#Comparison of surds

71 messages · Page 1 of 1 (latest)

still nimbus
#

In general, if I have to compare surds of some form like
$$ \sqrt{2} + 1 $$ and $$ \sqrt{17} - 1 $$ or
$$ \sqrt{32} - \sqrt{24} $$ and $$ \sqrt{50} - \sqrt{48} $$ or other forms of compound surds, whats the way to do it?

raw plumeBOT
#
  1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  2. Wait patiently for a helper to come along.
  3. Once someone helps you, say thank you and close the thread with:
    +close
  4. Feel free to nominate the person for helper of the week in #helper-nominations
  5. Do not ping the mods, unless someone is breaking the rules. If there is a conflict amongst multiple helpers feel free to ping “Helper Mod”
  6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:
compact rampartBOT
still nimbus
#

sometimes decimal approximations and bounding will work, usually you would try to square them

#

decimal approxmiation is not efficient

#

and not accurate

#

and time consuming

#

sad

#

i said "and bounding"

#

bounding as in?

#

rt(17) - 1 is a little bigger than 3

#

compare that to rt(2)+1, which is definitely smaller than 3

#

okay but what if they are closer in numerical value

#

like rt(14) - 1

#

now?

#

usually you would try to square them

#

you could also try to use difference of two squares

#

is there no surefire way

#

squaring them?

#

okay so

#

$$ \sqrt{2} + 1$$ $$ \sqrt{17} - \sqrt{2} $$
$$ 5 + 2\sqrt{2} $$ $$ 19 - 2\sqrt{34} $$

compact rampartBOT
still nimbus
#

ok, exactly which ones are you trying to compare

#

the top two

still nimbus
#

their squares are down

#

now even with squares

#

Its not amply clear

versed torrent
#

as he said, bounding

still nimbus
#

ok, take two of them

versed torrent
#

not always easy, depending on the given quantities

#

they might be very close to one another so you need very precise bound

still nimbus
#

Bro

#

I am only comparing the top two

#

I just squared them

#

also, did i mention that you should simplify these surds beforehand?

#

and the bottom two are the squares

#

sorry

#

$$ \sqrt{2} + 1$$ $$ \sqrt{17} - \sqrt{2} $$

#

we compare these

compact rampartBOT
still nimbus
#

your original was rt(17) - 1

#

upon squaring we get

#

$$ 5 + 2\sqrt{2} $$ $$ 19 - 2\sqrt{34} $$

compact rampartBOT
still nimbus
#

I just wanted method

#

this is the question from the book

still nimbus
#

you write it as rt(17) and 2rt(2)+1 first (adding rt(2) to both numbers will not change which is bigger)

#

then you square those

#

one gives you 17 and the other gives you, what, 9+4rt(2)?

#

now subtract 9 off both

#

8 vs 4rt(2)

#

guess which one of those is bigger

#

Wow

#

8 ofcourse

#

so then rt(17) - rt(2) is bigger than rt(2) + 1

#

no like super systematic way?

versed torrent
#

isn't that systematic?

still nimbus
#

alternatively, write as rt(17)-1 and 2rt(2) and note that only one of these is bigger than 3

versed torrent
#

apply arithmetic to get two easily comparable quantities?

still nimbus
#

maybe you are right

still nimbus
# still nimbus no like super systematic way?

if i were to describe this as an algorithm, it would be "start by minimizing the number and size of square roots you have, isolate the biggest one on one side, square both sides, and repeat until you have no more square roots"

#

i don't know how many systematic ways of comparing weird numbers exist

#

Thanks @still nimbus

random pulsarBOT
#

@ancient shoal has given 1 rep to @feral parrot

still nimbus
#

+close