> Concerning Q5:
> ---------------
>  I took this answer to mean that Diophantus did want to find all exact rational solutions to a given problem.  Whereas the Babylonians settled with approximations and irrational solutions.
>

According to our text, he was perfectly happy with just one positive rational solution
to the equation, so the first alternative is not quite correct.  Neither are the next
three, leaving the last 'none of the above' as the only logical answer.
 
 
 

Greek II
Concerning Q6:
---------------
 How can you tell which one is correct, just by looking at the pictures?  I chose the pink one because it looked like the sides of the pink one were approximately twice the size as the red one.

What does it mean to duplicate the cube?  It means to double the volume.  Look again.  (the volume of the pink one is about 4 times the volume of
the red one).

Concerning Q7:
---------------
There was a decline because there were fewer mathematicians working on solutions to the problems that were now being faced in their lives.

the phrase 'problems that were now being faces in their lives'  doesn't mean much to me.  Why were there fewer mathematicians?
 

Egyptian

Concerning Q3:
---------------
 7/8 cannot be written as the sum of 2 distinct unit fractions because if you subtract the greatest unit fraction (1/2), you end up with 3/8.  Then you subtract the next largest unit fraction, 1/4.  You end up with 1/8 but this is a sum of three unit fractions not two.

You have just proved that the Fibonacci decomposition into unit fractions gives three.  But there are other decompostions.  You need to prove that
none of them use just two unit fractions.   (Hint: The key is to note that  1/n + 1/m <= 1/2 + 1/3 < 7/8  for any distinct  integers m,n >1)

Concerning Q4:
---------------
 If you subtract the greatest unit fraction 1/2 you will not end up with another unit fraction.  Hence, you would need to simplify it more which would be over there requirement of only two distinct unit fractions.

See Q3

> Concerning Q6:
> ---------------
>  Suppose x=8
> 8+1/2(8)=16
> 12=16

In order to turn 12 into 16, multiply by 16/12, so   x = 16/12*8 = 32/3

False position arguments don't use trial and error, so the stuff you have below
doesn't count.

>
>
> Suppose x=10
> 10+1/2(10)=16
> 15=16
>
> Suppose x=11
> 11+1/2(11)=16
> 16.5=16
>
> Through the use of trail and error I found that the answer is between 10 and 11.  Through further use of this method the answer is 10.6666667 or 10/2/3.
>

 Concerning Q3:
> ---------------
>  7/8 cannot be written as the sum of two distinct fractions because it is greater than 5/6 and less than 1

That doesn't explain it.

>
>
> Concerning Q4:
> ---------------
>  because it cannot be written as a sum of unit fractions

False.  Every fraction can be written as a sum of unit fractions.

>
>
> Concerning Q6:
> ---------------
>  x= 32/3
>
>

Write in the steps.

> Concerning Q3:
> ---------------
>  Once you strip 1/2 off of 7/8, you are left with 3/8.  There is not unique decomposition of 3/8 int unit factions.

Uniqueness is not in question.  The question is whether out of all the ways to
write 7/8 as a sum of unit fractions, is there at least one which is a sum of
two unit fractions.
How do you know that it is not the case that if you strip off 1/n for some n, then 1/m is left for some m?

>
>
> Concerning Q4:
> ---------------
> There is not a unique decomposition of a fraction less that 1/6..........

Uniqueness is not the question.

>
>
> Concerning Q5:
> ---------------
>  I thought I marked TRUE...Eygptian fractions was around in 1950BC and Ptomery was about 150 AD...

If you marked true,  you were correct.  I neglected to put in a key for this question.

>
>
> Concerning Q6:
> ---------------
>  Suppose the quantity is 8.  Then add a half to get 12.  Now do the same thing to 8 and 12 until 12 turns into 16.   Like multiply 12 by 16/12.  So the answer is (16/12)*8= (32/3)=10 + (2/3).
>
>

Sehr gute!
 
 

Babylonian

> Concerning Q1:
> ---------------
>  I could not answer this one because the notes that you gave in class were incomplete, and the lecture when this topic was introduced was not fully explained.
>

I thought I gave a marvelous discussion of this topic, but in any case, there has been
ample time for you to work on this particular problem.   The worksheet on
babylonian mathematics is useful to check your work here.

To convert 18293 to base 60,  sucessively divide by 60 until the quotient is less than
than 60.  The remainders taken in reverse order form the 'digits' of the base 60
representation.
Thus  60 goes into 18293   304 times with a remainder 0f 53
         60 goes into     304     5 times with a remainder of 4
thus  18293 = 60(304)+53 = 5*60^2 + 4*60 + 53  = 5,4,53;

To convert  .2354 to base 60, sucessively multply by 60 until the fractional part is 0.
The sequence of integers will be the 'digits' in order of the base 60 representation
thus  60*.2345 = 14.07,   60*.07 = 4.20, 60*.2 = 12.0
thus  .2345 = ;14, 4, 12
and  18293.2345  = 5,4,53;14,4,12
 
 

Concerning Q4:
---------------
 for handy computational purposes

Not specific enough

Concerning Q5:
---------------
for handy computational purposes

Not specific enough

> Concerning Q5:
> ---------------
>  The Babylonians made tables of reciprocals handy for computation purposes.

Could you be more specific?  Which of the arithmetic operations of  * / + or -
would make use of a table of reciprocals?  (answer:  divison = multiplying by the reciprocal.)

>
>
> Concerning Q6:
> ---------------
>  The Babylonians prepared tables of square roots for use in solving quardratic equations.
>

correct.  Give and example of a problem they might work on.

> Concerning Q5:
> ---------------
> The clay tables showed evidence that they understood the pythagorean theorem

But that wouldn't account for their tables of reciprocals.    Reciprocal tables were used
to perform division.

>
>
> Concerning Q6:
> ---------------
>  Computing approximations of square roots was so time consuming that the tables provided the information for them to use
>

But what did  they need the square root for?  Answer:  to solve quadratic equations.
 

> Concerning Q4:
> ---------------
>  dectosexa(evalf(sqrt(2)))=>[1, `;`, 24, 51, 10, 7, 45, 48, 40, 19, 12].  Do I disagree with this question and answer...

Probably you're right.  I hate multiple choice questions for just this reason.  There
are two 'correct'  answers here.

>
>
> Concerning Q5:
> ---------------
>  They would most likely compiled thes tables because the math was so complex that they didn't want to have msake the calculations over & over...

But which particular operation (addition, subtraction, multiplication, or division) would
need a table of reciprocals?   Answer:  division.

>
>
> Concerning Q6:
> ---------------
>  They would most likely compiled thes tables because the math was so complex that they didn't want to have msake the calculations over & over...So they would have they readily available for use
>
>

Be more specific.   Give an example of a geometric problem which involves knowing the square root of 2.