EDB — 29Q

6.6 Approximation of irrational numbers[29Q]

In the next exercises we will use the following definitions.

(We define \(𝜑(x)=x-⌊ x ⌋\), note that \(𝜑(3,1415)=0,1415\) but \(𝜑(-4,222)=0,778\) because \(⌊ -4,222 ⌋=-5\)).

E20

[0BV]↺↻

E20

[0BW]↺↻

E20

[0BY]↺↻

E20

[0C1]↺↻

E20

[0C3]↺↻

E20

[0C5]↺↻

6.7 Algebraic

We note that every rational \(𝛼=n/m\) is algebraic, as the root of \(p(x)=mx-n\). We want to show that algebraic numbers are a field.

E22

[0C9]↺↻

E22

[0CB]↺↻

E22

[0CC]↺↻

E22

[0CD]↺↻

E22

[0CF]↺↻

E22

[0CH]↺↻

E22

[0CK]↺↻

The above shows that algebraic numbers are a field.