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1.2 Numbers

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Natural numbers N 1,2,3,…These are closed under Addition.
A set that can be put in correspondence with N or a subset of N is called countable.
Subtraction; makes us enlarge N to get  the Integers Z, positive or negative or 0.

Exercise 1.1  Is Z countable? Solution

To be closed under Division: we must enlarge Z to get the Rational Numbers Q, which are fractions, of the form Z / N.


1.2 Is Q countable? (see picture for hint) Solution

1.3 Prove or disprove: a countable set of countable sets is countable. Solution

Decimal form of numbers
All rational numbers repeat the same sequence of decimal digits forever.


1.4 Are there other decimal numbers? Solution

1.5 Are all decimal numbers countable? (see picture for hint) Solution

A number which differs from each number k in the kth decimal digit # cannot be on the list of numbers!

Algebraic numbers: these are solutions to polynomial equations, with integer coefficients.

Exercise 1.6 Are algebraic numbers countable? Solution

Real Numbers R = all decimal numbers,
You can add subtract multiply and divide them except dividing by 0 is not allowed.

Other numbers?
Numbers mod x
are remainders on dividing by x.
Complex numbers C are numbers of the form aib where i2 = -1 and multiplication and division are so defined, and a and b are in R.