Determine if 0, point, 949449444944449444449, point, point, point
0.949449444944449444449...
0.949449444944449444449... is rational or irrational and give a reason for your answer.
To determine if 0.949449444944449444449... is rational or irrational, let's analyze the pattern.
The number appears to consist of two parts:
1. The decimal is repeating the string "494" indefinitely.
2. Before the repeating block, there are three digits: "949".
Therefore, we can write the number as:
0.949 + 0.494494494494...
Now, let's examine the first part, 0.949. This is a terminating decimal since it has a finite number of digits. All terminating decimals are rational.
The second part, 0.494494494494..., is a repeating decimal. The repeating block "494" indicates that this part is also a repeating decimal, and all repeating decimals are rational.
Since both parts of the number are rational numbers, we can conclude that 0.949449444944449444449... is rational.
To determine if 0.949449444944449444449... is rational or irrational, we need to understand if it can be expressed as a fraction of two integers.
We can represent the repeating decimal 0.949449444944449444449... as follows:
Let x = 0.949449444944449444449...
Multiply both sides of the equation by 10:
10x = 9.49449444944449444449...
Subtract the original equation from this new equation:
10x - x = 9.49449444944449444449... - 0.949449444944449444449...
This simplifies to:
9x = 8.54504500450004555556...
Now, divide both sides of the equation by 9:
x = (8.54504500450004555556...) / 9
The number (8.54504500450004555556...) / 9 is a representation of a rational number, as it can be expressed as the fraction of two integers.
Therefore, 0.949449444944449444449... is a rational number.