The Golden Ratio is mentioned in Euclid's "Elements" in 300 B.C., then
throughout history, different mathematicians have done work with the
Golden Ratio, and in fact, here is one of the Islamic mathematicians working in
Baghdad, Iraq,
in the year 1150, and he gives the Golden Ratio as: √(125)-5
divided by 15 - √(125).
So here's our version of the simplified form of the Golden Ratio; what we want
to do his see if we can take this and turn it into this to show that he was
correct in his
expression for the Golden Ratio. So we can use our properties of radicals, let's
take his
expression right here, and the first thing we want to do is simplify
√(125).
So I know that
125 = 25(5), so I'll take that 25
out as √(25), which is 5.
√(125) in simplified form is 5√(5).
5√(5) - 5 over
15 - 5√(5).
So I'm just going to continue to simplify this expression until I get down to
something that looks like this Golden Ratio,
or be able to show that it's not the Golden Ratio
Well, let's see: I have a 5 common to the numerator and denominator, so let's take
this over here.
I'll factor out a 5 from the numerator, I get √(5) -1,
I'll factor out a 5 from the denominator, and I'll have 5(3 - √(5)).
Okay, so I can see that I can divide out those 5s right there, and now I have
left (√(5) - 1) over (3 - √(5)).
That is about as simple as this is going to get,
when I look back to my
expression for the Golden Ratio right here, I have a rational number,
just an integer in the denominator.
So when I look at this, I say, well we'll see if I can
do the same state gets and you've got your body religious group five
i think that by rationalizing that the dominant so i'm going to both vibrator
denominator
of this
by the college you get of the standoff so three-plus where fucked
i do that for the denominator will be the same thing numerator
so i'm a citizen involvement like this expression right here by the number one
set of changes value
now let's just go ahead multiplied
so i'm going to have screwed five times three three scary thought
square root five times square fire that'll be just fine
minus one pounds three
minus three minus one password fucked france where file
and the denominator you know what you multiply conjugates three minus forty
five pdt x three plus where five and get the first term square minus the second
trips where sold in the end up with nine minus square root five square which is
fine
okay so it looks like a giving them this is what we have here three square in
five minus three privacy two square fucked
five subtract three students
nominated here nine minus five seen before
but yes you have a two part of the numerator tended to go to the
denominator off after that to without
and two times
square root five
plus one
all by the fight to times too
although by those two to stop and helped change the order of the new boots so it
looks like my version of
golden ratio
cinder one-plus revival to okay
one-plus where if i were to exactly the same thing so he was right this is an
expression for the golden ratio
and i think she'll do so using my property so radicals now i'd like to use
this problem i classes because it really has to everything i want my students to
know about radicals
about the you know stops simplified by up factory contractor and numerator
denominator reducing the lowest terms
oh rationalizing denominators here multiplying
binomial expressions he was wrecked most of the u_n_ simplified by factory
numerator denominator dividing up the contactor
the more the same thing to do a class that there are questions are and will
keep them from trying to divide up just a little parts of this right here
biting at the correct way by fattening up liberator
and another dividing up com factors
so it's no problem
accident but the site about missus gore on it that the iraqis you're looking for
a city
so that i think you can tell the story of mathematics and algebra in particular
without going through
new baghdad iraq
in fact the first house report comes out of bed at about two hundred years before
the so it's very interesting story
but it would later be useful tool that sort of but
in any case to practice here with the properties right
and this situation it sure did actually you know
the golden rich