Which number produces an rational number when multiplied by 1/5

Mathematics

2 answers:

Nata [24]3 years ago

6 0

Answer with explanation:

Rational numbers are those numbers which can be written in the form of $\frac{a}{b}$ , where , b≠0.

Rational number can be written in the following way:

Integers + Terminating Decimal + Non terminating Repeating Decimal

When you will look at , fraction $\frac{1}{5}$ , it is terminating Decimal.

So, apart from any irrational number which is non terminating non repeating , if you multiply $\frac{1}{5}$ , with any number on the number line , that is any rational number ,because product of two rational is always a rational ,you will get a rational number.

Sedaia [141]3 years ago

5 0

Any rational number will satisfy the situation.

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Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) t

pashok25 [27]

Direct computation:

Parameterize the top part of the circle $x^2+y^2=1$ by

$\vec r(t)=(x(t),y(t))=(\cos t,\sin t)$

with $0\le t\le\pi$ , and the line segment by

$\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)$

with $0\le t\le1$ . Then

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)$

$=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt$

$=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}$

Using the fundamental theorem of calculus:

The integral can be written as

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}$

If there happens to be a scalar function $f$ such that $\vec F=\nabla f$ , then $\vec F$ is conservative and the integral is path-independent, so we only need to worry about the value of $f$ at the path's endpoints.