# 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?

Question
Decimals
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?

2020-11-25
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.

### Relevant Questions

True or False?
1) Let x and y real numbers. If $$\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}$$ and $$\displaystyle{x}\ne{y}$$, then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.
Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then $$\displaystyle{x}^{{2}}$$ is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational
Marks will be awarded for accuracy in the rounding of final answers where you are asked to round. To ensure that you receive these marks, take care in keeping more decimals in your intermediate steps than what the question is asking you to round your final answer to.
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(a) What is the probability that exactly one 3 is rolled?
(b) What is the probability that at least one 3 is rolled?
(c) What is the probability that exactly four of the rolls show an even number?
The rational numbers are dense in $$\displaystyle\mathbb{R}$$. This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in $$\displaystyle\mathbb{R}$$ as well.
Which of the following fractions are repeating decimals and which are terminating? How made decisions? $$a) \frac{2}{15}$$
$$b)\frac{11}{20}$$
$$c)\frac{17}{40}$$
$$d)\frac{1}{12}$$
factor in determining the usefulness of an examination as a measure of demonstrated ability is the amount of spread that occurs in the grades. If the spread or variation of examination scores is very small, it usually means that the examination was either too hard or too easy. However, if the variance of scores is moderately large, then there is a definite difference in scores between "better," "average," and "poorer" students. A group of attorneys in a Midwest state has been given the task of making up this year's bar examination for the state. The examination has 500 total possible points, and from the history of past examinations, it is known that a standard deviation of around 60 points is desirable. Of course, too large or too small a standard deviation is not good. The attorneys want to test their examination to see how good it is. A preliminary version of the examination (with slight modifications to protect the integrity of the real examination) is given to a random sample of 20 newly graduated law students. Their scores give a sample standard deviation of 70 points. Using a 0.01 level of significance, test the claim that the population standard deviation for the new examination is 60 against the claim that the population standard deviation is different from 60.
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H_{0}:\sigma=60,\ H_{1}:\sigma\ <\ 60H_{0}:\sigma\ >\ 60,\ H_{1}:\sigma=60H_{0}:\sigma=60,\ H_{1}:\sigma\ >\ 60H_{0}:\sigma=60,\ H_{1}:\sigma\ \neq\ 60$$
(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)
What are the degrees of freedom?
What assumptions are you making about the original distribution?
We assume a binomial population distribution.We assume a exponential population distribution. We assume a normal population distribution.We assume a uniform population distribution.
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let's suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. Give the distribution of X. Enter an exact number as an integer, fraction, or decimal.$$\displaystyle{f{{\left({x}\right)}}}=_{_}$$ where $$\displaystyle≤{X}≤.\mu=\sigma=$$. Find the probability that the individual lost more than 8 pounds in a month.Suppose it is known that the individual lost more than 9 pounds in a month. Find the probability that he lost less than 13 pounds in the month.
To check: Whether the set of numbers $$\displaystyle{\left\lbrace\sqrt{{{3}}},\pi,{\frac{{\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}}}{{{4}}}},\sqrt{{{5}}}\right\rbrace}$$ contains integers, rational numbers, and (or) irrational numbers.