. for all n{0,1,2,3,…}, n < . • Questions: • 1) Are there more than one infinity? • 2) Can we perform algebraic operations with ? • 3) Is  + 1 = ? Is  +  = ? Is  = ? • 4) How about   , /, 0? • 5) Can we write 1/ = 0? How about 1/0? • Answer: This depends on our definitions.

  4. Is  (or ) a real number? • We defined: For all n{0,1,2,3,…}, n < . • Likewise: For all n{0,1,2,3,…}, n > . • Question: Is  (or ) a real number? • Answer: This depends on our definition of real numbers. • Usually we define real numbers in a way that excludes both  and . • Question: What are real numbers?

  5. The Set of Real Numbers R: • Definition 1: This is the set of all points on the real line. • This definition is intuitive and visual, but faces the following problems: • What does a line mean? • Is a line physical? In this case are there infinitely many real numbers? • Does the line have a beginning and an end? I.e. do  and  belong to it?

  6. The Set of Real Numbers R: • Definition 2: This is the set of all decimal infinite sequences of digits (including a decimal point). • E.g.  = 3.14159265358979323846… • Also 1/3 = 0.3333333333333333333… • Caution: 1 = 1.000000… = 0.9999999… • This set does not contain a sequence like …9999.9999…. (infinite in both directions). • Thus, both ,R.

  7. The Set of Real Numbers R: • Definition 3: Instead of defining the set R, we define the structure (R,+,,<) by the following axioms: • 1) (R,+,) is a field, i.e. + and  satisfy the usual properties, e.g. x(y + z) = xy + xz. • 2) (R,<) is a linear order, i.e. for any x and y, either x < y or x = y or x > y, and the relation < is transitive, i.e. for all x,y, and z; x < y < z  x < z. • 3) < is congruent with respect to + and , i.e. for all x,y, and z; x < y  x + z < y + z. Also x < y and z > 0  xz < yz. • 4) Every nonempty subset of R that is bounded above, has a least upper bound.

  8. Do  and  belong to R? • Theorem: No. There is no real number that is larger than all natural numbers. • Proof: If there were such a number (called ), then the set N of all natural numbers is bounded above by . • Thus, using Axiom 4, we can get a least upper bound . • From 1 < 0, it follows that   1 < . • Since  is a least upper bound of N, it follows that   1 is NOT an upper bound of N. • Thus, there is an nN, such that  1  n. • It follows that  n + 1 < n + 2, contradicting the fact that  is an upper bound of N.

  9. The extended real numbers R* • In real arithmetic, we can choose to extend the set of real numbers R with the two new elements  and , i.e. we define: R* = R{,}. • Note: We have only one  and one . • (R*,<) is an extension of (R,<) by defining: • For all real numbers r,  < r <  , and  < . • Note: (R*,<) is a linear order as before, i.e. for any x and y, either x < y or x = y or x > y, and the relation < is transitive, i.e. for all x,y, and z; x < y < z  x < z.

  10. Algebraic Properties of  and  • We extend the algebraic operations by the following definitions: For all rR, pR+, nR, • r +  =  + r =  +  =  • r + () = () + r = () + () =  • p = p = n() = ()n =  = ()() =  • n = n = p() = ()p = () = () =  • 1/ = 1/() = 0 • Note: Since we can define x  y = x + (1)y, and also x/y = x(1/y), subtraction and division involving  and  can be defined.

  11. Example: Show that, for all rR, pR+, nR, • r   = ()  r = ()   =  •   r = r  () =   () =  • r/ = r/() = 0 • /p = ()/n =  • /n = ()/p = 

  12. Undefined Quantities involving  and  • The following quantities are left undefined: •  + (), () + , • and consequently   , ()  (), • 0, 0, 0(), ()0, • and consequently /, /(), ()/, ()/() • Also, 1/0 is still not defined (informally, we can not choose between + and ), • and consequently /0, ()/0 • Question: Why don’t we just define them as we like?

  13. Answer: • If we tried defining the previous undefined quantities, we will ruin the following theorem. • Theorem: Whenever defined, arithmetic expressions involving  and  obey Axioms 1 and 3 of Slide 8. • E.g., for all x,y,zR*=R{,}, the equation x + (y + z) = (x + y) + z holds, if both sides are defined. • Also, x < y and z > 0  xz < yz, etc.. • Proof: An exhaustive check of all properties.

  14. Defining the Undefined • What goes wrong if we defined  + () = 0, say? • Answer: The required properties will not hold anymore. If they did, we can derive the contradiction 0 = 1 as follows: • 0 =  + () = (1 + ) + () = 1 + ( + ()) = 1 + 0 = 1. • Exercise: Show that we can not define 0, without ruining Axiom 1. Hint: Use 0 = 1 + (1) • What about 1/0?

  15. Challenge! Give a definition of a set R** that contains all real numbers and (possibly many) infinities, with the operations of + and  defined for ALL elements, and obeying all of Axioms 1,2,3.

  16. Thank you for listening. Wafik

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