Distance between two points (Basic but confusing)

Baz_

I just found my old LC maths text book and decided to do a few of the problems to see how I get on.

I got to the question where the two points are
A.(1/2, 1/2) B.(2, 1)

So I first worked the problem out by converting the fractions to 0.5...

D= sqrt( ((2 - 0.5)squared) + ((1 - 0.5)squared) )
= sqrt( ((1.5)squared) + ((0.5)squared) )
= sqrt( 2.25 + 0.25 )
= sqrt( 2.5 )

I then checked the answer and what was expected was:
= sqrt( 10/2 )

And I can't understand why because to my small mind that is the sqrt( 5 ) which isn't even close to my first attempt. (Now don't get me wrong, I worked it through without converting the fractions and I know how to get to the expected answer.)

What I really am confused about is that whether you work it out with the fractions or the decimals you should come up with the same answer but you don't seem to.

Can anyone explain this? I know it's probably something simple and I presume I ran into the same problem in school and most likely asked about it then, but I have absolutely no recollection as to the answer...

Apologies for the basicness of my question, but it hopefully means it will be easily answered.

Baz_

Okay, having done some research it would seem that the book could be wrong, in particular in the way it seems to have squared the fractions, which would account for the error.

Thoughts still welcome...

im invisible

the answer is (sqrt 10)/2 which is sqrt 2.5

Yakuza

Agree with the above, chances are it was a misprint, they meant to print [LATEX]\sqrt{10}/2[/LATEX] but printed [LATEX]\sqrt{10/2}[/LATEX] instead.

Baz_

Nope, it was a simple misread on my behalf, to be fair to me the bar above the 2 didn't extend far enough in order to make it very clear but on closer inspection it looks like im invisible is spot on and it all makes sense again.

Thank you.

locum-motion

Why does the book express it in that way?
Why is it not sqrt2.5?
Or for that matter, why isn't it 1.58113883?
When asked "What is 10-6?", we don't express the answer as 2^-1 (mod7), do we?

Baz_

Absolutely spot on locum, and I presume one of the reasons that people struggle with mathematics in secondary school.

Even if they had published an either this or that scenario I imaging it would lead to much less frustration to students...

Yakuza

locum-motion wrote: »

Why does the book express it in that way?
Why is it not sqrt2.5?
Or for that matter, why isn't it 1.58113883?
When asked "What is 10-6?", we don't express the answer as 2^-1 (mod7), do we?

Perhaps, as the solution to the problem had fractions (and squares of fractions) in it, they decided to leave the answer in the form of a surd. Or perhaps it was done to remind the student of how to treat factions when you bring them out of a square root etc.

In a textbook, in my opinion, simply spewing out a decimal answer isn't much help for the weaker student as, if they get the answer wrong, they've no context to try to attempt to fix their error, whereas leaving it in the above form might reinforce the concept of ("oh, yes, I brought that 4 outside the square root sign, so I've to take the square root of it when I write it down in the solution"). This would help with a more holistic approach to the subject as factoring out squares and square roots is done all over the syllabus, not just in Line Geometry.

How old is the book? When I did my Leaving (in the late 80's), calculators were only being introduced and it was quite normal for problems to be answered in surd form, rather then writing out the decimal equivalent.

(Great clock, by the way - where do I get one? )

ozmo

locum-motion wrote: »

Why does the book express it in that way?
Why is it not sqrt2.5?
Or for that matter, why isn't it 1.58113883?

But then that wouldn't be correct
1.58113883 is not the correct answer - it is approximately 1.58113883 but the correct answer is sqrt(2.5) or better ...

locum-motion

I got 16% in my christmas exam in 5th year in higher level mathematics, and dropped down to ordinary level for my leaving. I got a B first time and an A second time, but basically a lot of this stuff leaves me baffled.

I understand how to reach sqrt2.5 as an answer.

I also understand that the decimal answer I gave is an aproximation, not the actual answer.

But I do not understand how to reach (sqrt10)/2.

Yakuza

Ah ok - the answer from the formula is [latex]\sqrt{10/4}[/latex] which is the same as [latex]\frac{\sqrt{10}}{\sqrt{4}}[/latex] which reduces down to [latex]\sqrt{10}/2[/latex].

Baz_

What you eventually get to if you leave the fractions intact is:

sqrt( 10/4 )

However, algebra tricks allow you to take the divisor from inside the square root, get the square root of that alone, and then divide the square root of the top of your fractional number by the square root of the divisor which will give you the same answer.

In other words:
sqrt(m/d) is equivalent to: sqrt(m)/sqrt(d)

Since in the example above 4 is a perfect square it boils down nicely to:
sqrt(10)/sqrt(4) -> sqrt(10)/2.

I equate these little algebraic tricks to magic. I have often sat in a maths class and followed along perfectly until BANG! a little bit of algebraic trickery is thrown in and then you're left scrambling to understand the trick used, lose track of the main problem being solved and by the time you've processed the trick it's nearly impossible to catch back up with the problem...

Maths! You tricky little Ba$tard!


Edit: Also, what yakuza said - way more elegantly than me. P.S. I must get me some of that mathtran.org action, looks good!

locum-motion

Yakuza wrote: »

Ah ok - the answer from the formula is [latex]\sqrt{10/4}[/latex] which is the same as [latex]\frac{\sqrt{10}}{\sqrt{4}}[/latex] which reduces down to [latex]\sqrt{10}/2[/latex].

Thanks, that seems to make sense. (To me, that is. I'm certain it makes sense to others!)

(Slightly off-topic: BTW, in the example I posted above of 2^-1 (mod7) being equal to 4: I have absolutely no idea what that means, or why it = 4. The only reason I know that 2^-1 (mod7) = 4 is because of the clock. That's why I posted the pic of the clock!)

locum-motion

Thanks. Yeah, I agree, those little tricks are magic - as in, they are completely baffling to me! Maybe that's why I got 16% in that exam!

OK, so I can now see why sqrt2.5 = sqrt(10/4) = sqrt10/sqrt4 = (sqrt10)/2

But why would you continue past the correct answer of sqrt2.5? What's wrong with expressing it that way? It's a perfectly correct answer, isn't it?

Baz_ wrote: »

What you eventually get to if you leave the fractions intact is:

sqrt( 10/4 )

However, algebra tricks allow you to take the divisor from inside the square root, get the square root of that alone, and then divide the square root of the top of your fractional number by the square root of the divisor which will give you the same answer.

In other words:
sqrt(m/d) is equivalent to: sqrt(m)/sqrt(d)

Since in the example above 4 is a perfect square it boils down nicely to:
sqrt(10)/sqrt(4) -> sqrt(10)/2.

I equate these little algebraic tricks to magic. I have often sat in a maths class and followed along perfectly until BANG! a little bit of algebraic trickery is thrown in and then you're left scrambling to understand the trick used, lose track of the main problem being solved and by the time you've processed the trick it's nearly impossible to catch back up with the problem...

Maths! You tricky little Ba$tard!


Edit: Also, what yakuza said - way more elegantly than me. P.S. I must get me some of that mathtran.org action, looks good!

locum-motion

Pardon me for being so thick, but I've just grabbed a piece of paper and I think I've got it now. Is this how you did it?

D = sqrt {[(3/2)^2] + [(1/2)^2] }
= sqrt {[3/2 * 3/2] + [1/2 * 1/2]}
= sqrt {[9/4]+[1/4]}
= sqrt {10/4}
= (sqrt 10) / (sqrt 4)
= (sqrt 10) / 2

ozmo

locum-motion wrote: »

(Slightly off-topic: BTW, in the example I posted above of 2^-1 (mod7) being equal to 4: I have absolutely no idea what that means, or why it = 4. The only reason I know that 2^-1 (mod7) = 4 is because of the clock. That's why I posted the pic of the clock!)

I found an article here that describes each item on that clock- too many hyperlinks to more detailed info on each to paste

http://boingboing.net/2009/01/29/clock-for-geeks-1.html

locum-motion

ozmo wrote: »

I found an article here that describes each item on that clock- too many hyperlinks to more detailed info on each to paste

http://boingboing.net/2009/01/29/clock-for-geeks-1.html

I saw the explanations before - back when I first saw the clock* I looked them up. Even with the explanations, some of them are Greek to me!

I would suggest we shouldn't discuss this further here. We're going off-topic. (And yes, I know it was me that started it! I'm sorry!)


* First time I saw the clock was in the 'Maths jokes' thread here.

Baz_

locum-motion wrote: »

Pardon me for being so thick, but I've just grabbed a piece of paper and I think I've got it now. Is this how you did it?

D = sqrt {[(3/2)^2] + [(1/2)^2] }
= sqrt {[3/2 * 3/2] + [1/2 * 1/2]}
= sqrt {[9/4]+[1/4]}
= sqrt {10/4}
= (sqrt 10) / (sqrt 4)
= (sqrt 10) / 2

That's exactly the way I solved it, well the second time anyway, when I decided to try to answer it using fractions...