Mathematical Reasoning True or False? There exist irrational numbers with repeating decimals? There exists a natural number x such that y > x for every naural number y?

Bergen 2020-11-24 Answered
Mathematical Reasoning
True or False?
There exist irrational numbers with repeating decimals?
There exists a natural number x such that y > x for every naural number y?
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Expert Answer

lamanocornudaW
Answered 2020-11-25 Author has 85 answers
Step 1
Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.
To find the given statement is true or false,
There exist irrational numbers with repeating decimals.
By definition of rational number,
A number that can be express as a ratio p/q where p and q are integers such that q  0 is called a rational number.
A number thet can not be expressed as a ratio of two integers is called an irrational number.
Step 2
Let a number with repeating digits, x=0.aaaa  where a is any digit from 1 to 9.
Multiply x by 10 on both sides:
10x=a.aaaa 
Now consider:
 10x  x=a.aaaa   0.aaaa 
 9x=a
 x= a9
Thus, we can write the repeating decimal number x=0.aaaa  as a ratio z= a9.
That means every repeating decimal number is a rational number.
Therefore, the given statement There exist irrational numbers with repeating decimals is False.
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