The Natural Numbers and the Integers

These slides helped facilitate discussion about chapters 1 & 2 of Mark Chu-Carroll's book "Good Math" at the Boulder Pivotal Labs book club.

Transcript

1. The Natural Numbers and the Integers Good Math, Chapters 1 & 2
2. The Natural Numbers ℕ = { 0, 1, 2, 3, 4, 5, 6, … } Semantically, we usually think of natural numbers in two different ways: Ordinals and Cardinals
3. Semantics Ordinals describe the order of items in a group. This is the sixth circle in the row
4. Semantics Cardinals count the number of items in a group. There are six circles in this row
5. Axioms An axiom is an unproven premise that is accepted as true. Axioms are used as a starting point to prove interesting facts. Peano Arithmetic is a set of 6 axioms defining the natural numbers.
6. Peano Arithmetic Initial Value Rule: 0 is a natural number. Successor Rule: Every number a has a unique successor, denoted s(a).
7. Peano Arithmetic Predecessor Rule: Every number except 0 has a predecessor. If b = s(a), then a is b’s predecessor. Uniqueness Rule: No two numbers have the same successor.
8. Peano Arithmetic Equality Rules: Numbers can be compared for equality. Equality is: reflexive: a = a symmetric: a = b implies b = a transitive: a = b and b = c implies a = c
9. Peano Arithmetic Induction Rule: For some statement P, P is true for all natural numbers if the following are true 1. P is true about 0 (that is, P(0) is true). 2. If you assume P is true for a natural number n (P(n) is true), then you can prove that P is true for the successor s(n) of n (that P(s(n)) is true).
10. Adding Structure The set ℕ = { 0, 1, 2, 3, 4, 5, 6, … } models Peano Arithmetic, but so far the result isn’t very interesting. Let’s add some structure.
11. Addition We define an operation + on ℕ with these rules: a + b = b + a; a + 0 = a; s(a + b) = a + s(b). In particular, s(a) = s(0 + a) = s(0) + a = 1 + a.
12. Addition s(a) = 1 + a 7 + 3 = 7 + s(2) = 7 + 1 + 2 = 7 + 1 + s(1) = 7 + 1 + 1 + 1 = s(7) + 1 + 1 = 8 + 1 + 1 = s(8) + 1 = 9 + 1 = s(9) = 10
13. Multiplication We define an operation × on ℕ with these rules: a × b = b × a; a × 0 = 0; a × s(b) = (a × b) + a.
14. Multiplication a × s(b) = (a × b) + a 7 × 3 = 7 × s(2) = (7 × 2) + 7 = (7 × s(1)) + 7 = ((7 × 1) + 7) + 7 = ((7 × s(0)) + 7) + 7 = (((7 × 0) + 7) + 7) + 7 = 0 + 7 + 7 + 7 = 21
15. Induction Prove that for each natural number n: 0 + 1 + 2 + 3 + 4 + ··· + n = n(n + 1)/2 True if n = 0: 0 = 0(0 + 1)/2
16. Induction Suppose true for n: 0 + 1 + 2 + 3 + ··· + n = n(n + 1)/2. In order for the statement to be true, we must show it is true for s(n) = n + 1: 0 + 1 + 2 + 3 + ··· + n + n + 1 = (n + 1)(n + 2)/2.
17. Induction For s(n) = n + 1: 0 + 1 + 2 + 3 + ··· + n = n(n + 1)/2 0 + 1 + 2 + 3 + ··· + n + n + 1 = n(n + 1)/2 + n + 1 = n(n + 1)/2 + 2(n + 1)/2 = (n(n + 1) + 2(n + 1))/2 = (n + 1)(n + 2)/2
18. Induction 0 + 1 + 2 + 3 + 4 + ··· + n = n(n + 1)/2 def sum(n) return 0 if n == 0 n + sum(n - 1) end
19. The Integers ℤ = … -4, -3, -2, -1, 0, 1, 2, 3, 4, … Allow us to express direction, differences, etc. 4 + ? = 0
20. The Integers Require two additions to Peano arithmetic: Additive Inverse: Any number a has an inverse -a such that a + -a = 0. Inverse Uniqueness: a = b if and only if -a = -b.
21. Modeling Integers Intuitively, we believe that the integers model Peano arithmetic, but how can we be sure? Are our axioms consistent? Does a model even exist? How can we better understand the integers?
22. Constructing the Integers Let’s create a model! We’ll look at the set of all ordered pairs (a, b) of natural numbers. (a, b) means a - b
23. Constructing the Integers Certain pairs are equivalent to others (5, 3) and (7, 5) both ≈ 2 so (5, 3) and (7, 5) are equivalent, or (5, 3) ≈ (7, 5). ≈ behaves similar =
24. Constructing the Integers This set (mostly) follows Peano’s axioms: Initial Value Rule: (0, 0) is an integer Successor Rule: s(a, b) = (s(a), b) Predecessor Rule: now 0 has a predecessor the predecessor of (a, b) is (a, s(b))
25. Constructing the Integers This set (mostly) follows Peano’s axioms: Uniqueness Rule: using ≈ instead of = Equality Rules: ≈ is symmetric, reflexive, and transitive Induction Rule: ???
26. Constructing the Integers We can define addition on the integers: (a, b) + (c, d) = (a + c, b + d) so (9, 1) + (7, 2) = (9 + 7, 1 + 2) = (16, 3) 8 + 5 = 13
27. Constructing the Integers We can define subtraction on the integers: (a, b) - (c, d) = (a + d, b + c) so (9, 1) - (7, 2) = (9 + 2, 7 + 1) = (11, 8) 8 - 5 = 3
28. Constructing the Integers Then additive inverses of an integer (a, b) is (b, a), since (a, b) + (b, a) = (a + b, a + b) ≈ (0, 0).
29. Constructing the Integers The inverse of (a, b) may be written in many different ways, (b + 1, a + 1), (b + 2, a + 2), (b + 3, a + 3), … but each is equivalent to (b, a).
30. Next Time Chapters 3 & 4: Rational and Real numbers like ½, π, ⅞, and √2. Questions?