0.9 = 1

0.9 = 1

Ne reamintim:

  • 1:10=0,11:10 = 0,1
  • 1:5=0,21:5 = 0,2
  • 1:4=0,251:4 = 0,25
  • 1:3=0,3333…=0,(3)1:3 = 0,3333\ldots = 0,(3)
  • 113:900=0,12555…=1,12(5)113:900 = 0,12555\ldots = 1,12(5)

Exerciții rezolvate:

  • 7.64  >  7.627.64 ~~ \boxed{>}~~ 7.62
  •   7.6  <  7.(6)~~7.6 ~~ \boxed{<}~~ 7.(6)

Exerciții propuse:

  •      1   ?   1.0~~~~~1~~ \boxed{~?~}~~ 1.0
  •   4.1   ?   4.10~~4.1~~ \boxed{~?~}~~ 4.10
  •   2.3   ?   2.(3)~~2.3~~ \boxed{~?~}~~ 2.(3)
  •      1   ?   0.(9)~~~~~1 ~~ \boxed{~?~}~~ 0.(9) - verifică cu formula
  • 0.25   ?   2.24(9)0.25 ~~ \boxed{~?~}~~ 2.24(9) - verifică cu formula

Formule magice (?)

  • 0.(5)=590.(5) = \dfrac{5}{9}
  • 0.12(5)=125−129000.12(5) = \dfrac{125-12}{900}

Demonstrații:

a) 0,(a)‾=a9\boxed{\overline{0,(a)}{} = \dfrac{a}{9}}

Demonstrație:
Notăm 0,(a)=x∣⋅100,(a)=x \qquad | \cdot 10
a,(a)=10⋅x∣−xa,(a) = 10 \cdot x \qquad | -x
a,(a)−0,(a)=9x⇒x=a9.a,(a) - 0,(a) = 9x \Rightarrow x=\dfrac{a}{9}.

b) a,(bc)‾=abc‾−a99\boxed{\overline{a,(bc)}{} = \dfrac{\overline{abc} - a}{99}}

Demonstrație:
Notăm a,(bc)‾=x∣⋅100\overline{a,(bc)}=x \qquad | \cdot 100
abc,(bc)‾=100⋅x∣−x\overline{abc,(bc)} = 100 \cdot x \qquad | -x
abc,(bc)‾−a,(bc)‾=99⋅x\overline{abc,(bc)} - \overline{a,(bc)} = 99 \cdot x
abc‾−a=99⋅x⇒x=abc‾−a9.\overline{abc}-a=99 \cdot x \Rightarrow x=\dfrac{\overline{abc}-a}{9}.