rational number
bassam karzeddin
I would like to ask a question of three parts:
a)we say that two is least integer that is greater than sqrt(three)
the question is what is the least rational number that is greater
than sqrt(three)?
b)what is the absolute value of{a+b*(-1)^(1/2)}
c)can we present the form {a+b*(-1)^(1/2)}^[sqrt(three)]
in another similar form,say, (A+Bi)
Thanking you
B.Karzeddin
A N Niel
wrote:
> Hi every body
>
> I would like to ask a question of three parts:
>
> a)we say that two is least integer that is greater than sqrt(three)
> the question is what is the least rational number that is greater
> than sqrt(three)?
There is none. Given any r=a/b with r > sqrt(3), consider
s=(a^2+3*b^2)/(2*a*b), so that sqrt(3) < s < r.
>
> b)what is the absolute value of{a+b*(-1)^(1/2)}
"Modulus" of complex number a+bi is sqrt(a^2+b^2), I used i = sqrt(-1).
>
> c)can we present the form {a+b*(-1)^(1/2)}^[sqrt(three)]
> in another similar form,say, (A+Bi)
Yes, use polar form. Since sqrt(3) is irrational, we get infinitely
many values. Maple says the principal value is:
> evalc((a+b*I)^sqrt(3));
exp(1/2*3^(1/2)*ln(a^2+b^2))*cos(3^(1/2)*arctan(b,a))
+I*exp(1/2*3^(1/2)*ln(a^2+b^2))*sin(3^(1/2)*arctan(b,a))
Maple uses I = sqrt(-1).
>
> Thanking you
> B.Karzeddin
>
>
>
>
>
Steve Bellenot
>
>I would like to ask a question of three parts:
>
> a)we say that two is least integer that is greater than sqrt(three)
> the question is what is the least rational number that is greater
> than sqrt(three)?
There is none. If sqrt(3) < x then there is a rational r with
sqrt(3) < r < x.
For example sqrt(3) = 1.7320508075688772935... so the rational sequence
2
1.8
1.74
1.733
1.7321
and so on ...
and so on converges to sqrt(3) and will eventually be smaller than any
particular x, with sqrt(3) < x.
The problem is the statement `the least rational number'. Just being able
to say the words doesn't mean it exists. Kind of like an oxymoron,
calling something `military intelligence' doesn't make it smart.
>
> b)what is the absolute value of{a+b*(-1)^(1/2)}
One usually writes i for sqrt(-1). |a + b*i| is defined to be
sqrt(a^2+b^2) = distance from the orgin to a+bi in the complex plane.
>
> c)can we present the form {a+b*(-1)^(1/2)}^[sqrt(three)]
> in another similar form,say, (A+Bi)
Yes, but it takes a bit of work. It is defined to be exp(sqrt(3)*log(a+bi))
where log(a+bi) = ln r + i (theta + 2*n*pi) for n = 0, +/- 1, +/- 2, ...
and a + bi = r e^(i theta).
--
Steven Bellenot http://www.math.fsu.edu/~bellenot
Department of Mathematics real....@line.below
Florida State University bellenot at math.fsu.edu
Tallahassee, FL 32306-4510 USA +1 (850) 644-7189 (FAX: 4053)
bassam karzeddin
Bkk