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 \(q\ \neq\ 0\) 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, \(x=0.aaaa\ \cdots\) where a is any digit from 1 to 9.

Multiply x by 10 on both sides:

\(10x = a.aaaa\ \cdots\)

Now consider:

\(\Rightarrow\ 10x\ -\ x=a.aaaa\ \cdots\ -\ 0.aaaa\ \cdots\)

\(\Rightarrow\ 9x=a\)

\(\Rightarrow\ x=\ \frac{a}{9}\)

Thus, we can write the repeating decimal number \(x=0.aaaa\ \cdots\ as\ a\ ratio\ z =\ \frac{a}{9}.\)

That means every repeating decimal number is a rational number.

Therefore, the given statement ''There exist irrational numbers with repeating decimals'' is False.

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 \(q\ \neq\ 0\) 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, \(x=0.aaaa\ \cdots\) where a is any digit from 1 to 9.

Multiply x by 10 on both sides:

\(10x = a.aaaa\ \cdots\)

Now consider:

\(\Rightarrow\ 10x\ -\ x=a.aaaa\ \cdots\ -\ 0.aaaa\ \cdots\)

\(\Rightarrow\ 9x=a\)

\(\Rightarrow\ x=\ \frac{a}{9}\)

Thus, we can write the repeating decimal number \(x=0.aaaa\ \cdots\ as\ a\ ratio\ z =\ \frac{a}{9}.\)

That means every repeating decimal number is a rational number.

Therefore, the given statement ''There exist irrational numbers with repeating decimals'' is False.