Explained: "Question which looks something similar to decimal expansionLetA={∑n=1∞ai5i:ai=0,1,2,3..."

Alfredo Holmes

Answered question

2022-03-31

Question which looks something similar to decimal expansion
Let

A = {\sum_{n = 1}^{\infty} \frac{a_{i}}{5^{i}} : a_{i} = 0, 1, 2, 3 or 4}

What is set A?

Jesse Gates

Beginner2022-04-01Added 19 answers

Step 1
You are right. Actually, if

x \in [0, 1]

and if you write x in base 5, then

x = 0 . a_{1} a_{2} a_{3} \dots;

besides,

1 = 0.444444444 \dots

Note that if

x \in [0, 1]

you can define

a_{1} = ⌊ 5 x ⌋

. Then

a_{2} = ⌊ 5^{2} x - 5 a_{1} ⌋, a_{3} = ⌊ 5^{3} x - 5^{2} a_{1} - 5 a_{2} ⌋

and so on. Then

x = \sum_{n = 1}^{\infty} \frac{a_{n}}{5^{n}} .

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