Numbers

Question 1

What is counterintuitive about the quotients?

Answer 1

It has loads of gaps so Q cant be used to solve all equations

Question 2

What are the axioms which are the basis for the real numbers called?

Answer 2

Field axioms

Question 3

How are the field axioms organised?

Answer 3

4 Additive
4 Multiplicative
1 Distributive

Question 4

What are the first 4 field axioms

Answer 4

Question 5

What are the middle 4 field axioms?

Answer 5

Question 6

What is the last field axiom?

Answer 6

Question 7

State the order axioms?

Answer 7

Question 8

What is useful about O2?

Answer 8

It could be used to prove that 2 numbers are equal

Question 9

How is a>b defined?

Answer 9

Question 10

Define the maximum or minimum

Answer 10

For min it is just the other way around

Question 11

What is the idea behind proving that a set has at most 1 Maximum?

Answer 11

Uniqueness so use contradiction
Use the order axioms and definitions

Question 12

What is the definition of an upper and lower bound?

Answer 12

These are inclusive inequalities

Question 13

What is the definition of bounded above or below and the definition of bounded?

Answer 13

A set S is called bounded above if there exists an upper bound …

A set S is bounded if it is bounded above and below

Question 14

Define the supremum or Infimum
Aka

Answer 14

The least upper bound or greatest lower bound

Question 15

What is the point of the supremum and infimum?

Answer 15

They will nearly always exist (not true for max and min

Question 16

How do we write the supremum if it is unbounded above?
What about the infimum

Answer 16

SupS=infinity
InfS=-ve infinity

Question 17

How have we defined the sup and inf of ø?

Answer 17

Supø=-ve infinity
Infø= +ve infinity

Question 18

What is the completeness axiom?

Answer 18

Every non empty set of real numbers that is bounded above has a supremum.
Every non empty set of real numbers that is bounded below has a infimum

Question 19

Why did we define the sup of unbounded and the empty sets earlier?

Answer 19

So that in conjunction with the completeness axioms every case is covered

Question 20

What is the Archimedean postulate

Answer 20

Question 21

What is the idea behind the proof of the Archimedean postulate?

Answer 21

Show that the natural numbers can’t be bounded above

Question 22

How is an nth root of unity defined?

Answer 22

A number that satisfies the equation z^n=1 for all natural numbers n the