All categories

3 years ago

Show that the sum of two irrational numbers can be rational

Mathematics

1 answer:

Cerrena [4.2K]3 years ago

7 0

Answer:

Hint: We will have to know about the rational numbers and irrational numbers. ... So, the sum of the given two irrational numbers is equal to 6 which is a rational number in the form of p/q where p=6 and q=1 both are integers. Therefore, it is proved that the sum of the two given irrational numbers is a rational number.

Step-by-step explanation:

Hope it helps u

FOLLOW MY ACCOUNT PLS PLS

You might be interested in

Find the laplace transform by intergration<br> f(t)=tcosh(3t)

Shkiper50 [21]

$\mathcal L_s\{t\cosh3t\}=\displaystyle\int_0^\infty t\cosh3t e^{-st}\,\mathrm dt$

Integrate by parts, setting

$u_1=t\implies\mathrm du_1=\mathrm dt$
$\mathrm dv_1=\cosh3t e^{-st}\,\mathrm dt\implies v_1=\displaystyle\int\cosh3t e^{-st}\,\mathrm dt$

To evaluate $v_1$ , integrate by parts again, this time setting

$u_2=\cosh3t\implies\mathrm du_2=3\sinh3t\,\mathrm dt$
$\mathrm dv_2=\displaystyle\int e^{-st}\,\mathrm dt\implies v_2=-\frac1se^{-st}$

$\implies\displaystyle\int\cosh3te^{-st}\,\mathrm dt=-\frac1s\cosh3te^{-st}+\frac3s\int \sinh3te^{-st}$

Integrate by parts yet again, with

$u_3=\sinh3t\implies\mathrm du_3=3\cosh3t\,\mathrm dt$
$\mathrm dv_3=e^{-st}\,\mathrm dt\implies v_3=-\dfrac1se^{-st}$

$\implies\displaystyle\int\cosh3te^{-st}\,\mathrm dt=-\frac1s\cosh3te^{-st}+\frac3s\left(-\frac1s\sinh3te^{-st}+\frac3s\int\cosh3te^{-st}\,\mathrm dt\right)$
$\displaystyle\int\cosh3te^{-st}\,\mathrm dt=-\frac1s\cosh3te^{-st}-\frac3{s^2}\sinh3te^{-st}+\frac9{s^2}\int\cosh3te^{-st}\,\mathrm dt$
$\displaystyle\frac{s^2-9}{s^2}\int\cosh3te^{-st}\,\mathrm dt=-\frac1s\cosh3te^{-st}-\frac3{s^2}\sinh3te^{-st}$
$\implies\displaystyle\underbrace{\int\cosh3te^{-st}\,\mathrm dt}_{v_1}=-\frac{(s\cosh3t+3\sinh3t)e^{-st}}{s^2-9}$

So we have

$\displaystyle\int_0^\infty t\cosh3t e^{-st}\,\mathrm dt=u_1v_1\big|_{t=0}^{t\to\infty}-\int_0^\infty v_1\,\mathrm du_1$
$=\displaystyle-\frac{t(s\cosh3t+3\sinh3t)e^{-st}}{s^2-9}\bigg|_{t=0}^{t\to\infty}-\int_0^\infty \left(-\frac{(s\cosh3t+3\sinh3t)e^{-st}}{s^2-9}\right)\,\mathrm dt$
$=\displaystyle\frac1{s^2-9}\int_0^\infty(s\cosh3t+3\sinh3t)e^{-st}\,\mathrm dt$

We already have the antiderivative for the first term:

$\displaystyle\frac s{s^2-9}\int_0^\infty \cosh3te^{-st}\,\mathrm dt=\frac s{s^2-9}\left(-\frac{(s\cosh3t+3\sinh3t)e^{-st}}{s^2-9}\right)\bigg|_{t=0}^{t\to\infty}$
$=\dfrac{s^2}{(s^2-9)^2}$

And we can easily find the remaining term's antiderivative by integrating by parts (for the last time!), or by simply exchanging $\cosh$ with $\sinh$ in the derivation of $v_1$ , so that we have

$\displaystyle\frac3{s^2-9}\int_0^\infty\sinh3te^{-st}\,\mathrm dt=\frac3{s^2-9}\left(-\frac{(s\sinh3t+3\cosh3t)e^{-st}}{s^2-9}\right)\bigg|_{t=0}^{t\to\infty}$
$=\dfrac9{(s^2-9)^2}$

(The exchanging is permissible because $(\sinh x)'=\cosh x$ and $(\cosh x)'=\sinh x$ ; there are no alternating signs to account for.)

And so we conclude that

$\mathcal L_s\{t\cosh3t\}=\dfrac{s^2+9}{(s^2-9)^2}$

8 0

3 years ago

Simplify this expression.

Ulleksa [173]

Answer: i think it is d

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

Find the amount paid for the loan.<br> $2400 at 10.5% for 5 years

KengaRu [80]

Loan amount=$2400.
has an interest rate of 10.5%.
That's PER year for FIVE years.
First, you need to get how much 10.5% of 2400$ is... which comes out to be 252.
$732 is the answer. (PER year).

6 0

3 years ago

I wanna know how to get this answer using the quadratic formula 4t2 + 9t − 3 = 0

motikmotik

$\text{The quadratic formula is:}\\\\x=\frac{-b\pm\sqrt(b^2-4ac)}{2a}\\\\\text{In the equation, your 4 is a, 9 is b, and -3 is c}\\\\\text{Plug in and solve:}\\\\t=\frac{-9\pm\sqrt(9^2-4(4)(-3))}{2(4)}\\\\t=\frac{-9\pm\sqrt(81-4(4)(-3))}{2(4)}\\\\t=\frac{-9\pm\sqrt(81+48)}{2(4)}\\\\t=\frac{-9\pm\sqrt129}{2(4)}\\\\t=\frac{-9\pm\sqrt129}{8}\\\\\text{Since you can't simplify any further, that'll be your answer}\\\\\boxed{t=\frac{-9+\sqrt129}{8}\,\,and\,\,t=\frac{-9-\sqrt129}{8}}$

6 0

3 years ago

A recipe requires the following ingredients to make 16 gingerbread men

aalyn [17]

Answer:

5,600 g of flour

10,000g of sugar

3,200g of ginger

4,400g of butter

Just have to multiply each ingredient by 40

8 0

2 years ago

Show that the sum of two irrational numbers can be rational ​

Show that the sum of two irrational numbers can be rational