The number 0.point(212112111211112111112...) is a binary number represented as 0.212112111211112111112...
To determine if this number is rational or irrational, we need to examine its nature.
Assuming the pattern continues indefinitely, we notice that after the decimal point there are two repeating segments: "21" and "1121". This repetition suggests that the number is rational.
To confirm this, let's express the repeating portion of the number as a fraction and simplify it. We'll use x to represent the repeating part:
x = 0.212112111211112111112...
10x = 2.12112111211112111112...
Subtracting the two equations: (10x - x = 21.x - 2)
9x = 19.x - 2
19x - 9x = 2
10x = 2
x = 2/10
x = 1/5
By expressing the repeating portion of the number (x) as a fraction, we deduce that the number 0.point(212112111211112111112...) can be expressed as 0.212112111211112111112... = 1/5, where both the numerator and denominator are integers.
Since the number can be expressed as a fraction, it is rational.
Determine if 0, point, 212112111211112111112... is rational or irrational and give a reason for your answer.
1 answer