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Ch. 11: Cantor’s Infinity!

Ch. 11: Cantor’s Infinity!. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers ” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rationals ” R = {all real numbers} “the real numbers ”.

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Ch. 11: Cantor’s Infinity!

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  1. Ch. 11: Cantor’s Infinity!

  2. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q= {all quotients “a/b” of integers with b≠0} “the rationals” R= {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size?

  3. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q= {all quotients “a/b” of integers with b≠0} “the rationals” R= {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size? OLD-FASHIONED DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if either (1) they are both finite and have the same number of members, or (2) they are both infinite. According to this definition, all of the sets above have the same size.

  4. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q= {all quotients “a/b” of integers with b≠0} “the rationals” R= {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size? OLD-FASHIONED DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if either (1) they are both finite and have the same number of members, or (2) they are both infinite. According to this definition, all of the sets above have the same size.

  5. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size…

  6. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers?

  7. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING!

  8. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING! When comparing two infinite sets, Your situation is similar to this child’s. You don’t know how to separately count each set … so you should try matching!

  9. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING! When comparing two infinite sets, Your situation is similar to this child’s. You don’t know how to separately count each set … so you should try matching! We have the same number of fingers!

  10. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. We have the same number of fingers.

  11. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers.

  12. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers”

  13. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” WRONG ANSWERS: “Yes, because they are both infinite” “No, because N contains all of E’s members plus more.”

  14. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” YES! Formula: n 2n

  15. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” YES! Formula: n 2n How strange that an infinite set can have the same size as a subset of itself!

  16. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Main Goal: to decide which pairs of these sets have the same size N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q= {all quotients “a/b” of integers with b≠0} “the rationals” R= {all real numbers} “the real numbers”

  17. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}“the integers”

  18. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}“the integers” WRONG ANSWERS: “Yes, because they are both infinite” “No, because Z goes to infinity in two directions.”

  19. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}“the integers” WRONG ANSWERS: “Yes, because they are both infinite” “No, because Z goes to infinity in two directions.” “No, because my first attempt “nn” is not a valid one-to-one correspondence.”

  20. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …}“the integers” BIG IDEA: Match the evens in N with the positives in Z…

  21. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z…

  22. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  23. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  24. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  25. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  26. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  27. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  28. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} YES! (even n) (odd n) n/2 -(n-1)/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z.

  29. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0,1,2,3,4,5, …} YES! (even n) (odd n) n/2 -(n-1)/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. The matching looks simpler if we re-order the members of Z…

  30. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1,2, -2,3, -3,4, -4,5,…} (even n) (odd n) n/2 -(n-1)/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. The matching looks simpler if we re-order the members of Z…

  31. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). (We just proved that Z is countable) N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1,2, -2,3, -3,4, -4,5,…}

  32. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). (We just proved that Z is countable) N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1,2, -2,3, -3,4, -4,5,…} 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th … To prove that an infinite set is countable, we must find an infinite listing of its members {1st, 2nd, 3rd, …} which is organized so as to eventually include each member.

  33. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). (We previously proved that Eis countable) N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} E= {0, 2, 4,6, 8,10, 12,14, 16,18,…} 1st 2nd 3rd4th 5th 6th 7th 8th9th 10th… To prove that an infinite set is countable, we must find an infinite listing of its members {1st, 2nd, 3rd, …} which is organized so as to eventually include each member.

  34. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? To prove that an infinite set is countable, we must find an infinite listing of its members {1st, 2nd, 3rd, …} which is organized so as to eventually include each member.

  35. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? To prove that an infinite set is countable, we must find an infinite listing of its members {1st, 2nd, 3rd, …} which is organized so as to eventually include each member.

  36. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … Does this pattern eventually include every fraction?

  37. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … Does this pattern eventually include every fraction? NO, it only includes the positive fractions with numerator 1.

  38. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … 2/3, 2/4, 3/4, 2/5, 3/5, 4/5, 2/6, 3/6, 4/6, 5/6, Idea: insert the other numerators

  39. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … 2/3, 2/4, 3/4, 2/5, 3/5, 4/5, 2/6, 3/6, 4/6, 5/6, Idea: insert the other numerators (but skip the ones that aren’t reduced).

  40. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, … Does this eventually include all of the fractions?

  41. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, … Does this eventually include all of the fractions? No, it only includes positive fractions whose numerators are smaller than their denominators.

  42. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, … 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5, Idea: Insert the reciprocals.

  43. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? THIRD ATTEMPT: 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, 2/5, 5/2, 3/5, 5/3, … Idea: Insert the reciprocals. Does this eventually include all of the fractions?

  44. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? THIRD ATTEMPT: 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, 2/5, 5/2, 3/5, 5/3, … 0, -1/1, -1/2, -2/1, -1/3, -3/1, and so on… Idea: Insert the reciprocals. Does this eventually include all of the fractions? No, but all that remains is to insert zero and intersperse the negatives!

  45. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? FOURTH ATTEMPT: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … Does this eventually include all of the fractions?

  46. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N(the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1st rational, 2nd rational, 3rd rational, …} is a manner that’s organized so as to eventually include each rational? YES! FOURTH ATTEMPT: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th … Does this eventually include all of the fractions? YES! THEOREM: The set of rational numbers, Q, is countable.

  47. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF:

  48. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Like a computer spreadsheet, this grid extends indefinitely right and down.

  49. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Next, organize the cells of this grid into an infinite list by meandering through it: The list goes: 1/1, 2/1, 2/2, 1/2, 1/3, 2/3, 3/3, 3/2, 3/1, 4/1, 4/2, 4/3, 4/4, 3/4,…

  50. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Next, organize the cells of this grid into an infinite list by meandering through it: The list goes: 1/1, 2/1, 2/2, 1/2, 1/3, 2/3, 3/3, 3/2, 3/1, 4/1, 4/2, 4/3, 4/4, 3/4,… Finally, insert zero and intersperse the negative, as before. That’s it!

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