let's say that we have 2y+10 over x + 3. We need to add that fraction to 2y+20 over 1/2x + 1.5 . the logical thing to do would be to multiply the second fraction by 2 to get common denominators, right? so we should multiply the top of the second fraction by 2 over the bottom half times 2. but if you multiply something by 2 over 2, isn't that kind of just like multiplying it by one? 2/2= 1...
2y+ 20 x 2
-------- --
1/2x+1.5 2... 2/2 is one, so why do we get a different fraction when multiplying by one?
what about simplifying? how can you simplify something like 20/4 by dividing it by 2 on each side? 2/2 is one
Originally posted by Duck Duck GooseThe whole POINT of multiplying by, say, 2/2 is that you're not changing the value of something, just the way of writing it. What's paradoxical about that?
let's say that we have 2y+10 over x + 3. We need to add that fraction to 2y+20 over 1/2x + 1.5 . the logical thing to do would be to multiply the second fraction by 2 to get common denominators, right? so we should multiply the ...[text shortened]... y something like 20/4 by dividing it by 2 on each side? 2/2 is one
20/4 = 5, and 10/2 = 5, and 5/1 = 5. The whole aim is to write something in the simplest way possible.
There are no 'sides' to your 20/4 example, by the way, an EQUATION has sides.
Originally posted by Duck Duck GooseEXACTLY! That's what I'm saying. What is your problem with that idea?
but isn't 2/2 one? that way, when you multiply the fraction by 1, it doesn't change anything
You said in your first post, "you get a different fraction". It's a differently written fraction WITH THE SAME VALUE.
Try putting any value of x in both of your fractions, you will get exactly the same answer.
Yeah, we're friendly...
You'll agree that your second fraction is 'ugly'. To write is in a simpler form you have to do something to it, without changing the value of the fraction in all points x.
EG instead of writing 1/(1/2) you simply write 2. What we did here is multiply 1/(1/2) with 1=2/2. The same trick can be done to other fractions as well, like yours for example.
Multiplying with 1 doesn't do anything, you're right about that. But i CAN change the way we write something. There are many way to write one single fraction. We resort to tricks like this to write them in the most convenient form.
Originally posted by TheMaster37it can be very useful, for instance 2/(2^0.5), i.e. two divided by root two, is actually root two! miltiply by (2^0.5)/(2^0.5),
Yeah, we're friendly...
You'll agree that your second fraction is 'ugly'. To write is in a simpler form you have to do something to it, without changing the value of the fraction in all points x.
EG instead of writing 1/(1/2) you simply write 2. What we did here is multiply 1/(1/2) with 1=2/2. The same trick can be done to other fractions as we ...[text shortened]... te one single fraction. We resort to tricks like this to write them in the most convenient form.
2 2^0.5
------- x -------
2^0.5 2^0.5
=2 x 2^0.5
--------------
. 2
the two's cancel, so your left with 2^0.5. i still find that beautiful...
Originally posted by geniusactualy this example isn't even as complex as that.
it can be very useful, for instance 2/(2^0.5), i.e. two divided by root two, is actually root two! miltiply by (2^0.5)/(2^0.5),
2 2^0.5
------- x -------
2^0.5 2^0.5
=2 x 2^0.5
--------------
. 2
the two's cancel, so your left with 2^0.5. i still find that beautiful...
x/(x^.5)=(x^.5)^2*(x^.5)^-1=(x^.5)^(2-1)=(x^.5)^1=x^.5