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# The rational numbers are dense in RR. 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 RR as well. # The rational numbers are dense in RR. 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 RR as well.

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Irrational numbers asked 2021-01-23
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.

## Answers (1) 2021-01-24
We need to prove that Irrational numbers are dense in $$\displaystyle\mathbb{R}$$ .We are given that Rational numbers are dense in $$\displaystyle\mathbb{R}$$ , which means between any two real numbers there exist a rational number. Now, $$\displaystyle\sqrt{{2}}$$ is an irrational number .
Let, a and b be two real numbers, such that, a Since $$\displaystyle{a},{b},\sqrt{{2}}$$ are real numbers therefore there division is also a real number
Therefore , $$\displaystyle\frac{{a}}{\sqrt{{2}}}$$ and b/sqrt2ZSK are also real numbers. We have,
$$\displaystyle{a}{<}{b}\Rightarrow\frac{{a}}{\sqrt{{2}}}{<}\frac{{b}}{\sqrt{{2}}}$$</span>
And we are given that between any two real numbers there exist a rational number .
Therefore , let c in $$\displaystyle\mathbb{Q}$$ be a rational number between $$\displaystyle\frac{{a}}{\sqrt{{2}}}$$ and $$\displaystyle\frac{{b}}{\sqrt{{2}}}\Rightarrow\frac{{a}}{\sqrt{{2}}}{<}{c}{<}\frac{{b}}{\sqrt{{2}}}$$</span>
We got,
$$\displaystyle\frac{{a}}{\sqrt{{2}}}{<}{c}{<}\frac{{b}}{\sqrt{{2}}}$$</span>
Now , multiply by $$\displaystyle\sqrt{{2}}$$ throughout, we get,
$$\displaystyle\sqrt{{2}}{\left(\frac{{a}}{\sqrt{{2}}}\right)}{<}\sqrt{{2}}\cdot{c}{<}\sqrt{{2}}{\left(\frac{{b}}{\sqrt{{2}}}\right)}\Rightarrow{a}{<}{c}\sqrt{{2}}{<}{b}$$</span>
Now, product of a rational and an irrational number is an irrational number , therefore , since c in $$\displaystyle\mathbb{Q}$$ and $$\displaystyle\sqrt{{2}}\in\frac{\mathbb{R}}{\mathbb{Q}}$$, therefore, csqrt2ZSK
Let, $$\displaystyle{c}\sqrt{{2}}={m}\Rightarrow{a}{<}{m}{<}{b}$$</span>
and a and b are real numbers and m is irrational number .
Therefore, we get that between any two real numbers a and b, with a Therefore, Irrational numbers are dense in real numbers .
Answer: Irrational numbers are dense in $$\displaystyle\mathbb{R}$$ .

### Relevant Questions asked 2021-05-12
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Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
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$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
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(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
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At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
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Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver. asked 2020-10-23
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Suspect was Armed:
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White - 1176
Hispanic - 378
Total - 2097
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White - 67
Hispanic - 38
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This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
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Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
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Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
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