Περιοδικοί αριθμοί §7.7 Α Γυμνασίου

1. x = 0, 2
x = 0,2222 …
10 ∙ x = 10 ∙ 0,2222 …
10x = 2,2222 …
10x = 2 + 0,2222 …
10x = 2 + x
10x − x = 2
9x = 2
x =
2
9
1η Περίπτωση
x = 1, 64
x = 1,646464 …
100 ∙ x = 100 ∙ 1,646464 …
100x = 164,646464 …
100x = 164 + 0,646464 …
100x = 164 + x − 1
100x − x = 164 − 1
99x = 163
x =
163
99

2. 2η Περίπτωση
x = 2,2954
x = 2,29545454 …
100 ∙ x = 100 ∙ 2,29545454 …
100x = 229,545454 …
100 ∙ 100x = 100 ∙ 229,545454 …
10000x = 22954 + 0,545454 …
10000x = 22954 + 100x − 229
10000x − 100x = 22954 − 229
9900x = 22725
x =
22725
9900
=
22725: 225
9900: 225
=
101
44
Άρα x =
𝟏𝟎𝟏
𝟒𝟒
= 2,29545454 …

3. 𝟓𝟕, 𝟗𝟐 =
5792: 4
100: 4
=
𝟏𝟒𝟒𝟖
𝟐𝟓
x = 2, 8
𝐱 = 𝟐, 𝟖𝟖𝟖 …
10 ∙ x = 10 ∙ 2,888 …
10x = 28,888 …
10x = 28 + 0,888 …
10x = 28 + x − 2
10x − x = 28 − 2
9x = 26
𝐱 =
𝟐𝟔
𝟗

4. x = 3, 83
𝐱 = 𝟑, 𝟖𝟑𝟖𝟑𝟖𝟑 …
100 ∙ x = 100 ∙ 3,838383 …
100x = 383,838383 …
100x = 383 + 0,838383 …
100x = 383 + x − 3
100x − x = 383 − 3
99x = 380
𝐱 =
𝟑𝟖𝟎
𝟗𝟗
x = 7,4561
𝐱 = 𝟕, 𝟒𝟓𝟔𝟏𝟓𝟔𝟏𝟓𝟔𝟏 …
10 ∙ x = 10 ∙ 7,4561561561 …
10x = 74,561561561 …
1000 ∙ 10x = 1000 ∙ 74,561561561 …
10000x = 74561,561561561 …
10000x = 74561 + 0,561561561 …
1000x = 74561 + 10x − 74
10000x − 10x = 74561 − 74
9990x = 74487
𝐱 =
𝟕𝟒𝟒𝟖𝟕
𝟗𝟗𝟗𝟎
15,399
𝐱 = 𝟏𝟓, 𝟑𝟗𝟗𝟗 …
10 ∙ x = 10 ∙ 15,3999…
10x=153,999…
10 ∙ 10x = 10 ∙ 153,999 …
100x = 1539,999 …
100x = 1539 + 0,999 …
100x = 1539 + 10x − 153
100x − 10x = 1539 − 153
90x = 1386
x=
1386
90
=
1386: 18
90: 18
=
77
5

5. α) 2, 9 = 2,999 … ≅ 3
β) 7,69 = 7,6999 … ≅ 7,7
γ) 7,3259 = 7,325999 … ≅ 7,326