# How to explain rational numbers, irrational numbers, and how they are different?

Question
Decimals
How to explain rational numbers, irrational numbers, and how they are different?

2020-11-09
Step 1 To explain: a) Rational number. b) Irrational number. c) Difference between rational number and irrational number. Step 2 a) Rational number: A Rational number is a number that can be expressed in the form of a ratio p/q, where p and q are integers and $$\displaystyle{q}\ne{q}{0}$$. Rational numbers are finite and has terminating or repeating decimals. Examples of rational numbers: $$\displaystyle{\frac{{{1}}}{{{2}}}},{\frac{{{7}}}{{{4}}}},\sqrt{{{81}}},{\frac{{{1}}}{{{100}}}}$$ etc. $$\displaystyle{\frac{{{3}}}{{{4}}}}$$ can be represented as 0.75 which is a terminating decimal. $$\displaystyle{\frac{{{2}}}{{{3}}}}$$ can be represented as 0.6666 .... which is a repeating decimal. Step 3 b) Irrational number: An irrational numbers is a number that is not rational. Irrational numbers cannot be expressed as a fraction with integer values in the numerator and denominator. Irrational numbers are infinite and has non - terminating and non-repeating terms. When an irrational number is expressed in decimal form, it goes on forever without repeating. But both the numbers are real numbers and can be represented in a number line. Examples of irrational numbers: $$\displaystyle{\frac{{\sqrt{{{3}}}}}{{{2}}}},\pi,{0.131331333}\ldots.,{5}+\sqrt{{{3}}}$$ Step 4 c) Difference between rational number and irrational number: PSK\begin{array}{|c|c|} \hline Rational\ number & Irrational\ number \\ \hline It\ is\ expressed\ in\ the\ ratio,\ where\ both\ numerator\ and\ denominator\ are\ the\ whole\ numbers & It\ is\ impossible\ to\ express\ irrational\ numbers\ as\ fractions\ or\ in\ a\ ratio\ of\ two\ integers \\ \hline It\ includes\ perfect\ squares & It\ includes\ surds\ ( we\ can't\ simplify\ a\ number\ to\ remove\ a\ square\ root ) \\ \hline The\ decimal\ expansion\ for\ rational\ number\ executes\ finite\ or\ recurring\ decimals & Here,\ non-terminating\ and\ non-recurring\ decimals\ are\ executed \\ \hline \end{array}ZSK

### Relevant Questions

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.
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.)
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Mathematical Reasoning
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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.
What is the level of precision for the solution to the addition problem below? Given: The numbers are 6.339m, 0.170m and 30.4m and the operation is addition.
Consider the quantity$$a^{2}\ -\ b^{2}$$ where a and b are real numbers.
(a) Under what conditions should one expect an unusually large relative error in the computed value of $$a^{2}\ -\ b^{2}$$ when this expression is evaluated in finite precision arithmetic?
(b)cWs 4-digit (decimal) rounding arithmetic to evaluate both $$a^{2}\ -\ b^{2}\ and\ (a\ +\ b)(a\ -\ b)\ with\ a\ = 995.1\ and\ b = 995.0.$$ Calculate th relative error in each result.
(c) The expression $$(a\ +\ b)(a\ -\ b)\ is\ algebraically\ equivalent\ to\ a^{2}\ -\ b^{2},$$ but it is a more accurate way to calculate this quantity if both a and b have exact floating point representations. Why?
Among 25- to 30-years-olds, $$28\%$$ say they have operated heavy machinery while under the influence of alcohol Suppose three 25- to 30-years-olds are selected at random. Complete psrts (a) through (d) below:
a) What is the probability that all three have operated heavy machinery while under the influence of alcohol?
b) What is the probability that at least one has not haavy operated heavy machinery while under the influence of alcohol?
c) What is the probability that none of three have operated heavy machinery while under the influence of alcohol?
d) What is the probability that at least one has operated heavy machinery while under the influence of alcohol?