Page 162 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

and by the LAMP again, the sequence

√
x n = t n + 2

converges to

√ √
0+ 2= 2.

√
But 2 is not a rational number! So, what happens to the mathematical sequence ants? Where do they
go? Do they disappear into a void or some black hole? To save the marching mathematical ants from
falling into missing least upper bounds, we need the real numbers.

7.2 Real numbers and friendly rational numbers

The set of real numbers contains Q and has the least upper bound and greatest lower bound properties.
This means that any non-empty set of real numbers which is bounded above has a least upper bound
which is in the set of real numbers, and similarly, any non-empty set of real numbers which is bounded
below has a greatest lower bound which is also in the set of real numbers.

Definition 7.2.1 The set of real numbers, which we write as R, is the unique set which satisfies the following.

1. Q ⊂ R .

2. Every non-empty subset of R which is bounded above has a least upper bound in R.

Assumptions (3–7) below guarantee that the binary operations addition and multiplication
work the same way in R as they do in Q .

3. R is closed under the binary operations addition and multiplication, and these operations
satisfy the associative, commutative, and distributive properties.
4. For any x ∈ R ,

x +0 = x. (The additive identity 0isthe same in R as in Q.)

5. For any x ∈ R ,

1 ∗ x = x. (The multiplicativeidentity1 is thesamein R as in Q.)

6. For any x ∈ R , there exists −x ∈ R such that

x + −x =0. (There areadditive inverses in R just likein Q.)

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