Aleah Avery

2022-11-21

Using a specific Lemma, how to prove that between any two distinct rational numbers there exists an irrational number?

Lemma: If $m/n$ and $r/s$ are rational, with $r/s\ne 0$, then $r/s\ne 0$ is irrational.

Lemma: If $m/n$ and $r/s$ are rational, with $r/s\ne 0$, then $r/s\ne 0$ is irrational.

martinmommy26nv8

Beginner2022-11-22Added 16 answers

The lemma simply says that if $p$ and $q$ are rational, and $q\ne 0$, then $p+q\sqrt{2}$ is irrational; it doesn’t matter which rational numbers $p$ and $q$ are (as long as $q\ne 0$). Now take $p=\frac{m}{n}$ and $q=\frac{r}{s}-\frac{m}{n}$; these are rational, and by hypothesis $\frac{m}{n}<\frac{r}{s}$, so $q\ne 0$, so the lemma applies to let us conclude that $p+q\sqrt{2}=\frac{m}{n}+\sqrt{2}(\frac{r}{s}-\frac{m}{n})$ is irrational.

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.