# Define rational and irrational number. Give examples.

Question
Decimals
Define rational and irrational number. Give examples.

2020-12-26
Rational Number: A number is said to be a rational number if we can write it as a fraction where the numerator of the fraction and denominator of the fraction both are whole numbers, denominator cannot equal to zero and also rational number has finite or recurring decimals.
Example are given below:
$$\frac{1}{9},\ 4=\frac{4}{1},\ \sqrt{9}=\ \pm\ 3$$
All numbers above are rational numbers.
Irrational Number: An irrational number is a number which is not a rational number. It is a number that cannot be written as a ratio of two integers or can’t be expressed as fraction form and irrational number has infinite or non-recurring decimals.
Examples are below:
$$\sqrt{2},\ \sqrt{3},\ 5 \sqrt{2},\ \pi,\ e.$$

### Relevant Questions

How to explain rational numbers, irrational numbers, and how they are different?
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.
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?
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.
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.
How do you calculate the following retaining the correct number of significant figures $$\displaystyle{12.432}\times{3}=$$ and $$\displaystyle{208}\times{62.1}=$$
$$x = 233,\ \mu = 223,\ \sigma = 12.2$$
$$z = ?$$